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Versions: (draft-enyedi-rtgwg-mrt-frr-algorithm) 00 01 02 03 04 05 06 07 08 09 RFC 7811

Routing Area Working Group                                     G. Enyedi
Internet-Draft                                                A. Csaszar
Intended status: Standards Track                                Ericsson
Expires: August 19, 2016                                        A. Atlas
                                                               C. Bowers
                                                        Juniper Networks
                                                              A. Gopalan
                                                   University of Arizona
                                                       February 16, 2016


 An Algorithm for Computing Maximally Redundant Trees for IP/LDP Fast-
                                Reroute
                 draft-ietf-rtgwg-mrt-frr-algorithm-09

Abstract

   A solution for IP and LDP Fast-Reroute using Maximally Redundant
   Trees is presented in draft-ietf-rtgwg-mrt-frr-architecture.  This
   document defines the associated MRT Lowpoint algorithm that is used
   in the Default MRT profile to compute both the necessary Maximally
   Redundant Trees with their associated next-hops and the alternates to
   select for MRT-FRR.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
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   Internet-Drafts are draft documents valid for a maximum of six months
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   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on August 19, 2016.

Copyright Notice

   Copyright (c) 2016 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents



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   (http://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
   2.  Requirements Language . . . . . . . . . . . . . . . . . . . .   5
   3.  Terminology and Definitions . . . . . . . . . . . . . . . . .   5
   4.  Algorithm Key Concepts  . . . . . . . . . . . . . . . . . . .   6
     4.1.  Partial Ordering for Disjoint Paths . . . . . . . . . . .   7
     4.2.  Finding an Ear and the Correct Direction  . . . . . . . .   8
     4.3.  Low-Point Values and Their Uses . . . . . . . . . . . . .  11
     4.4.  Blocks in a Graph . . . . . . . . . . . . . . . . . . . .  14
     4.5.  Determining Local-Root and Assigning Block-ID . . . . . .  16
   5.  MRT Lowpoint Algorithm Specification  . . . . . . . . . . . .  18
     5.1.  Interface Ordering  . . . . . . . . . . . . . . . . . . .  18
     5.2.  MRT Island Identification . . . . . . . . . . . . . . . .  21
     5.3.  GADAG Root Selection  . . . . . . . . . . . . . . . . . .  21
     5.4.  Initialization  . . . . . . . . . . . . . . . . . . . . .  22
     5.5.  Constructing the GADAG using lowpoint inheritance . . . .  23
     5.6.  Augmenting the GADAG by directing all links . . . . . . .  25
     5.7.  Compute MRT next-hops . . . . . . . . . . . . . . . . . .  29
       5.7.1.  MRT next-hops to all nodes ordered with respect to
               the computing node  . . . . . . . . . . . . . . . . .  29
       5.7.2.  MRT next-hops to all nodes not ordered with respect
               to the computing node . . . . . . . . . . . . . . . .  30
       5.7.3.  Computing Redundant Tree next-hops in a 2-connected
               Graph . . . . . . . . . . . . . . . . . . . . . . . .  31
       5.7.4.  Generalizing for a graph that isn't 2-connected . . .  33
       5.7.5.  Complete Algorithm to Compute MRT Next-Hops . . . . .  34
     5.8.  Identify MRT alternates . . . . . . . . . . . . . . . . .  36
     5.9.  Named Proxy-Nodes . . . . . . . . . . . . . . . . . . . .  43
       5.9.1.  Determining Proxy-Node Attachment Routers . . . . . .  43
       5.9.2.  Computing if an Island Neighbor (IN) is loop-free . .  44
       5.9.3.  Computing MRT Next-Hops for Proxy-Nodes . . . . . . .  46
       5.9.4.  Computing MRT Alternates for Proxy-Nodes  . . . . . .  52
   6.  MRT Lowpoint Algorithm: Next-hop conformance  . . . . . . . .  60
   7.  Broadcast interfaces  . . . . . . . . . . . . . . . . . . . .  60
     7.1.  Computing MRT next-hops on broadcast networks . . . . . .  61
     7.2.  Using MRT next-hops as alternates in the event of
           failures on broadcast networks  . . . . . . . . . . . . .  62
   8.  Evaluation of Alternative Methods for Constructing GADAGs . .  63
   9.  Implementation Status . . . . . . . . . . . . . . . . . . . .  64



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   10. Operational Considerations  . . . . . . . . . . . . . . . . .  65
     10.1.  GADAG Root Selection . . . . . . . . . . . . . . . . . .  65
     10.2.  Destination-rooted GADAGs  . . . . . . . . . . . . . . .  65
   11. Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  66
   12. IANA Considerations . . . . . . . . . . . . . . . . . . . . .  66
   13. Security Considerations . . . . . . . . . . . . . . . . . . .  66
   14. References  . . . . . . . . . . . . . . . . . . . . . . . . .  66
     14.1.  Normative References . . . . . . . . . . . . . . . . . .  66
     14.2.  Informative References . . . . . . . . . . . . . . . . .  66
   Appendix A.  Python Implementation of MRT Lowpoint Algorithm  . .  67
   Appendix B.  Constructing a GADAG using SPFs  . . . . . . . . . . 108
   Appendix C.  Constructing a GADAG using a hybrid method . . . . . 113
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . . 115

1.  Introduction

   MRT Fast-Reroute requires that packets can be forwarded not only on
   the shortest-path tree, but also on two Maximally Redundant Trees
   (MRTs), referred to as the MRT-Blue and the MRT-Red.  A router which
   experiences a local failure must also have pre-determined which
   alternate to use.  This document defines how to compute these three
   things for use in MRT-FRR and describes the algorithm design
   decisions and rationale.  The algorithm is based on those presented
   in [MRTLinear] and expanded in [EnyediThesis].  The MRT Lowpoint
   algorithm is required for implementation when the Default MRT profile
   is implemented.

   Just as packets routed on a hop-by-hop basis require that each router
   compute a shortest-path tree which is consistent, it is necessary for
   each router to compute the MRT-Blue next-hops and MRT-Red next-hops
   in a consistent fashion.  This document defines the MRT Lowpoint
   algorithm to be used as a standard in the default MRT profile for
   MRT-FRR.

   As now, a router's FIB will contain primary next-hops for the current
   shortest-path tree for forwarding traffic.  In addition, a router's
   FIB will contain primary next-hops for the MRT-Blue for forwarding
   received traffic on the MRT-Blue and primary next-hops for the MRT-
   Red for forwarding received traffic on the MRT-Red.

   What alternate next-hops a point-of-local-repair (PLR) selects need
   not be consistent - but loops must be prevented.  To reduce
   congestion, it is possible for multiple alternate next-hops to be
   selected; in the context of MRT alternates, each of those alternate
   next-hops would be equal-cost paths.

   This document defines an algorithm for selecting an appropriate MRT
   alternate for consideration.  Other alternates, e.g.  LFAs that are



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   downstream paths, may be preferred when available.  See the
   Operational Considerations section of
   [I-D.ietf-rtgwg-mrt-frr-architecture] for a more detailed discussion
   of combining MRT alternates with those produced by other FRR
   technologies.

   [E]---[D]---|           [E]<--[D]<--|                [E]-->[D]
    |     |    |            |     ^    |                       |
    |     |    |            V     |    |                       V
   [R]   [F]  [C]          [R]   [F]  [C]               [R]   [F]  [C]
    |     |    |                  ^                      ^     |    |
    |     |    |                  |                      |     V    |
   [A]---[B]---|           [A]-->[B]                    [A]---[B]<--|

         (a)                     (b)                         (c)
   a 2-connected graph     MRT-Blue towards R          MRT-Red towards R

                                 Figure 1

   The MRT Lowpoint algorithm can handle arbitrary network topologies
   where the whole network graph is not 2-connected, as in Figure 2, as
   well as the easier case where the network graph is 2-connected
   (Figure 1).  Each MRT is a spanning tree.  The pair of MRTs provide
   two paths from every node X to the root of the MRTs.  Those paths
   share the minimum number of nodes and the minimum number of links.
   Each such shared node is a cut-vertex.  Any shared links are cut-
   links.
























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                        [E]---[D]---|     |---[J]
                         |     |    |     |    |
                         |     |    |     |    |
                        [R]   [F]  [C]---[G]   |
                         |     |    |     |    |
                         |     |    |     |    |
                        [A]---[B]---|     |---[H]

                       (a) a graph that isn't 2-connected

         [E]<--[D]<--|         [J]        [E]-->[D]---|     |---[J]
          |     ^    |          |                |    |     |    ^
          V     |    |          |                V    V     V    |
         [R]   [F]  [C]<--[G]   |         [R]   [F]  [C]<--[G]   |
                ^    ^     ^    |          ^     |    |          |
                |    |     |    V          |     V    |          |
         [A]-->[B]---|     |---[H]        [A]<--[B]<--|         [H]

          (b) MRT-Blue towards R          (c) MRT-Red towards R

                                 Figure 2

2.  Requirements Language

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in [RFC2119].

3.  Terminology and Definitions

   Please see the Terminology section of
   [I-D.ietf-rtgwg-mrt-frr-architecture] for a complete list of
   terminology relevant to this draft.  The list below does not repeat
   terminology introduced in that draft.

   spanning tree:   A tree containing links that connects all nodes in
      the network graph.

   back-edge:   In the context of a spanning tree computed via a depth-
      first search, a back-edge is a link that connects a descendant of
      a node x with an ancestor of x.

   partial ADAG:   A subset of an ADAG that doesn't yet contain all the
      nodes in the block.  A partial ADAG is created during the MRT
      algorithm and then expanded until all nodes in the block are
      included and it is an ADAG.

   DFS:    Depth-First Search



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   DFS ancestor:    A node n is a DFS ancestor of x if n is on the DFS-
      tree path from the DFS root to x.

   DFS descendant:    A node n is a DFS descendant of x if x is on the
      DFS-tree path from the DFS root to n.

   ear:    A path along not-yet-included-in-the-GADAG nodes that starts
      at a node that is already-included-in-the-GADAG and that ends at a
      node that is already-included-in-the-GADAG.  The starting and
      ending nodes may be the same node if it is a cut-vertex.

   X >> Y or Y << X:   Indicates the relationship between X and Y in a
      partial order, such as found in a GADAG.  X >> Y means that X is
      higher in the partial order than Y.  Y << X means that Y is lower
      in the partial order than X.

   X > Y or Y < X:   Indicates the relationship between X and Y in the
      total order, such as found via a topological sort.  X > Y means
      that X is higher in the total order than Y.  Y < X means that Y is
      lower in the total order than X.

   X ?? Y:   Indicates that X is unordered with respect to Y in the
      partial order.

   UNDIRECTED:   In the GADAG, each link is marked as OUTGOING, INCOMING
      or both.  Until the directionality of the link is determined, the
      link is marked as UNDIRECTED to indicate that its direction hasn't
      been determined.

   OUTGOING:   A link marked as OUTGOING has direction in the GADAG from
      the interface's router to the remote end.

   INCOMING:   A link marked as INCOMING has direction in the GADAG from
      the remote end to the interface's router.

4.  Algorithm Key Concepts

   There are five key concepts that are critical for understanding the
   MRT Lowpoint algorithm.  The first is the idea of partially ordering
   the nodes in a network graph with regard to each other and to the
   GADAG root.  The second is the idea of finding an ear of nodes and
   adding them in the correct direction.  The third is the idea of a
   Low-Point value and how it can be used to identify cut-vertices and
   to find a second path towards the root.  The fourth is the idea that
   a non-2-connected graph is made up of blocks, where a block is a
   2-connected cluster, a cut-link or an isolated node.  The fifth is
   the idea of a local-root for each node; this is used to compute ADAGs
   in each block.



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4.1.  Partial Ordering for Disjoint Paths

   Given any two nodes X and Y in a graph, a particular total order
   means that either X < Y or X > Y in that total order.  An example
   would be a graph where the nodes are ranked based upon their unique
   IP loopback addresses.  In a partial order, there may be some nodes
   for which it can't be determined whether X << Y or X >> Y.  A partial
   order can be captured in a directed graph, as shown in Figure 3.  In
   a graphical representation, a link directed from X to Y indicates
   that X is a neighbor of Y in the network graph and X << Y.


         [A]<---[R]    [E]       R << A << B << C << D << E
          |             ^        R << A << B << F << G << H << D << E
          |             |
          V             |        Unspecified Relationships:
         [B]--->[C]--->[D]             C and F
          |             ^              C and G
          |             |              C and H
          V             |
         [F]--->[G]--->[H]


             Figure 3: Directed Graph showing a Partial Order

   To compute MRTs, the root of the MRTs is at both the very bottom and
   the very top of the partial ordering.  This means that from any node
   X, one can pick nodes higher in the order until the root is reached.
   Similarly, from any node X, one can pick nodes lower in the order
   until the root is reached.  For instance, in Figure 4, from G the
   higher nodes picked can be traced by following the directed links and
   are H, D, E and R.  Similarly, from G the lower nodes picked can be
   traced by reversing the directed links and are F, B, A, and R.  A
   graph that represents this modified partial order is no longer a DAG;
   it is termed an Almost DAG (ADAG) because if the links directed to
   the root were removed, it would be a DAG.















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     [A]<---[R]<---[E]      R << A << B << C << R
      |      ^      ^       R << A << B << C << D << E << R
      |      |      |       R << A << B << F << G << H << D << E << R
      V      |      |
     [B]--->[C]--->[D]      Unspecified Relationships:
      |             ^              C and F
      |             |              C and G
      V             |              C and H
     [F]--->[G]--->[H]


     Figure 4: ADAG showing a Partial Order with R lowest and highest

   Most importantly, if a node Y >> X, then Y can only appear on the
   increasing path from X to the root and never on the decreasing path.
   Similarly, if a node Z << X, then Z can only appear on the decreasing
   path from X to the root and never on the inceasing path.

   When following the increasing paths, it is possible to pick multiple
   higher nodes and still have the certainty that those paths will be
   disjoint from the decreasing paths.  E.g. in the previous example
   node B has multiple possibilities to forward packets along an
   increasing path: it can either forward packets to C or F.

4.2.  Finding an Ear and the Correct Direction

   For simplicity, the basic idea of creating a GADAG by adding ears is
   described assuming that the network graph is a single 2-connected
   cluster so that an ADAG is sufficient.  Generalizing to multiple
   blocks is done by considering the block-roots instead of the GADAG
   root - and the actual algorithm is given in Section 5.5.

   In order to understand the basic idea of finding an ADAG, first
   suppose that we have already a partial ADAG, which doesn't contain
   all the nodes in the block yet, and we want to extend it to cover all
   the nodes.  Suppose that we find a path from a node X to Y such that
   X and Y are already contained by our partial ADAG, but all the
   remaining nodes along the path are not added to the ADAG yet.  We
   refer to such a path as an ear.

   Recall that our ADAG is closely related to a partial order.  More
   precisely, if we remove root R, the remaining DAG describes a partial
   order of the nodes.  If we suppose that neither X nor Y is the root,
   we may be able to compare them.  If one of them is definitely lesser
   with respect to our partial order (say X<<Y), we can add the new path
   to the ADAG in a direction from X to Y.  As an example consider
   Figure 5.




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           E---D---|              E<--D---|           E<--D<--|
           |   |   |              |   ^   |           |   ^   |
           |   |   |              V   |   |           V   |   |
           R   F   C              R   F   C           R   F   C
           |   |   |              |   ^   |           |   ^   ^
           |   |   |              V   |   |           V   |   |
           A---B---|              A-->B---|           A-->B---|

              (a)                    (b)                 (c)

                            (a) A 2-connected graph
                      (b) Partial ADAG (C is not included)
           (c) Resulting ADAG after adding path (or ear) B-C-D

                                 Figure 5

   In this partial ADAG, node C is not yet included.  However, we can
   find path B-C-D, where both endpoints are contained by this partial
   ADAG (we say those nodes are "ready" in the following text), and the
   remaining node (node C) is not contained yet.  If we remove R, the
   remaining DAG defines a partial order, and with respect to this
   partial order we can say that B<<D, so we can add the path to the
   ADAG in the direction from B to D (arcs B->C and C->D are added).  If
   B >> D, we would add the same path in reverse direction.

   If in the partial order where an ear's two ends are X and Y, X << Y,
   then there must already be a directed path from X to Y in the ADAG.
   The ear must be added in a direction such that it doesn't create a
   cycle; therefore the ear must go from X to Y.

   In the case, when X and Y are not ordered with each other, we can
   select either direction for the ear.  We have no restriction since
   neither of the directions can result in a cycle.  In the corner case
   when one of the endpoints of an ear, say X, is the root (recall that
   the two endpoints must be different), we could use both directions
   again for the ear because the root can be considered both as smaller
   and as greater than Y.  However, we strictly pick that direction in
   which the root is lower than Y.  The logic for this decision is
   explained in Section 5.7

   A partial ADAG is started by finding a cycle from the root R back to
   itself.  This can be done by selecting a non-ready neighbor N of R
   and then finding a path from N to R that doesn't use any links
   between R and N.  The direction of the cycle can be assigned either
   way since it is starting the ordering.

   Once a partial ADAG is already present, it will always have a node
   that is not the root R in it.  As a brief proof that a partial ADAG



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   can always have ears added to it: just select a non-ready neighbor N
   of a ready node Q, such that Q is not the root R, find a path from N
   to the root R in the graph with Q removed.  This path is an ear where
   the first node of the ear is Q, the next is N, then the path until
   the first ready node the path reached (that ready node is the other
   endpoint of the path).  Since the graph is 2-connected, there must be
   a path from N to R without Q.

   It is always possible to select a non-ready neighbor N of a ready
   node Q so that Q is not the root R.  Because the network is
   2-connected, N must be connected to two different nodes and only one
   can be R.  Because the initial cycle has already been added to the
   ADAG, there are ready nodes that are not R.  Since the graph is
   2-connected, while there are non-ready nodes, there must be a non-
   ready neighbor N of a ready node that is not R.

   Generic_Find_Ears_ADAG(root)
      Create an empty ADAG.  Add root to the ADAG.
      Mark root as IN_GADAG.
      Select an arbitrary cycle containing root.
      Add the arbitrary cycle to the ADAG.
      Mark cycle's nodes as IN_GADAG.
      Add cycle's non-root nodes to process_list.
      while there exists connected nodes in graph that are not IN_GADAG
         Select a new ear.  Let its endpoints be X and Y.
         if Y is root or (Y << X)
            add the ear towards X to the ADAG
         else // (a) X is root or (b)X << Y or (c) X, Y not ordered
            Add the ear towards Y to the ADAG

      Figure 6: Generic Algorithm to find ears and their direction in
                             2-connected graph

   The algorithm in Figure 6 merely requires that a cycle or ear be
   selected without specifying how.  Regardless of the method for
   selecting the path, we will get an ADAG.  The method used for finding
   and selecting the ears is important; shorter ears result in shorter
   paths along the MRTs.  The MRT Lowpoint algorithm uses the Low-Point
   Inheritance method for constructing an ADAG (and ultimately a GADAG).
   This method is defined in Section 5.5.  Other methods for
   constructing GADAGs are described in Appendix B and Appendix C.  An
   evaluation of these different methods is given in Section 8

   As an example, consider Figure 5 again.  First, we select the
   shortest cycle containing R, which can be R-A-B-F-D-E (uniform link
   costs were assumed), so we get to the situation depicted in Figure 5
   (b).  Finally, we find a node next to a ready node; that must be node
   C and assume we reached it from ready node B.  We search a path from



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   C to R without B in the original graph.  The first ready node along
   this is node D, so the open ear is B-C-D.  Since B<<D, we add arc
   B->C and C->D to the ADAG.  Since all the nodes are ready, we stop at
   this point.

4.3.  Low-Point Values and Their Uses

   A basic way of computing a spanning tree on a network graph is to run
   a depth-first-search, such as given in Figure 7.  This tree has the
   important property that if there is a link (x, n), then either n is a
   DFS ancestor of x or n is a DFS descendant of x.  In other words,
   either n is on the path from the root to x or x is on the path from
   the root to n.

                        global_variable: dfs_number

                        DFS_Visit(node x, node parent)
                           D(x) = dfs_number
                           dfs_number += 1
                           x.dfs_parent = parent
                           for each link (x, w)
                             if D(w) is not set
                               DFS_Visit(w, x)

                        Run_DFS(node gadag_root)
                           dfs_number = 0
                           DFS_Visit(gadag_root, NONE)

               Figure 7: Basic Depth-First Search algorithm

   Given a node x, one can compute the minimal DFS number of the
   neighbours of x, i.e. min( D(w) if (x,w) is a link).  This gives the
   earliest attachment point neighbouring x.  What is interesting,
   though, is what is the earliest attachment point from x and x's
   descendants.  This is what is determined by computing the Low-Point
   value.

   In order to compute the low point value, the network is traversed
   using DFS and the vertices are numbered based on the DFS walk.  Let
   this number be represented as DFS(x).  All the edges that lead to
   already visited nodes during DFS walk are back-edges.  The back-edges
   are important because they give information about reachability of a
   node via another path.

   The low point number is calculated by finding:

   Low(x) = Minimum of (  (DFS(x),
      Lowest DFS(n, x->n is a back-edge),



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      Lowest Low(n, x->n is tree edge in DFS walk) ).

   A detailed algorithm for computing the low-point value is given in
   Figure 8.  Figure 9 illustrates how the lowpoint algorithm applies to
   a example graph.

            global_variable: dfs_number

            Lowpoint_Visit(node x, node parent, interface p_to_x)
               D(x) = dfs_number
               L(x) = D(x)
               dfs_number += 1
               x.dfs_parent = parent
               x.dfs_parent_intf = p_to_x.remote_intf
               x.lowpoint_parent = NONE
               for each ordered_interface intf of x
                 if D(intf.remote_node) is not set
                   Lowpoint_Visit(intf.remote_node, x, intf)
                   if L(intf.remote_node) < L(x)
                      L(x) = L(intf.remote_node)
                      x.lowpoint_parent = intf.remote_node
                      x.lowpoint_parent_intf = intf
                 else if intf.remote_node is not parent
                   if D(intf.remote_node) < L(x)
                      L(x) = D(intf.remote_node)
                      x.lowpoint_parent = intf.remote_node
                      x.lowpoint_parent_intf = intf

            Run_Lowpoint(node gadag_root)
               dfs_number = 0
               Lowpoint_Visit(gadag_root, NONE, NONE)

                    Figure 8: Computing Low-Point value


















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            [E]---|    [J]-------[I]   [P]---[O]
             |    |     |         |     |     |
             |    |     |         |     |     |
            [R]  [D]---[C]--[F]  [H]---[K]   [N]
             |          |    |    |     |     |
             |          |    |    |     |     |
            [A]--------[B]  [G]---|    [L]---[M]

               (a) a non-2-connected graph

             [E]----|    [J]---------[I]    [P]------[O]
            (5, )   |  (10, )       (9, ) (16,  ) (15,  )
              |     |     |           |      |        |
              |     |     |           |      |        |
             [R]   [D]---[C]---[F]   [H]----[K]      [N]
            (0, ) (4, ) (3, ) (6, ) (8, ) (11, )  (14, )
              |           |     |     |      |        |
              |           |     |     |      |        |
             [A]---------[B]   [G]----|     [L]------[M]
            (1, )       (2, ) (7, )       (12,  )  (13,  )

               (b) with DFS values assigned   (D(x), L(x))

             [E]----|    [J]---------[I]    [P]------[O]
            (5,0)   |  (10,3)       (9,3) (16,11) (15,11)
              |     |     |           |      |        |
              |     |     |           |      |        |
             [R]   [D]---[C]---[F]   [H]----[K]      [N]
            (0,0) (4,0) (3,0) (6,3) (8,3) (11,11) (14,11)
              |           |     |     |      |        |
              |           |     |     |      |        |
             [A]---------[B]   [G]----|     [L]------[M]
            (1,0)       (2,0) (7,3)       (12,11)  (13,11)

                (c) with low-point values assigned (D(x), L(x))


               Figure 9: Example lowpoint value computation

   From the low-point value and lowpoint parent, there are three very
   useful things which motivate our computation.

   First, if there is a child c of x such that L(c) >= D(x), then there
   are no paths in the network graph that go from c or its descendants
   to an ancestor of x - and therefore x is a cut-vertex.  In Figure 9,
   this can be seen by looking at the DFS children of C.  C has two
   children - D and F and L(F) = 3 = D(C) so it is clear that C is a
   cut-vertex and F is in a block where C is the block's root.  L(D) = 0



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   < 3 = D(C) so D has a path to the ancestors of C; in this case, D can
   go via E to reach R.  Comparing the low-point values of all a node's
   DFS-children with the node's DFS-value is very useful because it
   allows identification of the cut-vertices and thus the blocks.

   Second, by repeatedly following the path given by lowpoint_parent,
   there is a path from x back to an ancestor of x that does not use the
   link [x, x.dfs_parent] in either direction.  The full path need not
   be taken, but this gives a way of finding an initial cycle and then
   ears.

   Third, as seen in Figure 9, even if L(x) < D(x), there may be a block
   that contains both the root and a DFS-child of a node while other
   DFS-children might be in different blocks.  In this example, C's
   child D is in the same block as R while F is not.  It is important to
   realize that the root of a block may also be the root of another
   block.

4.4.  Blocks in a Graph

   A key idea for the MRT Lowpoint algorithm is that any non-2-connected
   graph is made up by blocks (e.g. 2-connected clusters, cut-links,
   and/or isolated nodes).  To compute GADAGs and thus MRTs, computation
   is done in each block to compute ADAGs or Redundant Trees and then
   those ADAGs or Redundant Trees are combined into a GADAG or MRT.


























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                  [E]---|    [J]-------[I]   [P]---[O]
                   |    |     |         |     |     |
                   |    |     |         |     |     |
                  [R]  [D]---[C]--[F]  [H]---[K]   [N]
                   |          |    |    |     |     |
                   |          |    |    |     |     |
                  [A]--------[B]  [G]---|    [L]---[M]

                  (a)  A graph with four blocks that are:
                       three 2-connected clusters
                       and one cut-link


                  [E]<--|    [J]<------[I]   [P]<--[O]
                   |    |     |         ^     |     ^
                   V    |     V         |     V     |
                  [R]  [D]<--[C]  [F]  [H]<---[K]  [N]
                              ^    |    ^           ^
                              |    V    |           |
                  [A]------->[B]  [G]---|     [L]-->[M]

                    (b) MRT-Blue for destination R


                  [E]---|    [J]-------->[I]    [P]-->[O]
                        |                 |            |
                        V                 V            V
                  [R]  [D]-->[C]<---[F]  [H]<---[K]   [N]
                   ^          |      ^    |      ^     |
                   |          V      |    |      |     V
                  [A]<-------[B]    [G]<--|     [L]<--[M]

                     (c) MRT-Red for destination R


                                 Figure 10

   Consider the example depicted in Figure 10 (a).  In this figure, a
   special graph is presented, showing us all the ways 2-connected
   clusters can be connected.  It has four blocks: block 1 contains R,
   A, B, C, D, E, block 2 contains C, F, G, H, I, J, block 3 contains K,
   L, M, N, O, P, and block 4 is a cut-link containing H and K.  As can
   be observed, the first two blocks have one common node (node C) and
   blocks 2 and 3 do not have any common node, but they are connected
   through a cut-link that is block 4.  No two blocks can have more than
   one common node, since two blocks with at least two common nodes
   would qualify as a single 2-connected cluster.




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   Moreover, observe that if we want to get from one block to another,
   we must use a cut-vertex (the cut-vertices in this graph are C, H,
   K), regardless of the path selected, so we can say that all the paths
   from block 3 along the MRTs rooted at R will cross K first.  This
   observation means that if we want to find a pair of MRTs rooted at R,
   then we need to build up a pair of RTs in block 3 with K as a root.
   Similarly, we need to find another pair of RTs in block 2 with C as a
   root, and finally, we need the last pair of RTs in block 1 with R as
   a root.  When all the trees are selected, we can simply combine them;
   when a block is a cut-link (as in block 4), that cut-link is added in
   the same direction to both of the trees.  The resulting trees are
   depicted in Figure 10 (b) and (c).

   Similarly, to create a GADAG it is sufficient to compute ADAGs in
   each block and connect them.

   It is necessary, therefore, to identify the cut-vertices, the blocks
   and identify the appropriate local-root to use for each block.

4.5.  Determining Local-Root and Assigning Block-ID

   Each node in a network graph has a local-root, which is the cut-
   vertex (or root) in the same block that is closest to the root.  The
   local-root is used to determine whether two nodes share a common
   block.

               Compute_Localroot(node x, node localroot)
                   x.localroot = localroot
                   for each DFS child node c of x
                       if L(c) < D(x)   //x is not a cut-vertex
                           Compute_Localroot(c, x.localroot)
                       else
                           mark x as cut-vertex
                           Compute_Localroot(c, x)

               Compute_Localroot(gadag_root, gadag_root)

               Figure 11: A method for computing local-roots

   There are two different ways of computing the local-root for each
   node.  The stand-alone method is given in Figure 11 and better
   illustrates the concept; it is used by the GADAG construction methods
   given in Appendix B and Appendix C.  The MRT Lowpoint algorithm
   computes the local-root for a block as part of computing the GADAG
   using lowpoint inheritance; the essence of this computation is given
   in Figure 12.  Both methods for computing the local-root produce the
   same results.




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            Get the current node, s.
            Compute an ear(either through lowpoint inheritance
            or by following dfs parents) from s to a ready node e.
            (Thus, s is not e, if there is such ear.)
            if s is e
               for each node x in the ear that is not s
                   x.localroot = s
            else
               for each node x in the ear that is not s or e
                   x.localroot = e.localroot

           Figure 12: Ear-based method for computing local-roots

   Once the local-roots are known, two nodes X and Y are in a common
   block if and only if one of the following three conditions apply.

   o  Y's local-root is X's local-root : They are in the same block and
      neither is the cut-vertex closest to the root.

   o  Y's local-root is X: X is the cut-vertex closest to the root for
      Y's block

   o  Y is X's local-root: Y is the cut-vertex closest to the root for
      X's block

   Once we have computed the local-root for each node in the network
   graph, we can assign for each node, a block id that represents the
   block in which the node is present.  This computation is shown in
   Figure 13.

                 global_var: max_block_id

                 Assign_Block_ID(x, cur_block_id)
                   x.block_id = cur_block_id
                   foreach DFS child c of x
                      if (c.local_root is x)
                         max_block_id += 1
                         Assign_Block_ID(c, max_block_id)
                      else
                        Assign_Block_ID(c, cur_block_id)

                 max_block_id = 0
                 Assign_Block_ID(gadag_root, max_block_id)

             Figure 13: Assigning block id to identify blocks






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5.  MRT Lowpoint Algorithm Specification

   The MRT Lowpoint algorithm computes one GADAG that is then used by a
   router to determine its MRT-Blue and MRT-Red next-hops to all
   destinations.  Finally, based upon that information, alternates are
   selected for each next-hop to each destination.  The different parts
   of this algorithm are described below.

   o  Order the interfaces in the network graph.  [See Section 5.1]

   o  Compute the local MRT Island for the particular MRT Profile.  [See
      Section 5.2]

   o  Select the root to use for the GADAG.  [See Section 5.3]

   o  Initialize all interfaces to UNDIRECTED.  [See Section 5.4]

   o  Compute the DFS value,e.g.  D(x), and lowpoint value, L(x).  [See
      Figure 8]

   o  Construct the GADAG.  [See Section 5.5]

   o  Assign directions to all interfaces that are still UNDIRECTED.
      [See Section 5.6]

   o  From the computing router x, compute the next-hops for the MRT-
      Blue and MRT-Red. [See Section 5.7]

   o  Identify alternates for each next-hop to each destination by
      determining which one of the blue MRT and the red MRT the
      computing router x should select.  [See Section 5.8]

   A Python implementation of this algorithm is given in Appendix A.

5.1.  Interface Ordering

   To ensure consistency in computation, all routers MUST order
   interfaces identically down to the set of links with the same metric
   to the same neighboring node.  This is necessary for the DFS in
   Lowpoint_Visit in Section 4.3, where the selection order of the
   interfaces to explore results in different trees.  Consistent
   interface ordering is also necessary for computing the GADAG, where
   the selection order of the interfaces to use to form ears can result
   in different GADAGs.  It is also necessary for the topological sort
   described in Section 5.8, where different topological sort orderings
   can result in undirected links being added to the GADAG in different
   directions.




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   The required ordering between two interfaces from the same router x
   is given in Figure 14.

      Interface_Compare(interface a, interface b)
        if a.metric < b.metric
           return A_LESS_THAN_B
        if b.metric < a.metric
           return B_LESS_THAN_A
        if a.neighbor.mrt_node_id < b.neighbor.mrt_node_id
           return A_LESS_THAN_B
        if b.neighbor.mrt_node_id < a.neighbor.mrt_node_id
           return B_LESS_THAN_A
        // Same metric to same node, so the order doesn't matter for
        // interoperability.
        return A_EQUAL_TO_B


   Figure 14: Rules for ranking multiple interfaces.  Order is from low
                                 to high.

   In Figure 14, if two interfaces on a router connect to the same
   remote router with the same metric, the Interface_Compare function
   returns A_EQUAL_TO_B.  This is because the order in which those
   interfaces are initially explored does not affect the final GADAG
   produced by the algorithm described here.  While only one of the
   links will be added to the GADAG in the initial traversal, the other
   parallel links will be added to the GADAG with the same direction
   assigned during the procedure for assigning direction to UNDIRECTED
   links described in Section 5.6.  An implementation is free to apply
   some additional criteria to break ties in interface ordering in this
   situation, but that criteria is not specified here since it will not
   affect the final GADAG produced by the algorithm.

   The Interface_Compare function in Figure 14 relies on the
   interface.metric and the interface.neighbor.mrt_node_id values to
   order interfaces.  The exact source of these values for different
   IGPs and applications is specified in Figure 15.  The metric and
   mrt_node_id values for OSPFv2, OSPFv3, and IS-IS provided here is
   normative.  The metric and mrt_node_id values for ISIS-PCR in this
   table should be considered informational.  The normative values are
   specified in [IEEE8021Qca] .










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  +--------------+-----------------------+-----------------------------+
  | IGP/flooding | mrt_node_id           | metric of                   |
  | protocol     | of neighbor           | interface                   |
  | and          | on interface          |                             |
  | application  |                       |                             |
  +--------------+-----------------------+-----------------------------+
  | OSPFv2 for   | 4 octet Neighbor      | 2 octet Metric field        |
  | IP/LDP FRR   | Router ID in          | for corresponding           |
  |              | Link ID field for     | point-to-point link         |
  |              | corresponding         | in Router-LSA               |
  |              | point-to-point link   |                             |
  |              | in Router-LSA         |                             |
  +--------------+-----------------------+-----------------------------+
  | OSPFv3 for   | 4 octet Neighbor      | 2 octet Metric field        |
  | IP/LDP FRR   | Router ID field       | for corresponding           |
  |              | for corresponding     | point-to-point link         |
  |              | point-to-point link   | in Router-LSA               |
  |              | in Router-LSA         |                             |
  +--------------+-----------------------+-----------------------------+
  | IS-IS for    | 7 octet neighbor      | 3 octet metric field        |
  | IP/LDP FRR   | system ID and         | in Extended IS              |
  |              | pseudonode number     | Reachability TLV #22        |
  |              | in Extended IS        | or Multi-Topology           |
  |              | Reachability TLV #22  | IS Neighbor TLV #222        |
  |              | or Multi-Topology     |                             |
  |              | IS Neighbor TLV #222  |                             |
  +--------------+-----------------------+-----------------------------+
  | ISIS-PCR for | 8 octet Bridge ID     | 3 octet SPB-LINK-METRIC in  |
  | protection   | created from  2 octet | SPB-Metric sub-TLV (type 29)|
  | of traffic   | Bridge Priority in    | in Extended IS Reachability |
  | in bridged   | SPB Instance sub-TLV  | TLV #22 or Multi-Topology   |
  | networks     | (type 1) carried in   | Intermediate Systems        |
  |              | MT-Capability TLV     | TLV #222.  In the case      |
  |              | #144 and 6 octet      | of asymmetric link metrics, |
  |              | neighbor system ID in | the larger link metric      |
  |              | Extended IS           | is used for both link       |
  |              | Reachability TLV #22  | directions.                 |
  |              | or Multi-Topology     | (informational)             |
  |              | Intermediate Systems  |                             |
  |              | TLV #222              |                             |
  |              | (informational)       |                             |
  +--------------+-----------------------+-----------------------------+

          Figure 15: value of interface.neighbor.mrt_node_id and
     interface.metric to be used for ranking interfaces, for different
                    flooding protocols and applications





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   The metrics are unsigned integers and MUST be compared as unsigned
   integers.  The results of mrt_node_id comparisons MUST be the same as
   would be obtained by converting the mrt_node_ids to unsigned integers
   using network byte order and performing the comparison as unsigned
   integers.  In the case of IS-IS for IP/LDP FRR with point-to-point
   links, the pseudonode number (the 7th octet) is zero.  Broadcast
   interfaces will be discussed in Section 7.

5.2.  MRT Island Identification

   The local MRT Island for a particular MRT profile can be determined
   by starting from the computing router in the network graph and doing
   a breadth-first-search (BFS).  The BFS explores only links that are
   in the same area/level, are not IGP-excluded, and are not MRT-
   ineligible.  The BFS explores only nodes that are are not IGP-
   excluded, and that support the particular MRT profile.  See section 7
   of [I-D.ietf-rtgwg-mrt-frr-architecture] for more precise definitions
   of these criteria.

   MRT_Island_Identification(topology, computing_rtr, profile_id, area)
     for all routers in topology
         rtr.IN_MRT_ISLAND = FALSE
     computing_rtr.IN_MRT_ISLAND = TRUE
     explore_list = { computing_rtr }
     while (explore_list is not empty)
        next_rtr = remove_head(explore_list)
        for each intf in next_rtr
           if (not intf.MRT-ineligible
              and not intf.remote_intf.MRT-ineligible
              and not intf.IGP-excluded and (intf in area)
              and (intf.remote_node supports profile_id) )
              intf.IN_MRT_ISLAND = TRUE
              intf.remote_node.IN_MRT_ISLAND = TRUE
              if (not intf.remote_node.IN_MRT_ISLAND))
                 intf.remote_node.IN_MRT_ISLAND = TRUE
                 add_to_tail(explore_list, intf.remote_node)

                   Figure 16: MRT Island Identification

5.3.  GADAG Root Selection

   In Section 8.3 of [I-D.ietf-rtgwg-mrt-frr-architecture], the GADAG
   Root Selection Policy is described for the MRT default profile.  This
   selection policy allows routers to consistently select a common GADAG
   Root inside the local MRT Island, based on advertised priority
   values.  The MRT Lowpoint algorithm simply requires that all routers
   in the MRT Island MUST select the same GADAG Root; the mechanism can
   vary based upon the MRT profile description.  Before beginning



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   computation, the network graph is reduced to contain only the set of
   routers that support the specific MRT profile whose MRTs are being
   computed.

   As noted in Section 7, pseudonodes MUST NOT be considered for GADAG
   root selection.

   It is expected that an operator will designate a set of routers as
   good choices for selection as GADAG root by setting the GADAG Root
   Selection Priority for that set of routers to lower (more preferred)
   numerical values.  For guidance on setting the GADAG Root Selection
   Priority values, refer to Section 10.1.

5.4.  Initialization

   Before running the algorithm, there is the standard type of
   initialization to be done, such as clearing any computed DFS-values,
   lowpoint-values, DFS-parents, lowpoint-parents, any MRT-computed
   next-hops, and flags associated with algorithm.

   It is assumed that a regular SPF computation has been run so that the
   primary next-hops from the computing router to each destination are
   known.  This is required for determining alternates at the last step.

   Initially, all interfaces MUST be initialized to UNDIRECTED.  Whether
   they are OUTGOING, INCOMING or both is determined when the GADAG is
   constructed and augmented.

   It is possible that some links and nodes will be marked using
   standard IGP mechanisms to discourage or prevent transit traffic.
   Section 7.3.1 of [I-D.ietf-rtgwg-mrt-frr-architecture] describes how
   those links and nodes are excluded from MRT Island formation.

   MRT-FRR also has the ability to advertise links MRT-Ineligible, as
   described in Section 7.3.2 of [I-D.ietf-rtgwg-mrt-frr-architecture].
   These links are excluded from the MRT Island and the GADAG.
   Computation of MRT next-hops will therefore not use any MRT-
   ineligible links.  The MRT algorithm does still need to consider MRT-
   ineligible links when computing FRR alternates, because an MRT-
   ineligible link can still be the shortest-path next-hop to reach a
   destination.

   When a broadcast interface is advertised as MRT-ineligible, then the
   pseudo-node representing the entire broadcast network MUST NOT be
   included in the MRT Island.  This is equivalent to excluding all of
   the broadcast interfaces on that broadcast network from the MRT
   Island.




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5.5.  Constructing the GADAG using lowpoint inheritance

   As discussed in Section 4.2, it is necessary to find ears from a node
   x that is already in the GADAG (known as IN_GADAG).  Two different
   methods are used to find ears in the algorithm.  The first is by
   going to a not IN_GADAG DFS-child and then following the chain of
   low-point parents until an IN_GADAG node is found.  The second is by
   going to a not IN_GADAG neighbor and then following the chain of DFS
   parents until an IN_GADAG node is found.  As an ear is found, the
   associated interfaces are marked based on the direction taken.  The
   nodes in the ear are marked as IN_GADAG.  In the algorithm, first the
   ears via DFS-children are found and then the ears via DFS-neighbors
   are found.

   By adding both types of ears when an IN_GADAG node is processed, all
   ears that connect to that node are found.  The order in which the
   IN_GADAG nodes is processed is, of course, key to the algorithm.  The
   order is a stack of ears so the most recent ear is found at the top
   of the stack.  Of course, the stack stores nodes and not ears, so an
   ordered list of nodes, from the first node in the ear to the last
   node in the ear, is created as the ear is explored and then that list
   is pushed onto the stack.

   Each ear represents a partial order (see Figure 4) and processing the
   nodes in order along each ear ensures that all ears connecting to a
   node are found before a node higher in the partial order has its ears
   explored.  This means that the direction of the links in the ear is
   always from the node x being processed towards the other end of the
   ear.  Additionally, by using a stack of ears, this means that any
   unprocessed nodes in previous ears can only be ordered higher than
   nodes in the ears below it on the stack.

   In this algorithm that depends upon Low-Point inheritance, it is
   necessary that every node have a low-point parent that is not itself.
   If a node is a cut-vertex, that may not yet be the case.  Therefore,
   any nodes without a low-point parent will have their low-point parent
   set to their DFS parent and their low-point value set to the DFS-
   value of their parent.  This assignment also properly allows an ear
   between two cut-vertices.

   Finally, the algorithm simultaneously computes each node's local-
   root, as described in Figure 12.  This is further elaborated as
   follows.  The local-root can be inherited from the node at the end of
   the ear unless the end of the ear is x itself, in which case the
   local-root for all the nodes in the ear would be x.  This is because
   whenever the first cycle is found in a block, or an ear involving a
   bridge is computed, the cut-vertex closest to the root would be x
   itself.  In all other scenarios, the properties of lowpoint/dfs



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   parents ensure that the end of the ear will be in the same block, and
   thus inheriting its local-root would be the correct local-root for
   all newly added nodes.

   The pseudo-code for the GADAG algorithm (assuming that the adjustment
   of lowpoint for cut-vertices has been made) is shown in Figure 17.

           Construct_Ear(x, Stack, intf, ear_type)
              ear_list = empty
              cur_node = intf.remote_node
              cur_intf = intf
              not_done = true

              while not_done
                 cur_intf.UNDIRECTED = false
                 cur_intf.OUTGOING = true
                 cur_intf.remote_intf.UNDIRECTED = false
                 cur_intf.remote_intf.INCOMING = true

                 if cur_node.IN_GADAG is false
                    cur_node.IN_GADAG = true
                    add_to_list_end(ear_list, cur_node)
                    if ear_type is CHILD
                       cur_intf = cur_node.lowpoint_parent_intf
                       cur_node = cur_node.lowpoint_parent
                    else  // ear_type must be NEIGHBOR
                       cur_intf = cur_node.dfs_parent_intf
                       cur_node = cur_node.dfs_parent
                 else
                    not_done = false

              if (ear_type is CHILD) and (cur_node is x)
                 // x is a cut-vertex and the local root for
                 // the block in which the ear is computed
                 x.IS_CUT_VERTEX = true
                 localroot = x
              else
                 // Inherit local-root from the end of the ear
                 localroot = cur_node.localroot
              while ear_list is not empty
                 y = remove_end_item_from_list(ear_list)
                 y.localroot = localroot
                 push(Stack, y)

           Construct_GADAG_via_Lowpoint(topology, gadag_root)
             gadag_root.IN_GADAG = true
             gadag_root.localroot = None
             Initialize Stack to empty



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             push gadag_root onto Stack
             while (Stack is not empty)
                x = pop(Stack)
                foreach ordered_interface intf of x
                   if ((intf.remote_node.IN_GADAG == false) and
                       (intf.remote_node.dfs_parent is x))
                       Construct_Ear(x, Stack, intf, CHILD)
                foreach ordered_interface intf of x
                   if ((intf.remote_node.IN_GADAG == false) and
                       (intf.remote_node.dfs_parent is not x))
                       Construct_Ear(x, Stack, intf, NEIGHBOR)

           Construct_GADAG_via_Lowpoint(topology, gadag_root)

             Figure 17: Low-point Inheritance GADAG algorithm

5.6.  Augmenting the GADAG by directing all links

   The GADAG, regardless of the method used to construct it, at this
   point could be used to find MRTs, but the topology does not include
   all links in the network graph.  That has two impacts.  First, there
   might be shorter paths that respect the GADAG partial ordering and so
   the alternate paths would not be as short as possible.  Second, there
   may be additional paths between a router x and the root that are not
   included in the GADAG.  Including those provides potentially more
   bandwidth to traffic flowing on the alternates and may reduce
   congestion compared to just using the GADAG as currently constructed.

   The goal is thus to assign direction to every remaining link marked
   as UNDIRECTED to improve the paths and number of paths found when the
   MRTs are computed.

   To do this, we need to establish a total order that respects the
   partial order described by the GADAG.  This can be done using Kahn's
   topological sort[Kahn_1962_topo_sort] which essentially assigns a
   number to a node x only after all nodes before it (e.g. with a link
   incoming to x) have had their numbers assigned.  The only issue with
   the topological sort is that it works on DAGs and not ADAGs or
   GADAGs.

   To convert a GADAG to a DAG, it is necessary to remove all links that
   point to a root of block from within that block.  That provides the
   necessary conversion to a DAG and then a topological sort can be
   done.  When adding undirected links to the GADAG, links connecting
   the block root to other nodes in that block need special handling
   because the topological order will not always give the right answer
   for those links.  There are three cases to consider.  If the
   undirected link in question has another parallel link between the



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   same two nodes that is already directed, then the direction of the
   undirected link can be inherited from the previously directed link.
   In the case of parallel cut links, we set all of the parallel links
   to both INCOMING and OUTGOING.  Otherwise, the undirected link in
   question is set to OUTGOING from the block root node.  A cut-link can
   then be identified by the fact that it will be directed both INCOMING
   and OUTGOING in the GADAG.  The exact details of this whole process
   are captured in Figure 18

     Add_Undirected_Block_Root_Links(topo, gadag_root)
         foreach node x in topo
             if x.IS_CUT_VERTEX or x is gadag_root
                 foreach interface i of x
                     if (i.remote_node.localroot is not x
                                         or i.PROCESSED )
                         continue
                     Initialize bundle_list to empty
                     bundle.UNDIRECTED = true
                     bundle.OUTGOING = false
                     bundle.INCOMING = false
                     foreach interface i2 in x
                         if i2.remote_node is i.remote_node
                             add_to_list_end(bundle_list, i2)
                             if not i2.UNDIRECTED:
                                 bundle.UNDIRECTED = false
                                 if i2.INCOMING:
                                     bundle.INCOMING = true
                                 if i2.OUTGOING:
                                     bundle.OUTGOING = true
                     if bundle.UNDIRECTED
                         foreach interface i3 in bundle_list
                             i3.UNDIRECTED = false
                             i3.remote_intf.UNDIRECTED = false
                             i3.PROCESSED = true
                             i3.remote_intf.PROCESSED = true
                             i3.OUTGOING = true
                             i3.remote_intf.INCOMING = true
                     else
                         if (bundle.OUTGOING and bundle.INCOMING)
                             foreach interface i3 in bundle_list
                                 i3.UNDIRECTED = false
                                 i3.remote_intf.UNDIRECTED = false
                                 i3.PROCESSED = true
                                 i3.remote_intf.PROCESSED = true
                                 i3.OUTGOING = true
                                 i3.INCOMING = true
                                 i3.remote_intf.INCOMING = true
                                 i3.remote_intf.OUTGOING = true



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                         else if bundle.OUTGOING
                             foreach interface i3 in bundle_list
                                 i3.UNDIRECTED = false
                                 i3.remote_intf.UNDIRECTED = false
                                 i3.PROCESSED = true
                                 i3.remote_intf.PROCESSED = true
                                 i3.OUTGOING = true
                                 i3.remote_intf.INCOMING = true
                         else if bundle.INCOMING
                             foreach interface i3 in bundle_list
                                 i3.UNDIRECTED = false
                                 i3.remote_intf.UNDIRECTED = false
                                 i3.PROCESSED = true
                                 i3.remote_intf.PROCESSED = true
                                 i3.INCOMING = true
                                 i3.remote_intf.OUTGOING = true

     Modify_Block_Root_Incoming_Links(topo, gadag_root)
         foreach node x in topo
             if x.IS_CUT_VERTEX or x is gadag_root
                 foreach interface i of x
                     if i.remote_node.localroot is x
                         if i.INCOMING:
                             i.INCOMING = false
                             i.INCOMING_STORED = true
                             i.remote_intf.OUTGOING = false
                             i.remote_intf.OUTGOING_STORED = true

     Revert_Block_Root_Incoming_Links(topo, gadag_root)
         foreach node x in topo
             if x.IS_CUT_VERTEX or x is gadag_root
                 foreach interface i of x
                     if i.remote_node.localroot is x
                         if i.INCOMING_STORED
                             i.INCOMING = true
                             i.remote_intf.OUTGOING = true
                             i.INCOMING_STORED = false
                             i.remote_intf.OUTGOING_STORED = false

     Run_Topological_Sort_GADAG(topo, gadag_root)
         Modify_Block_Root_Incoming_Links(topo, gadag_root)
         foreach node x in topo
             node.unvisited = 0
             foreach interface i of x
                 if (i.INCOMING)
                     node.unvisited += 1
         Initialize working_list to empty
         Initialize topo_order_list to empty



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         add_to_list_end(working_list, gadag_root)
         while working_list is not empty
             y = remove_start_item_from_list(working_list)
             add_to_list_end(topo_order_list, y)
             foreach ordered_interface i of y
                 if intf.OUTGOING
                     i.remote_node.unvisited -= 1
                     if i.remote_node.unvisited is 0
                         add_to_list_end(working_list, i.remote_node)
         next_topo_order = 1
         while topo_order_list is not empty
             y = remove_start_item_from_list(topo_order_list)
             y.topo_order = next_topo_order
             next_topo_order += 1
         Revert_Block_Root_Incoming_Links(topo, gadag_root)

     def Set_Other_Undirected_Links_Based_On_Topo_Order(topo)
         foreach node x in topo
             foreach interface i of x
                 if i.UNDIRECTED:
                     if x.topo_order < i.remote_node.topo_order
                         i.OUTGOING = true
                         i.UNDIRECTED = false
                         i.remote_intf.INCOMING = true
                         i.remote_intf.UNDIRECTED = false
                     else
                         i.INCOMING = true
                         i.UNDIRECTED = false
                         i.remote_intf.OUTGOING = true
                         i.remote_intf.UNDIRECTED = false

     Add_Undirected_Links(topo, gadag_root)
         Add_Undirected_Block_Root_Links(topo, gadag_root)
         Run_Topological_Sort_GADAG(topo, gadag_root)
         Set_Other_Undirected_Links_Based_On_Topo_Order(topo)

     Add_Undirected_Links(topo, gadag_root)

            Figure 18: Assigning direction to UNDIRECTED links

   Proxy-nodes do not need to be added to the network graph.  They
   cannot be transited and do not affect the MRTs that are computed.
   The details of how the MRT-Blue and MRT-Red next-hops are computed
   for proxy-nodes and how the appropriate alternate next-hops are
   selected is given in Section 5.9.






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5.7.  Compute MRT next-hops

   As was discussed in Section 4.1, once a ADAG is found, it is
   straightforward to find the next-hops from any node X to the ADAG
   root.  However, in this algorithm, we will reuse the common GADAG and
   find not only the one pair of MRTs rooted at the GADAG root with it,
   but find a pair rooted at each node.  This is useful since it is
   significantly faster to compute.

   The method for computing differently rooted MRTs from the common
   GADAG is based on two ideas.  First, if two nodes X and Y are ordered
   with respect to each other in the partial order, then an SPF along
   OUTGOING links (an increasing-SPF) and an SPF along INCOMING links (a
   decreasing-SPF) can be used to find the increasing and decreasing
   paths.  Second, if two nodes X and Y aren't ordered with respect to
   each other in the partial order, then intermediary nodes can be used
   to create the paths by increasing/decreasing to the intermediary and
   then decreasing/increasing to reach Y.

   As usual, the two basic ideas will be discussed assuming the network
   is two-connected.  The generalization to multiple blocks is discussed
   in Section 5.7.4.  The full algorithm is given in Section 5.7.5.

5.7.1.  MRT next-hops to all nodes ordered with respect to the computing
        node

   To find two node-disjoint paths from the computing router X to any
   node Y, depends upon whether Y >> X or Y << X.  As shown in
   Figure 19, if Y >> X, then there is an increasing path that goes from
   X to Y without crossing R; this contains nodes in the interval [X,Y].
   There is also a decreasing path that decreases towards R and then
   decreases from R to Y; this contains nodes in the interval
   [X,R-small] or [R-great,Y].  The two paths cannot have common nodes
   other than X and Y.


                     [Y]<---(Cloud 2)<--- [X]
                      |                    ^
                      |                    |
                      V                    |
                   (Cloud 3)--->[R]--->(Cloud 1)

                  MRT-Blue path: X->Cloud 2->Y
                  MRT-Red path: X->Cloud 1->R->Cloud 3->Y

                             Figure 19: Y >> X





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   Similar logic applies if Y << X, as shown in Figure 20.  In this
   case, the increasing path from X increases to R and then increases
   from R to Y to use nodes in the intervals [X,R-great] and [R-small,
   Y].  The decreasing path from X reaches Y without crossing R and uses
   nodes in the interval [Y,X].


                    [X]<---(Cloud 2)<--- [Y]
                     |                    ^
                     |                    |
                     V                    |
                  (Cloud 3)--->[R]--->(Cloud 1)

                 MRT-Blue path: X->Cloud 3->R->Cloud 1->Y
                 MRT-Red path: X->Cloud 2->Y

                             Figure 20: Y << X

5.7.2.  MRT next-hops to all nodes not ordered with respect to the
        computing node

   When X and Y are not ordered, the first path should increase until we
   get to a node G, where G >> Y.  At G, we need to decrease to Y.  The
   other path should be just the opposite: we must decrease until we get
   to a node H, where H << Y, and then increase.  Since R is smaller and
   greater than Y, such G and H must exist.  It is also easy to see that
   these two paths must be node disjoint: the first path contains nodes
   in interval [X,G] and [Y,G], while the second path contains nodes in
   interval [H,X] and [H,Y].  This is illustrated in Figure 21.  It is
   necessary to decrease and then increase for the MRT-Blue and increase
   and then decrease for the MRT-Red; if one simply increased for one
   and decreased for the other, then both paths would go through the
   root R.


















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                 (Cloud 6)<---[Y]<---(Cloud 5)<------------|
                   |                                       |
                   |                                       |
                   V                                       |
                  [G]--->(Cloud 4)--->[R]--->(Cloud 1)--->[H]
                   ^                                       |
                   |                                       |
                   |                                       |
                  (Cloud 3)<---[X]<---(Cloud 2)<-----------|

              MRT-Blue path: decrease to H and increase to Y
                   X->Cloud 2->H->Cloud 5->Y
              MRT-Red path:  increase to G and decrease to Y
                   X->Cloud 3->G->Cloud 6->Y

                       Figure 21: X and Y unordered

   This gives disjoint paths as long as G and H are not the same node.
   Since G >> Y and H << Y, if G and H could be the same node, that
   would have to be the root R.  This is not possible because there is
   only one incoming interface to the root R which is created when the
   initial cycle is found.  Recall from Figure 6 that whenever an ear
   was found to have an end that was the root R, the ear was directed
   from R so that the associated interface on R is outgoing and not
   incoming.  Therefore, there must be exactly one node M which is the
   largest one before R, so the MRT-Red path will never reach R; it will
   turn at M and decrease to Y.

5.7.3.  Computing Redundant Tree next-hops in a 2-connected Graph

   The basic ideas for computing RT next-hops in a 2-connected graph
   were given in Section 5.7.1 and Section 5.7.2.  Given these two
   ideas, how can we find the trees?

   If some node X only wants to find the next-hops (which is usually the
   case for IP networks), it is enough to find which nodes are greater
   and less than X, and which are not ordered; this can be done by
   running an increasing-SPF and a decreasing-SPF rooted at X and not
   exploring any links from the ADAG root.

   In principle, an traversal method other than SPF could be used to
   traverse the GADAG in the process of determining blue and red next-
   hops that result in maximally redundant trees.  This will be the case
   as long as one traversal uses the links in the direction specified by
   the GADAG and the other traversal uses the links in the direction
   opposite of that specified by the GADAG.  However, a different
   traversal algorithm will generally result in different blue and red
   next-hops.  Therefore, the algorithm specified here requires the use



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   of SPF to traverse the GADAG to generate MRT blue and red next-hops,
   as described below.

   An increasing-SPF rooted at X and not exploring links from the root
   will find the increasing next-hops to all Y >> X.  Those increasing
   next-hops are X's next-hops on the MRT-Blue to reach Y.  A
   decreasing-SPF rooted at X and not exploring links from the root will
   find the decreasing next-hops to all Z << X.  Those decreasing next-
   hops are X's next-hops on the MRT-Red to reach Z.  Since the root R
   is both greater than and less than X, after this increasing-SPF and
   decreasing-SPF, X's next-hops on the MRT-Blue and on the MRT-Red to
   reach R are known.  For every node Y >> X, X's next-hops on the MRT-
   Red to reach Y are set to those on the MRT-Red to reach R.  For every
   node Z << X, X's next-hops on the MRT-Blue to reach Z are set to
   those on the MRT-Blue to reach R.

   For those nodes which were not reached by either the increasing-SPF
   or the decreasing-SPF, we can determine the next-hops as well.  The
   increasing MRT-Blue next-hop for a node which is not ordered with
   respect to X is the next-hop along the decreasing MRT-Red towards R,
   and the decreasing MRT-Red next-hop is the next-hop along the
   increasing MRT-Blue towards R.  Naturally, since R is ordered with
   respect to all the nodes, there will always be an increasing and a
   decreasing path towards it.  This algorithm does not provide the
   complete specific path taken but just the appropriate next-hops to
   use.  The identities of G and H are not determined by the computing
   node X.

   The final case to consider is when the GADAG root R computes its own
   next-hops.  Since the GADAG root R is << all other nodes, running an
   increasing-SPF rooted at R will reach all other nodes; the MRT-Blue
   next-hops are those found with this increasing-SPF.  Similarly, since
   the GADAG root R is >> all other nodes, running a decreasing-SPF
   rooted at R will reach all other nodes; the MRT-Red next-hops are
   those found with this decreasing-SPF.

                 E---D---|              E<--D<--|
                 |   |   |              |   ^   |
                 |   |   |              V   |   |
                 R   F   C              R   F   C
                 |   |   |              |   ^   ^
                 |   |   |              V   |   |
                 A---B---|              A-->B---|

                    (a)                    (b)
            A 2-connected graph    A spanning ADAG rooted at R

                                 Figure 22



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   As an example consider the situation depicted in Figure 22.  Node C
   runs an increasing-SPF and a decreasing-SPF on the ADAG.  The
   increasing-SPF reaches D, E and R and the decreasing-SPF reaches B, A
   and R.  E>>C.  So towards E the MRT-Blue next-hop is D, since E was
   reached on the increasing path through D.  And the MRT-Red next-hop
   towards E is B, since R was reached on the decreasing path through B.
   Since E>>D, D will similarly compute its MRT-Blue next-hop to be E,
   ensuring that a packet on MRT-Blue will use path C-D-E.  B, A and R
   will similarly compute the MRT-Red next-hops towards E (which is
   ordered less than B, A and R), ensuring that a packet on MRT-Red will
   use path C-B-A-R-E.

   C can determine the next-hops towards F as well.  Since F is not
   ordered with respect to C, the MRT-Blue next-hop is the decreasing
   one towards R (which is B) and the MRT-Red next-hop is the increasing
   one towards R (which is D).  Since F>>B, for its MRT-Blue next-hop
   towards F, B will use the real increasing next-hop towards F.  So a
   packet forwarded to B on MRT-Blue will get to F on path C-B-F.
   Similarly, D will use the real decreasing next-hop towards F as its
   MRT-Red next-hop, a packet on MRT-Red will use path C-D-F.

5.7.4.  Generalizing for a graph that isn't 2-connected

   If a graph isn't 2-connected, then the basic approach given in
   Section 5.7.3 needs some extensions to determine the appropriate MRT
   next-hops to use for destinations outside the computing router X's
   blocks.  In order to find a pair of maximally redundant trees in that
   graph we need to find a pair of RTs in each of the blocks (the root
   of these trees will be discussed later), and combine them.

   When computing the MRT next-hops from a router X, there are three
   basic differences:

   1.  Only nodes in a common block with X should be explored in the
       increasing-SPF and decreasing-SPF.

   2.  Instead of using the GADAG root, X's local-root should be used.
       This has the following implications:

       A.  The links from X's local-root should not be explored.

       B.  If a node is explored in the outgoing SPF so Y >> X, then X's
           MRT-Red next-hops to reach Y uses X's MRT-Red next-hops to
           reach X's local-root and if Z << X, then X's MRT-Blue next-
           hops to reach Z uses X's MRT-Blue next-hops to reach X's
           local-root.





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       C.  If a node W in a common block with X was not reached in the
           increasing-SPF or decreasing-SPF, then W is unordered with
           respect to X.  X's MRT-Blue next-hops to W are X's decreasing
           (aka MRT-Red) next-hops to X's local-root.  X's MRT-Red next-
           hops to W are X's increasing (aka MRT-Blue) next-hops to X's
           local-root.

   3.  For nodes in different blocks, the next-hops must be inherited
       via the relevant cut-vertex.

   These are all captured in the detailed algorithm given in
   Section 5.7.5.

5.7.5.  Complete Algorithm to Compute MRT Next-Hops

   The complete algorithm to compute MRT Next-Hops for a particular
   router X is given in Figure 23.  In addition to computing the MRT-
   Blue next-hops and MRT-Red next-hops used by X to reach each node Y,
   the algorithm also stores an "order_proxy", which is the proper cut-
   vertex to reach Y if it is outside the block, and which is used later
   in deciding whether the MRT-Blue or the MRT-Red can provide an
   acceptable alternate for a particular primary next-hop.

   In_Common_Block(x, y)
     if ( (x.block_id is y.block_id)
          or (x is y.localroot) or (y is x.localroot) )
        return true
     return false

   Store_Results(y, direction)
      if direction is FORWARD
         y.higher = true
         y.blue_next_hops = y.next_hops
      if direction is REVERSE
         y.lower = true
         y.red_next_hops = y.next_hops

   SPF_No_Traverse_Block_Root(spf_root, block_root, direction)
      Initialize spf_heap to empty
      Initialize nodes' spf_metric to infinity and next_hops to empty
      spf_root.spf_metric = 0
      insert(spf_heap, spf_root)
      while (spf_heap is not empty)
          min_node = remove_lowest(spf_heap)
          Store_Results(min_node, direction)
          if ((min_node is spf_root) or (min_node is not block_root))
             foreach interface intf of min_node
                   if ( ( ((direction is FORWARD) and intf.OUTGOING) or



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                       ((direction is REVERSE) and intf.INCOMING) )
                       and In_Common_Block(spf_root, intf.remote_node) )
                   path_metric = min_node.spf_metric + intf.metric
                   if path_metric < intf.remote_node.spf_metric
                      intf.remote_node.spf_metric = path_metric
                      if min_node is spf_root
                        intf.remote_node.next_hops = make_list(intf)
                      else
                        intf.remote_node.next_hops = min_node.next_hops
                      insert_or_update(spf_heap, intf.remote_node)
                   else if path_metric == intf.remote_node.spf_metric
                      if min_node is spf_root
                         add_to_list(intf.remote_node.next_hops, intf)
                      else
                         add_list_to_list(intf.remote_node.next_hops,
                                          min_node.next_hops)

   SetEdge(y)
     if y.blue_next_hops is empty and y.red_next_hops is empty
        SetEdge(y.localroot)
        y.blue_next_hops = y.localroot.blue_next_hops
        y.red_next_hops = y.localroot.red_next_hops
        y.order_proxy = y.localroot.order_proxy

   Compute_MRT_NextHops(x, gadag_root)
      foreach node y
        y.higher = y.lower = false
        clear y.red_next_hops and y.blue_next_hops
        y.order_proxy = y
      SPF_No_Traverse_Block_Root(x, x.localroot, FORWARD)
      SPF_No_Traverse_Block_Root(x, x.localroot, REVERSE)

      // red and blue next-hops are stored to x.localroot as different
      // paths are found via the SPF and reverse-SPF.
      // Similarly any nodes whose local-root is x will have their
      // red_next_hops and blue_next_hops already set.

      // Handle nodes in the same block that aren't the local-root
      foreach node y
        if (y.IN_MRT_ISLAND and (y is not x) and
             (y.block_id is x.block_id) )
           if y.higher
              y.red_next_hops = x.localroot.red_next_hops
           else if y.lower
              y.blue_next_hops = x.localroot.blue_next_hops
           else
              y.blue_next_hops = x.localroot.red_next_hops
              y.red_next_hops = x.localroot.blue_next_hops



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      // Inherit next-hops and order_proxies to other components
      if (x is not gadag_root) and (x.localroot is not gadag_root)
         gadag_root.blue_next_hops = x.localroot.blue_next_hops
         gadag_root.red_next_hops = x.localroot.red_next_hops
         gadag_root.order_proxy = x.localroot
      foreach node y
         if (y is not gadag_root) and (y is not x) and y.IN_MRT_ISLAND
           SetEdge(y)

   max_block_id = 0
   Assign_Block_ID(gadag_root, max_block_id)
   Compute_MRT_NextHops(x, gadag_root)

                                 Figure 23

5.8.  Identify MRT alternates

   At this point, a computing router S knows its MRT-Blue next-hops and
   MRT-Red next-hops for each destination in the MRT Island.  The
   primary next-hops along the SPT are also known.  It remains to
   determine for each primary next-hop to a destination D, which of the
   MRTs avoids the primary next-hop node F.  This computation depends
   upon data set in Compute_MRT_NextHops such as each node y's
   y.blue_next_hops, y.red_next_hops, y.order_proxy, y.higher, y.lower
   and topo_orders.  Recall that any router knows only which are the
   nodes greater and lesser than itself, but it cannot decide the
   relation between any two given nodes easily; that is why we need
   topological ordering.

   For each primary next-hop node F to each destination D, S can call
   Select_Alternates(S, D, F, primary_intf) to determine whether to use
   the MRT-Blue or MRT-Red next-hops as the alternate next-hop(s) for
   that primary next hop.  The algorithm is given in Figure 24 and
   discussed afterwards.

  Select_Alternates_Internal(D, F, primary_intf,
                                 D_lower, D_higher, D_topo_order):
      if D_higher and D_lower
          if F.HIGHER and F.LOWER
              if F.topo_order < D_topo_order
                  return USE_RED
              else
                  return USE_BLUE
          if F.HIGHER
              return USE_RED
          if F.LOWER
              return USE_BLUE
          //F unordered wrt S



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          return USE_RED_OR_BLUE

      else if D_higher
          if F.HIGHER and F.LOWER
              return USE_BLUE
          if F.LOWER
              return USE_BLUE
          if F.HIGHER
              if (F.topo_order > D_topo_order)
                  return USE_BLUE
              if (F.topo_order < D_topo_order)
                  return USE_RED
          //F unordered wrt S
          return USE_RED_OR_BLUE

      else if D_lower
          if F.HIGHER and F.LOWER
              return USE_RED
          if F.HIGHER
              return USE_RED
          if F.LOWER
              if F.topo_order > D_topo_order
                  return USE_BLUE
              if F.topo_order < D_topo_order
                  return USE_RED
          //F unordered wrt S
          return USE_RED_OR_BLUE

      else  //D is unordered wrt S
          if F.HIGHER and F.LOWER
              if primary_intf.OUTGOING and primary_intf.INCOMING
                  return USE_RED_OR_BLUE
              if primary_intf.OUTGOING
                  return USE_BLUE
              if primary_intf.INCOMING
                  return USE_RED
              //primary_intf not in GADAG
              return USE_RED
          if F.LOWER
              return USE_RED
          if F.HIGHER
              return USE_BLUE
          //F unordered wrt S
          if F.topo_order > D_topo_order:
              return USE_BLUE
          else:
              return USE_RED




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  Select_Alternates(D, F, primary_intf)
      if not In_Common_Block(F, S)
          return PRIM_NH_IN_DIFFERENT_BLOCK
      if (D is F) or (D.order_proxy is F)
          return PRIM_NH_IS_D_OR_OP_FOR_D
      D_lower = D.order_proxy.LOWER
      D_higher = D.order_proxy.HIGHER
      D_topo_order = D.order_proxy.topo_order
      return Select_Alternates_Internal(D, F, primary_intf,
                                        D_lower, D_higher, D_topo_order)

      Figure 24: Select_Alternates() and Select_Alternates_Internal()

   It is useful to first handle the case where where F is also D, or F
   is the order proxy for D.  In this case, only link protection is
   possible.  The MRT that doesn't use the failed primary next-hop is
   used.  If both MRTs use the primary next-hop, then the primary next-
   hop must be a cut-link, so either MRT could be used but the set of
   MRT next-hops must be pruned to avoid the failed primary next-hop
   interface.  To indicate this case, Select_Alternates returns
   PRIM_NH_IS_D_OR_OP_FOR_D.  Explicit pseudo-code to handle the three
   sub-cases above is not provided.

   The logic behind Select_Alternates_Internal is described in
   Figure 25.  As an example, consider the first case described in the
   table, where the D>>S and D<<S.  If this is true, then either S or D
   must be the block root, R.  If F>>S and F<<S, then S is the block
   root.  So the blue path from S to D is the increasing path to D, and
   the red path S to D is the decreasing path to D.  If the
   F.topo_order<D.topo_order, then either F is ordered higher than D or
   F is unordered with respect to D.  Therefore, F is either on a
   decreasing path from S to D, or it is on neither an increasing nor a
   decreasing path from S to D.  In either case, it is safe to take an
   increasing path from S to D to avoid F.  We know that when S is R,
   the increasing path is the blue path, so it is safe to use the blue
   path to avoid F.

   If instead F.topo_order>D.topo_order, then either F is ordered lower
   than D, or F is unordered with respect to D.  Therefore, F is either
   on an increasing path from S to D, or it is on neither an increasing
   nor a decreasing path from S to D.  In either case, it is safe to
   take a decreasing path from S to D to avoid F.  We know that when S
   is R, the decreasing path is the red path, so it is safe to use the
   red path to avoid F.

   If F>>S or F<<S (but not both), then D is the block root.  We then
   know that the blue path from S to D is the increasing path to R, and
   the red path is the decreasing path to R.  When F>>S, we deduce that



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   F is on an increasing path from S to R.  So in order to avoid F, we
   use a decreasing path from S to R, which is the red path.  Instead,
   when F<<S, we deduce that F is on a decreasing path from S to R.  So
   in order to avoid F, we use an increasing path from S to R, which is
   the blue path.

   All possible cases are systematically described in the same manner in
   the rest of the table.

+------+------------+------+------------------------------+------------+
| D    | MRT blue   | F    | additional      | F          | Alternate  |
| wrt  | and red    | wrt  | criteria        | wrt        |            |
| S    | path       | S    |                 | MRT        |            |
|      | properties |      |                 | (deduced)  |            |
+------+------------+------+-----------------+------------+------------+
| D>>S | Blue path: | F>>S | additional      | F on an    | Use Red    |
| and  | Increasing | only | criteria        | increasing | to avoid   |
| D<<S,| path to R. |      | not needed      | path from  | F          |
| D is | Red path:  |      |                 | S to R     |            |
| R,   | Decreasing +------+-----------------+------------+------------+
|      | path to R. | F<<S | additional      | F on a     | Use Blue   |
|      |            | only | criteria        | decreasing | to avoid   |
|      |            |      | not needed      | path from  | F          |
| or   |            |      |                 | S to R     |            |
|      |            +------+-----------------+------------+------------+
|      |            | F>>S | topo(F)>topo(D) | F on a     | Use Blue   |
| S is | Blue path: | and  | implies that    | decreasing | to avoid   |
| R    | Increasing | F<<S,| F>>D or F??D    | path from  | F          |
|      | path to D. | F is |                 | S to D or  |            |
|      | Red path:  | R    |                 | neither    |            |
|      | Decreasing |      +-----------------+------------+------------+
|      | path to D. |      | topo(F)<topo(D) | F on an    | Use Red    |
|      |            |      | implies that    | increasing | to avoid   |
|      |            |      | F<<D or F??D    | path from  | F          |
|      |            |      |                 | S to D or  |            |
|      |            |      |                 | neither    |            |
|      |            +------+-----------------+------------+------------+
|      |            | F??S | Can only occur  | F is on    | Use Red    |
|      |            |      | when link       | neither    | or Blue    |
|      |            |      | between         | increasing | to avoid   |
|      |            |      | F and S         | nor decr.  | F          |
|      |            |      | is marked       | path from  |            |
|      |            |      | MRT_INELIGIBLE  | S to D or R|            |
+------+------------+------+-----------------+------------+------------+
| D>>S | Blue path: | F<<S | additional      | F on       | Use Blue   |
| only | Increasing | only | criteria        | decreasing | to avoid   |
|      | shortest   |      | not needed      | path from  | F          |
|      | path from  |      |                 | S to R     |            |



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|      | S to D.    +------+-----------------+------------+------------+
|      | Red path:  | F>>S | topo(F)>topo(D) | F on       | Use Blue   |
|      | Decreasing | only | implies that    | decreasing | to avoid   |
|      | shortest   |      | F>>D or F??D    | path from  | F          |
|      | path from  |      |                 | R to D     |            |
|      | S to R,    |      |                 | or         |            |
|      | then       |      |                 | neither    |            |
|      | decreasing |      +-----------------+------------+------------+
|      | shortest   |      | topo(F)<topo(D) | F on       | Use Red    |
|      | path from  |      | implies that    | increasing | to avoid   |
|      | R to D.    |      | F<<D or F??D    | path from  | F          |
|      |            |      |                 | S to D     |            |
|      |            |      |                 | or         |            |
|      |            |      |                 | neither    |            |
|      |            +------+-----------------+------------+------------+
|      |            | F>>S | additional      | F on Red   | Use Blue   |
|      |            | and  | criteria        |            | to avoid   |
|      |            | F<<S,| not needed      |            | F          |
|      |            | F is |                 |            |            |
|      |            | R    |                 |            |            |
|      |            +------+-----------------+------------+------------+
|      |            | F??S | Can only occur  | F is on    | Use Red    |
|      |            |      | when link       | neither    | or Blue    |
|      |            |      | between         | increasing | to avoid   |
|      |            |      | F and S         | nor decr.  | F          |
|      |            |      | is marked       | path from  |            |
|      |            |      | MRT_INELIGIBLE  | S to D or R|            |
+------+------------+------+-----------------+------------+------------+
| D<<S | Blue path: | F>>S | additional      | F on       | Use Red    |
| only | Increasing | only | criteria        | increasing | to avoid   |
|      | shortest   |      | not needed      | path from  | F          |
|      | path from  |      |                 | S to R     |            |
|      | S to R,    +------+-----------------+------------+------------+
|      | then       | F<<S | topo(F)>topo(D) | F on       | Use Blue   |
|      | increasing | only | implies that    | decreasing | to avoid   |
|      | shortest   |      | F>>D or F??D    | path from  | F          |
|      | path from  |      |                 | R to D     |            |
|      | R to D.    |      |                 | or         |            |
|      | Red path:  |      |                 | neither    |            |
|      | Decreasing |      +-----------------+------------+------------+
|      | shortest   |      | topo(F)<topo(D) | F on       | Use Red    |
|      | path from  |      | implies that    | increasing | to avoid   |
|      | S to D.    |      | F<<D or F??D    | path from  | F          |
|      |            |      |                 | S to D     |            |
|      |            |      |                 | or         |            |
|      |            |      |                 | neither    |            |
|      |            +------+-----------------+------------+------------+
|      |            | F>>S | additional      | F on Blue  | Use Red    |



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|      |            | and  | criteria        |            | to avoid   |
|      |            | F<<S,| not             |            | F          |
|      |            | F is | needed          |            |            |
|      |            | R    |                 |            |            |
|      |            +------+-----------------+------------+------------+
|      |            | F??S | Can only occur  | F is on    | Use Red    |
|      |            |      | when link       | neither    | or Blue    |
|      |            |      | between         | increasing | to avoid   |
|      |            |      | F and S         | nor decr.  | F          |
|      |            |      | is marked       | path from  |            |
|      |            |      | MRT_INELIGIBLE  | S to D or R|            |
+------+------------+------+-----------------+------------+------------+
| D??S | Blue path: | F<<S | additional      | F on a     | Use Red    |
|      | Decr. from | only | criteria        | decreasing | to avoid   |
|      | S to first |      | not needed      | path from  | F          |
|      | node K<<D, |      |                 | S to K.    |            |
|      | then incr. +------+-----------------+------------+------------+
|      | to D.      | F>>S | additional      | F on an    | Use Blue   |
|      | Red path:  | only | criteria        | increasing | to avoid   |
|      | Incr. from |      | not needed      | path from  | F          |
|      | S to first |      |                 | S to L     |            |
|      | node L>>D, |      |                 |            |            |
|      | then decr. |      |                 |            |            |
|      |            +------+-----------------+------------+------------+
|      |            | F??S | F<-->S link is  |            |            |
|      |            |      | MRT_INELIGIBLE  |            |            |
|      |            |      +-----------------+------------+------------+
|      |            |      | topo(F)>topo(D) | F on decr. | Use Blue   |
|      |            |      | implies that    | path from  | to avoid   |
|      |            |      | F>>D or F??D    | L to D or  | F          |
|      |            |      |                 | neither    |            |
|      |            |      +-----------------+------------+------------+
|      |            |      | topo(F)<topo(D) | F on incr. | Use Red    |
|      |            |      | implies that    | path from  | to avoid   |
|      |            |      | F<<D or F??D    | K to D or  | F          |
|      |            |      |                 | neither    |            |
|      |            +------+-----------------+------------+------------+
|      |            | F>>S | GADAG link      | F on an    | Use Blue   |
|      |            | and  | direction       | incr. path | to avoid   |
|      |            | F<<S,| S->F            | from S     | F          |
|      |            | F is +-----------------+------------+------------+
|      |            | R    | GADAG link      | F on a     | Use Red    |
|      |            |      | direction       | decr. path | to avoid   |
|      |            |      | S<-F            | from S     | F          |
|      |            |      +-----------------+------------+------------+
|      |            |      | GADAG link      | Either F is the order   |
|      |            |      | direction       | proxy for D (case       |
|      |            |      | S<-->F          | already handled) or D   |



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|      |            |      |                 | is in a different block |
|      |            |      |                 | from F, in which case   |
|      |            |      |                 | Red or Blue avoids F    |
|      |            |      +-----------------+-------------------------+
|      |            |      | S-F link not    | Relies on special       |
|      |            |      | in GADAG,       | construction of GADAG   |
|      |            |      | only when       | to demonstrate that     |
|      |            |      | S-F link is     | using Red avoids F      |
|      |            |      | MRT_INELIGIBLE  | (see text)              |
+------+------------+------+-----------------+-------------------------+

     Figure 25: determining MRT next-hops and alternates based on the
       partial order and topological sort relationships between the
    source(S), destination(D), primary next-hop(F), and block root(R).
       topo(N) indicates the topological sort value of node N.  X??Y
     indicates that node X is unordered with respect to node Y.  It is
   assumed that the case where F is D, or where F is the order proxy for
                       D, has already been handled.

   The last case in Figure 25 requires additional explanation.  The fact
   that the red path from S to D in this case avoids F relies on a
   special property of the GADAGs that we have constructed in this
   algorithm, a property not shared by all GADAGs in general.  When D is
   unordered with respect to S, and F is the localroot for S, it can
   occur that the link between S and F is not in the GADAG only when
   that link has been marked MRT_INELIGIBLE.  For an arbitrary GADAG, S
   doesn't have enough information based on the computed order
   relationships to determine if the red path or blue path will hit F
   (which is also the localroot) before hitting K or L, and making it
   safely to D.  However, the GADAGs that we construct using the
   algorithm in this document are not arbitrary GADAGs.  They have the
   additional property that incoming links to a localroot come from only
   one other node in the same block.  This is a result of the method of
   construction.  This additional property guarantees that the red path
   from S to D will never pass through the localroot of S.  (That would
   require the localroot to play the role of L, the first node in the
   path ordered higher than D, which would in turn require the localroot
   to have two incoming links in the GADAG, which cannot happen.)
   Therefore it is safe to use the red path to avoid F with these
   specially constructed GADAGs.

   As an example of how Select_Alternates_Internal() operates, consider
   the ADAG depicted in Figure 26 and first suppose that G is the
   source, D is the destination and H is the failed next-hop.  Since
   D>>G, we need to compare H.topo_order and D.topo_order.  Since
   D.topo_order>H.topo_order, D must be either higher than H or
   unordered with respect to H, so we should select the decreasing path
   towards the root.  If, however, the destination were instead J, we



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   must find that H.topo_order>J.topo_order, so we must choose the
   increasing Blue next-hop to J, which is I.  In the case, when instead
   the destination is C, we find that we need to first decrease to avoid
   using H, so the Blue, first decreasing then increasing, path is
   selected.

                            [E]<-[D]<-[H]<-[J]
                             |    ^    ^    ^
                             V    |    |    |
                            [R]  [C]  [G]->[I]
                             |    ^    ^    ^
                             V    |    |    |
                            [A]->[B]->[F]---|

                          (a)ADAG rooted at R for
                            a 2-connected graph

                                 Figure 26

5.9.  Named Proxy-Nodes

   As discussed in Section 11.2 of
   [I-D.ietf-rtgwg-mrt-frr-architecture], it is necessary to find MRT-
   Blue and MRT-Red next-hops and MRT-FRR alternates for named proxy-
   nodes.  An example use case is for a router that is not part of that
   local MRT Island, when there is only partial MRT support in the
   domain.

5.9.1.  Determining Proxy-Node Attachment Routers

   Section 11.2 of [I-D.ietf-rtgwg-mrt-frr-architecture] discusses
   general considerations for determining the two proxy-node attachment
   routers for a given proxy-node, corresponding to a prefix.  A router
   in the MRT Island that advertises the prefix is a candidate for being
   a proxy-node attachment router, with the associated named-proxy-cost
   equal to the advertised cost to the prefix.

   An Island Border Router (IBR) is a router in the MRT Island that is
   connected to an Island Neighbor(IN), which is a router not in the MRT
   Island but in the same area/level.  An (IBR,IN) pair is a candidate
   for being a proxy-node attachment router, if the shortest path from
   the IN to the prefix does not enter the MRT Island.  A method for
   identifying such loop-free Island Neighbors(LFINs) is given below.
   The named-proxy-cost assigned to each (IBR, IN) pair is cost(IBR, IN)
   + D_opt(IN, prefix).

   From the set of prefix-advertising routers and the set of IBRs with
   at least one LFIN, the two routers with the lowest named-proxy-cost



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   are selected.  Ties are broken based upon the lowest Router ID.  For
   ease of discussion, the two selected routers will be referred to as
   proxy-node attachment routers.

5.9.2.  Computing if an Island Neighbor (IN) is loop-free

   As discussed above, the Island Neighbor needs to be loop-free with
   respect to the whole MRT Island for the destination.  This can be
   accomplished by running the usual SPF algorithm while keeping track
   of which shortest paths have passed through the MRT island.  Pseudo-
   code for this is shown in Figure 27.  The Island_Marking_SPF() is run
   for each IN that needs to be evaluated for the loop-free condition,
   with the IN as the spf_root.  Whether or not an IN is loop-free with
   respect to the MRT island can then be determined by evaluating
   node.PATH_HITS_ISLAND for each destination of interest.




































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    Island_Marking_SPF(spf_root)
        Initialize spf_heap to empty
        Initialize nodes' spf_metric to infinity and next_hops to empty
            and PATH_HITS_ISLAND to false
        spf_root.spf_metric = 0
        insert(spf_heap, spf_root)
        while (spf_heap is not empty)
            min_node = remove_lowest(spf_heap)
            foreach interface intf of min_node
                path_metric = min_node.spf_metric + intf.metric
                if path_metric < intf.remote_node.spf_metric
                    intf.remote_node.spf_metric = path_metric
                    if min_node is spf_root
                        intf.remote_node.next_hops = make_list(intf)
                    else
                        intf.remote_node.next_hops = min_node.next_hops
                    if intf.remote_node.IN_MRT_ISLAND
                        intf.remote_node.PATH_HITS_ISLAND = true
                    else
                        intf.remote_node.PATH_HITS_ISLAND =
                            min_node.PATH_HITS_ISLAND
                    insert_or_update(spf_heap, intf.remote_node)
                else if path_metric == intf.remote_node.spf_metric
                    if min_node is spf_root
                        add_to_list(intf.remote_node.next_hops, intf)
                    else
                        add_list_to_list(intf.remote_node.next_hops,
                                         min_node.next_hops)
                    if intf.remote_node.IN_MRT_ISLAND
                        intf.remote_node.PATH_HITS_ISLAND = true
                    else
                        intf.remote_node.PATH_HITS_ISLAND =
                            min_node.PATH_HITS_ISLAND

    Figure 27: Island_Marking_SPF for determining if an Island Neighbor
                               is loop-free

   It is also possible that a given prefix is originated by a
   combination of non-island routers and island routers.  The results of
   the Island_Marking_SPF computation can be used to determine if the
   shortest path from an IN to reach that prefix hits the MRT island.
   The shortest path for the IN to reach prefix P is determined by the
   total cost to reach prefix P, which is the sum of the cost for the IN
   to reach a prefix-advertising node and the cost with which that node
   advertises the prefix.  The path with the minimum total cost to
   prefix P is chosen.  If the prefix-advertising node for that minimum
   total cost path has PATH_HITS_ISLAND set to True, then the IN is not
   loop-free with respect to the MRT Island for reaching prefix P.  If



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   there multiple minimum total cost paths to reach prefix P, then all
   of the prefix-advertising routers involved in the minimum total cost
   paths MUST have PATH_HITS_ISLAND set to False for the IN to be
   considered loop-free to reach P.

   Note that there are other computations that could be used to
   determine if paths from a given IN _might_ pass through the MRT
   Island for a given prefix or destination.  For example, a previous
   version of this draft specified running the SPF algorithm on modified
   topology which treats the MRT island as a single node (with intra-
   island links set to zero cost) in order to provide input to
   computations to determine if the path from IN to non-island
   destination hits the MRT island in this modified topology.  This
   computation is enough to guarantee that a path will not hit the MRT
   island in the original topology.  However, it is possible that a path
   which is disqualified for hitting the MRT island in the modified
   topology will not actually hit the MRT Island in the original
   topology.  The algorithm described in Island_Marking_SPF() above does
   not modify the original topology, and will only disqualify a path if
   the actual path does in fact hit the MRT island.

   Since all routers need to come to the same conclusion about which
   routers qualify as LFINs, this specification requires that all
   routers computing LFINs MUST use an algorithm whose result is
   identical to that of the Island_Marking_SPF() in Figure 27.

5.9.3.  Computing MRT Next-Hops for Proxy-Nodes

   Determining the MRT next-hops for a proxy-node in the degenerate case
   where the proxy-node is attached to only one node in the GADAG is
   trivial, as all needed information can be derived from that proxy
   node attachment router.  If there are multiple interfaces connecting
   the proxy node to the single proxy node attachment router, then some
   can be assigned to MRT-Red and others to MRT_Blue.

   Now, consider the proxy-node P that is attached to two proxy-node
   attachment routers.  The pseudo-code for Select_Proxy_Node_NHs(P,S)
   in Figure 28 specifies how a computing-router S MUST compute the MRT
   red and blue next-hops to reach proxy-node P.  The proxy-node
   attachment router with the lower value of mrt_node_id (as defined in
   Figure 15) is assigned to X, and the other proxy-node attachment
   router is assigned to Y.  We will be using the relative order of X,Y,
   and S in the partial order defined by the GADAG to determine the MRT
   red and blue next-hops to reach P, so we also define A and B as the
   order proxies for X and Y, respectively, with respect to S.  The
   order proxies for all nodes with respect to S were already computed
   in Compute_MRT_NextHops().




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 def Select_Proxy_Node_NHs(P,S):
     if P.pnar1.node.node_id < P.pnar2.node.node_id:
         X = P.pnar1.node
         Y = P.pnar2.node
     else:
         X = P.pnar2.node
         Y = P.pnar1.node
     P.pnar_X = X
     P.pnar_Y = Y
     A = X.order_proxy
     B = Y.order_proxy
     if (A is S.localroot
         and B is S.localroot):
         // case 1.0
         Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
         Copy_List_Items(P.red_next_hops, Y.red_next_hops)
         return
     if (A is S.localroot
         and B is not S.localroot):
         // case 2.0
         if B.LOWER:
             // case 2.1
             Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
             Copy_List_Items(P.red_next_hops, Y.red_next_hops)
             return
         if B.HIGHER:
             // case 2.2
             Copy_List_Items(P.blue_next_hops, X.red_next_hops)
             Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
             return
         else:
             // case 2.3
             Copy_List_Items(P.blue_next_hops, X.red_next_hops)
             Copy_List_Items(P.red_next_hops, Y.red_next_hops)
             return
     if (A is not S.localroot
         and B is S.localroot):
         // case 3.0
         if A.LOWER:
             // case 3.1
             Copy_List_Items(P.blue_next_hops, X.red_next_hops)
             Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
             return
         if A.HIGHER:
             // case 3.2
             Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
             Copy_List_Items(P.red_next_hops, Y.red_next_hops)
             return



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         else:
             // case 3.3
             Copy_List_Items(P.blue_next_hops, X.red_next_hops)
             Copy_List_Items(P.red_next_hops, Y.red_next_hops)
             return
     if (A is not S.localroot
         and B is not S.localroot):
         // case 4.0
         if (S is A.localroot or S is B.localroot):
             // case 4.05
             if A.topo_order < B.topo_order:
                 // case 4.05.1
                 Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                 Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                 return
             else:
                 // case 4.05.2
                 Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                 Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                 return
         if A.LOWER:
             // case 4.1
             if B.HIGHER:
                 // case 4.1.1
                 Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                 Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                 return
             if B.LOWER:
                 // case 4.1.2
                 if A.topo_order < B.topo_order:
                     // case 4.1.2.1
                     Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                     Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                     return
                 else:
                     // case 4.1.2.2
                     Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                     Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                     return
             else:
                 // case 4.1.3
                 Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                 Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                 return
         if A.HIGHER:
             // case 4.2
             if B.HIGHER:
                 // case 4.2.1



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                 if A.topo_order < B.topo_order:
                     // case 4.2.1.1
                     Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                     Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                     return
                 else:
                     // case 4.2.1.2
                     Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                     Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                     return
             if B.LOWER:
                 // case 4.2.2
                 Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                 Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                 return
             else:
                 // case 4.2.3
                 Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                 Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                 return
         else:
             // case 4.3
             if B.LOWER:
                 // case 4.3.1
                 Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                 Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                 return
             if B.HIGHER:
                 // case 4.3.2
                 Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                 Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                 return
             else:
                 // case 4.3.3
                 if A.topo_order < B.topo_order:
                     // case 4.3.3.1
                     Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                     Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                     return
                 else:
                     // case 4.3.3.2
                     Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                     Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                     return
     assert(False)

                    Figure 28: Select_Proxy_Node_NHs()




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   It is useful to understand up front that the blue next-hops to reach
   proxy-node P produced by Select_Proxy_Node_NHs() will always be the
   next-hops that reach proxy-node attachment router X, while the red
   next-hops to reach proxy-node P will always be the next-hops that
   reach proxy-node attachment router Y.  This is different from the red
   and blue next-hops produced by Compute_MRT_NextHops() where, for
   example, blue next-hops to a destination that is ordered with respect
   to the source will always correspond to an INCREASING next-hop on the
   GADAG.  The exact choice of which next-hops chosen by
   Select_Proxy_Node_NHs() as the blue next-hops to reach P (which will
   necessarily go through X on its way to P) does depend on the GADAG,
   but the relationship is more complex than was the case with
   Compute_MRT_NextHops().

   There are twenty-one different relative order relationships between
   A, B and S that Select_Proxy_Node_NHs() uses to determine red and
   blue next-hops to P.  This document does not attempt to provide an
   exhaustive description of each case considered in
   Select_Proxy_Node_NHs().  Instead we provide a high level overview of
   the different cases, and we consider a few cases in detail to give an
   example of the reasoning that can be used to understand each case.

   At the highest level, Select_Proxy_Node_NHs() distinguishes between
   four different cases depending on whether or not A or B is the
   localroot for S.  For example, for case 4.0, neither A nor B is the
   localroot for S.  Case 4.05 addresses the case where S is the
   localroot for either A or B, while cases 4.1, 4.2, and 4.3 address
   the cases where A is ordered lower than S, A is ordered higher than
   S, or A is unordered with respect to S on the GADAG.  In general,
   each of these cases is then further subdivided into whether or not B
   is ordered lower than S, B is ordered higher than S, or B is
   unordered with respect to S.  In some cases we also need a further
   level of discrimination, where we use the topological sort order of A
   with respect to B.

   As a detailed example, let's consider case 4.1 and all of its sub-
   cases, and explain why the red and blue next-hops to reach P are
   chosen as they are in Select_Proxy_Node_NHs().  In case 4.1, neither
   A nor B is the localroot for S, S is not the localroot for A or B,
   and A is ordered lower than S on the GADAG.  In this situation, we
   know that the red path to reach X (as computed in
   Compute_MRT_NextHops()) will follow DECREASING next-hops towards A,
   while the blue path to reach X will follow INCREASING next-hops to
   the localroot, and then INCREASING next-hops to A.

   Now consider sub-case 4.1.1 where B is ordered higher than S.  In
   this situation, we know that the blue path to reach Y will follow
   INCREASING next-hops towards B, while the red next-hops to reach Y



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   will follow DECREASING next-hops to the localroot, and then
   DECREASING next-hops to B.  So to reach X and Y by two disjoint
   paths, we can choose the red next-hops to X and the blue next-hops to
   Y.  We have chosen the convention that blue next-hops to P are those
   that pass through X, and red next-hops to P are those that pass
   through Y, so we can see that case 4.1.1 produces the desired result.
   Choosing blue to X and red to Y does not produce disjoint paths
   because the paths intersect at least at the localroot.

   Now consider sub-case 4.1.2 where B is ordered lower than S.  In this
   situation, we know that the red path to reach Y will follow
   DECREASING next-hops towards B, while the BLUE next-hops to reach Y
   will follow INCREASING next-hops to the localroot, and then
   INCREASING next-hops to A.  The choice here is more difficult than in
   4.1.1 because A and B are both on the DECREASING path from S towards
   the localroot.  We want to use the direct DECREASING(red) path to the
   one that is nearer to S on the GADAG.  We get this extra information
   by comparing the topological sort order of A and B.  If
   A.topo_order<B.topo_order, then we use red to Y and blue to X, since
   the red path to Y will DECREASE to B without hitting A, and the blue
   path to X will INCREASE to A without hitting B.  Instead, if
   A.topo_order>B.topo_order, then we use red to X and blue to Y.

   Note that when A is unordered with respect to B, the result of
   comparing A.topo_order with B.topo_order could be greater than or
   less than.  In this case, the result doesn't matter because either
   choice (red to Y and blue to X or red to X and blue to Y) would work.
   What is required is that all nodes in the network give the same
   result when comparing A.topo_order with B.topo_order.  This is
   guarantee by having all nodes run the same algorithm
   (Run_Topological_Sort_GADAG()) to compute the topological sort order.

   Finally we consider case 4.1.3, where B is unordered with respect to
   S.  In this case, the blue path to reach Y will follow the DECREASING
   next-hops towards the localroot until it reaches some node (K) which
   is ordered less than B, after which it will take INCREASING next-hops
   to B.  The red path to reach Y will follow the INCREASING next-hops
   towards the localroot until it reaches some node (L) which is ordered
   greater than B, after which it will take DECREASING next-hops to B.
   Both K and A are reached by DECREASING from S, but we don't have
   information about whether or not that DECREASING path will hit K or A
   first.  Instead, we do know that the INCREASING path from S will hit
   L before reaching A.  Therefore, we use the red path to reach Y and
   the red path to reach X.

   Similar reasoning can be applied to understand the other seventeen
   cases used in Select_Proxy_Node_NHs().  However, cases 2.3 and 3.3
   deserve special attention because the correctness of the solution for



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   these two cases relies on a special property of the GADAGs that we
   have constructed in this algorithm, a property not shared by all
   GADAGs in general.  Focusing on case 2.3, we consider the case where
   A is the localroot for S, while B is not, and B is unordered with
   respect to S.  The red path to X DECREASES from S to the localroot A,
   while the blue path to X INCREASES from S to the localroot A.  The
   blue path to Y DECREASES towards the localroot A until it reaches
   some node (K) which is ordered less than B, after which the path
   INCREASES to B.  The red path to Y INCREASES towards the localroot A
   until it reaches some node (L) which is ordered greater than B, after
   which the path DECREASES to B.  It can be shown that for an arbitrary
   GADAG, with only the ordering relationships computed so far, we don't
   have enough information to choose a pair of paths to reach X and Y
   that are guaranteed to be disjoint.  In some topologies, A will play
   the role of K, the first node ordered less than B on the blue path to
   Y.  In other topologies, A will play the role of L, the first node
   ordered greater than B on the red path to Y.  The basic problem is
   that we cannot distinguish between these two cases based on the
   ordering relationships.

   As discussed Section 5.8, the GADAGs that we construct using the
   algorithm in this document are not arbitrary GADAGs.  They have the
   additional property that incoming links to a localroot come from only
   one other node in the same block.  This is a result of the method of
   construction.  This additional property guarantees that localroot A
   will never play the role of L in the red path to Y, since L must have
   at least two incoming links from different nodes in the same block in
   the GADAG.  This in turn allows Select_Proxy_Node_NHs() to choose the
   red path to Y and the red path to X as the disjoint MRT paths to
   reach P.

5.9.4.  Computing MRT Alternates for Proxy-Nodes

   After finding the red and the blue next-hops for a given proxy-node
   P, it is necessary to know which one of these to use in the case of
   failure.  This can be done by Select_Alternates_Proxy_Node(), as
   shown in the pseudo-code in Figure 29.

  def Select_Alternates_Proxy_Node(P,F,primary_intf):
      S = primary_intf.local_node
      X = P.pnar_X
      Y = P.pnar_Y
      A = X.order_proxy
      B = Y.order_proxy
      if F is A and F is B:
          return 'PRIM_NH_IS_OP_FOR_BOTH_X_AND_Y'
      if F is A:
          return 'USE_RED'



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      if F is B:
          return 'USE_BLUE'

      if not In_Common_Block(A, B):
          if In_Common_Block(F, A):
              return 'USE_RED'
          elif In_Common_Block(F, B):
              return 'USE_BLUE'
          else:
              return 'USE_RED_OR_BLUE'
      if (not In_Common_Block(F, A)
          and not In_Common_Block(F, A) ):
          return 'USE_RED_OR_BLUE'

      alt_to_X = Select_Alternates(X, F, primary_intf)
      alt_to_Y = Select_Alternates(Y, F, primary_intf)

      if (alt_to_X == 'USE_RED_OR_BLUE'
          and alt_to_Y == 'USE_RED_OR_BLUE'):
          return 'USE_RED_OR_BLUE'
      if alt_to_X == 'USE_RED_OR_BLUE':
          return 'USE_BLUE'
      if alt_to_Y == 'USE_RED_OR_BLUE':
          return 'USE_RED'

      if (A is S.localroot
          and B is S.localroot):
          // case 1.0
          if (alt_to_X == 'USE_BLUE' and alt_to_Y == 'USE_RED'):
              return 'USE_RED_OR_BLUE'
          if alt_to_X == 'USE_BLUE':
              return 'USE_BLUE'
          if alt_to_Y == 'USE_RED':
              return 'USE_RED'
          assert(False)
      if (A is S.localroot
          and B is not S.localroot):
          // case 2.0
          if B.LOWER:
              // case 2.1
              if (alt_to_X == 'USE_BLUE' and alt_to_Y == 'USE_RED'):
                  return 'USE_RED_OR_BLUE'
              if alt_to_X == 'USE_BLUE':
                  return 'USE_BLUE'
              if alt_to_Y == 'USE_RED':
                  return 'USE_RED'
              assert(False)
          if B.HIGHER:



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              // case 2.2
              if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_BLUE'):
                  return 'USE_RED_OR_BLUE'
              if alt_to_X == 'USE_RED':
                  return 'USE_BLUE'
              if alt_to_Y == 'USE_BLUE':
                  return 'USE_RED'
              assert(False)
          else:
              // case 2.3
              if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_RED'):
                  return 'USE_RED_OR_BLUE'
              if alt_to_X == 'USE_RED':
                  return 'USE_BLUE'
              if alt_to_Y == 'USE_RED':
                  return 'USE_RED'
              assert(False)
      if (A is not S.localroot
          and B is S.localroot):
          // case 3.0
          if A.LOWER:
              // case 3.1
              if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_BLUE'):
                  return 'USE_RED_OR_BLUE'
              if alt_to_X == 'USE_RED':
                  return 'USE_BLUE'
              if alt_to_Y == 'USE_BLUE':
                  return 'USE_RED'
              assert(False)
          if A.HIGHER:
              // case 3.2
              if (alt_to_X == 'USE_BLUE' and alt_to_Y == 'USE_RED'):
                  return 'USE_RED_OR_BLUE'
              if alt_to_X == 'USE_BLUE':
                  return 'USE_BLUE'
              if alt_to_Y == 'USE_RED':
                  return 'USE_RED'
              assert(False)
          else:
              // case 3.3
              if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_RED'):
                  return 'USE_RED_OR_BLUE'
              if alt_to_X == 'USE_RED':
                  return 'USE_BLUE'
              if alt_to_Y == 'USE_RED':
                  return 'USE_RED'
              assert(False)
      if (A is not S.localroot



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          and B is not S.localroot):
          // case 4.0
          if (S is A.localroot or S is B.localroot):
              // case 4.05
              if A.topo_order < B.topo_order:
                  // case 4.05.1
                  if (alt_to_X == 'USE_BLUE' and alt_to_Y == 'USE_RED'):
                      return 'USE_RED_OR_BLUE'
                  if alt_to_X == 'USE_BLUE':
                      return 'USE_BLUE'
                  if alt_to_Y == 'USE_RED':
                      return 'USE_RED'
                  assert(False)
              else:
                  // case 4.05.2
                  if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_BLUE'):
                      return 'USE_RED_OR_BLUE'
                  if alt_to_X == 'USE_RED':
                      return 'USE_BLUE'
                  if alt_to_Y == 'USE_BLUE':
                      return 'USE_RED'
                  assert(False)
          if A.LOWER:
              // case 4.1
              if B.HIGHER:
                  // case 4.1.1
                  if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_BLUE'):
                      return 'USE_RED_OR_BLUE'
                  if alt_to_X == 'USE_RED':
                      return 'USE_BLUE'
                  if alt_to_Y == 'USE_BLUE':
                      return 'USE_RED'
                  assert(False)
              if B.LOWER:
                  // case 4.1.2
                  if A.topo_order < B.topo_order:
                      // case 4.1.2.1
                      if (alt_to_X == 'USE_BLUE'
                          and alt_to_Y == 'USE_RED'):
                          return 'USE_RED_OR_BLUE'
                      if alt_to_X == 'USE_BLUE':
                          return 'USE_BLUE'
                      if alt_to_Y == 'USE_RED':
                          return 'USE_RED'
                      assert(False)
                  else:
                      // case 4.1.2.2
                      if (alt_to_X == 'USE_RED'



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                          and alt_to_Y == 'USE_BLUE'):
                          return 'USE_RED_OR_BLUE'
                      if alt_to_X == 'USE_RED':
                          return 'USE_BLUE'
                      if alt_to_Y == 'USE_BLUE':
                          return 'USE_RED'
                      assert(False)
              else:
                  // case 4.1.3
                  if (F.LOWER and not F.HIGHER
                      and F.topo_order > A.topo_order):
                      // case 4.1.3.1
                      return 'USE_RED'
                  else:
                      // case 4.1.3.2
                      return 'USE_BLUE'
          if A.HIGHER:
              // case 4.2
              if B.HIGHER:
                  // case 4.2.1
                  if A.topo_order < B.topo_order:
                      // case 4.2.1.1
                      if (alt_to_X == 'USE_BLUE'
                          and alt_to_Y == 'USE_RED'):
                          return 'USE_RED_OR_BLUE'
                      if alt_to_X == 'USE_BLUE':
                          return 'USE_BLUE'
                      if alt_to_Y == 'USE_RED':
                          return 'USE_RED'
                      assert(False)
                  else:
                      // case 4.2.1.2
                      if (alt_to_X == 'USE_RED'
                          and alt_to_Y == 'USE_BLUE'):
                          return 'USE_RED_OR_BLUE'
                      if alt_to_X == 'USE_RED':
                          return 'USE_BLUE'
                      if alt_to_Y == 'USE_BLUE':
                          return 'USE_RED'
                      assert(False)
              if B.LOWER:
                  // case 4.2.2
                  if (alt_to_X == 'USE_BLUE'
                      and alt_to_Y == 'USE_RED'):
                      return 'USE_RED_OR_BLUE'
                  if alt_to_X == 'USE_BLUE':
                      return 'USE_BLUE'
                  if alt_to_Y == 'USE_RED':



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                      return 'USE_RED'
                  assert(False)
              else:
                  // case 4.2.3
                  if (F.HIGHER and not F.LOWER
                      and F.topo_order < A.topo_order):
                      return 'USE_RED'
                  else:
                      return 'USE_BLUE'
          else:
              // case 4.3
              if B.LOWER:
                  // case 4.3.1
                  if (F.LOWER and not F.HIGHER
                      and F.topo_order > B.topo_order):
                      return 'USE_BLUE'
                  else:
                      return 'USE_RED'
              if B.HIGHER:
                  // case 4.3.2
                  if (F.HIGHER and not F.LOWER
                      and F.topo_order < B.topo_order):
                      return 'USE_BLUE'
                  else:
                      return 'USE_RED'
              else:
                  // case 4.3.3
                  if A.topo_order < B.topo_order:
                      // case 4.3.3.1
                      if (alt_to_X == 'USE_BLUE'
                          and alt_to_Y == 'USE_RED'):
                          return 'USE_RED_OR_BLUE'
                      if alt_to_X == 'USE_BLUE':
                          return 'USE_BLUE'
                      if alt_to_Y == 'USE_RED':
                          return 'USE_RED'
                      assert(False)
                  else:
                      // case 4.3.3.2
                      if (alt_to_X == 'USE_RED'
                          and alt_to_Y == 'USE_BLUE'):
                          return 'USE_RED_OR_BLUE'
                      if alt_to_X == 'USE_RED':
                          return 'USE_BLUE'
                      if alt_to_Y == 'USE_BLUE':
                          return 'USE_RED'
                      assert(False)
      assert(False)



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                 Figure 29: Select_Alternates_Proxy_Node()

   Select_Alternates_Proxy_Node(P,F,primary_intf) determines whether it
   is safe to use the blue path to P (which goes through X), the red
   path to P (which goes through Y), or either, when the primary_intf to
   node F (and possibly node F) fails.  The basic approach is to run
   Select_Alternates(X,F,primary_intf) and
   Select_Alternates(Y,F,primary_intf) to determine which of the two MRT
   paths to X and which of the two MRT paths to Y is safe to use in the
   event of the failure of F.  In general, we will find that if it is
   safe to use a particular path to X or Y when F fails, and
   Select_Proxy_Node_NHs() used that path when constructing the red or
   blue path to reach P, then it will also be safe to use that path to
   reach P when F fails.  This rule has one exception which is covered
   below.  First, we give a concrete example of how
   Select_Alternates_Proxy_Node() works in the common case.

   The twenty one ordering relationships used in Select_Proxy_Node_NHs()
   are repeated in Select_Alternates_Proxy_Node().  We focus on case
   4.1.1 to give give a detailed example of the reasoning used in
   Select_Alternates_Proxy_Node().  In Select_Proxy_Node_NHs(), we
   determined for case 4.1.1 that the red next-hops to X and the blue
   next-hops to Y allow us to reach X and Y by disjoint paths, and are
   thus the blue and red next-hops to reach P.  Therefore, if we run
   Select_Alternates(X, F, primary_intf) and we find that it is safe to
   USE_RED to reach X, then we also conclude that it is safe to use the
   MRT path through X to reach P (the blue path to P) when F fails.
   Similarly, if run Select_Alternates(X, F, primary_intf) and we find
   that it is safe to USE_BLUE to reach Y, then we also conclude that it
   is safe to use the MRT path through Y to reach P (the red path to P)
   when F fails.  If both of the paths that were used in
   Select_Proxy_Node_NHs() to construct the blue and red paths to P are
   found to be safe to use to use to reach X and Y, t then we conclude
   that we can use either the red or the blue path to P.

   This simple reasoning gives the correct answer in most of the cases.
   However, additional logic is needed when either A or B (but not both
   A and B) is unordered with respect to S.  This applies to cases
   4.1.3, 4.2.3, 4.3.1, and 4.3.2.  Looking at case 4.1.3 in more
   detail, A is ordered less than S, but B is unordered with respect to
   S.  In the discussion of case 4.1.3 above, we saw that
   Select_Proxy_Node_NHs() chose the red path to reach Y and the red
   path to reach X.  We also saw that the red path to reach Y will
   follow the INCREASING next-hops towards the localroot until it
   reaches some node (L) which is ordered greater than B, after which it
   will take DECREASING next-hops to B.  The problem is that the red
   path to reach P (the one that goes through Y) won't necessarily be
   the same as the red path to reach Y.  This is because the next-hop



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   that node L computes for its red next-hop to reach P may be different
   from the next-hop it computes for its red next-hop to reach Y.  This
   is because B is ordered lower than L, so L applies case 4.1.2 of
   Select_Proxy_Node_NHs() in order to determine its next-hops to reach
   P.  If A.topo_order<B.topo_order (case 4.1.2.1), then L will choose
   DECREASING next-hops directly to B, which is the same result that L
   computes in Compute_MRT_NextHops() to reach Y.  However, if
   A.topo_order>B.topo_order (case 4.1.2.2), then L will choose
   INCREASING next-hops to reach B, which is different from what L
   computes in Compute_MRT_NextHops() to reach Y.  So testing the safety
   of the path for S to reach Y on failure of F as a surrogate for the
   safety of using the red path to reach P is not reliable in this case.
   It is possible construct topologies where the red path to P hits F
   even though the red path to Y does not hit F.

   Fortunately there is enough information in the order relationships
   that we have already computed to still figure out which alternate to
   choose in these four cases.  The basic idea is to always choose the
   path involving the ordered node, unless that path would hit F.
   Returning to case 4.1.3, we see that since A is ordered lower than S,
   the only way for S to hit F using a simple DECREASING path to A is
   for F to lie between A and S on the GADAG.  This scenario is covered
   by requiring that F be lower than S (but not also higher than S) and
   that F.topo_order>A.topo_order in case 4.1.3.1.

   We just need to confirm that it is safe to use the path involving B
   in this scenario.  In case 4.1.3.1, either F is between A and S on
   the GADAG, or F is unordered with respect to A and lies on the
   DECREASING path from S to the localroot.  When F is between A and S
   on the GADAG, then the path through B chosen to avoid A in
   Select_Proxy_Node_NHs() will also avoid F.  When F is unordered with
   respect to A and lies on the DECREASING path from S to the localroot,
   then we consider two cases.  Either F.topo_order<B.topo_order or
   F.topo_order>B.topo_order.  In the first case, since
   F.topo_order<B.topo_order and F.topo_order>A.topo_order, it must be
   the case that A.topo_order<B.topo_order.  Therefore, L will choose
   DECREASING next-hops directly to B (case 4.1.2.1), which cannot hit F
   since F.topo_order<B.topo_order.  In the second case, where
   F.topo_order>B.topo_order, the only way for the path involving B to
   hit F is if it DECREASES from L to B through F , ie. it must be that
   L>>F>>B.  However, since S>>F, this would imply that S>>B.  However,
   we know that S is unordered with respect to B, so the second case
   cannot occur.  So we have demonstrated that the red path to P (which
   goes via B and Y) is safe to use under the conditions of 4.1.3.1.
   Similar reasoning can be applied to the other three special cases
   where either A or B is unordered with respect to S.





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6.  MRT Lowpoint Algorithm: Next-hop conformance

   This specification defines the MRT Lowpoint Algorithm, which include
   the construction of a common GADAG and the computation of MRT-Red and
   MRT-Blue next-hops to each node in the graph.  An implementation MAY
   select any subset of next-hops for MRT-Red and MRT-Blue that respect
   the available nodes that are described in Section 5.7 for each of the
   MRT-Red and MRT-Blue and the selected next-hops are further along in
   the interval of allowed nodes towards the destination.

   For example, the MRT-Blue next-hops used when the destination Y >> X,
   the computing router, MUST be one or more nodes, T, whose topo_order
   is in the interval [X.topo_order, Y.topo_order] and where Y >> T or Y
   is T.  Similarly, the MRT-Red next-hops MUST be have a topo_order in
   the interval [R-small.topo_order, X.topo_order] or [Y.topo_order,
   R-big.topo_order].

   Implementations SHOULD implement the Select_Alternates() function to
   pick an MRT-FRR alternate.

7.  Broadcast interfaces

   When broadcast interfaces are used to connect nodes, the broadcast
   network MUST be represented as a pseudonode, where each real node
   connects to the pseudonode.  The interface metric in the direction
   from real node to pseudonode is the non-zero interface metric, while
   the interface metric in the direction from the pseudonode to the real
   node is set to zero.  This is consistent with the way that broadcast
   interfaces are represented as pseudonodes in IS-IS and OSPF.

   Pseudonodes MUST be treated as equivalent to real nodes in the
   network graph used in the MRT algorithm with a few exceptions
   detailed below.

   The pseudonodes MUST be included in the computation of the GADAG.
   The neighbors of the pseudonode need to know the mrt_node_id of the
   pseudonode in order to consistently order interfaces, which is needed
   to compute the GADAG.  The mrt_node_id for IS-IS is the 7 octet
   neighbor system ID and pseudonode number in TLV #22 or TLV#222.  The
   mrt_node_id for OSPFv2 is the 4 octet interface address of the
   Designated Router found in the Link ID field for the link type 2
   (transit network) in the Router-LSA.  The mrt_node_id for OSPFv3 is
   the 4 octet interface address of the Designated Router found in the
   Neighbor Interface ID field for the link type 2 (transit network) in
   the Router-LSA. pseudonodes MUST NOT be considered as candidates for
   GADAG root selection.  Note that this is different from the Neighbor
   Router ID field used for the mrt_node_id for point-to-point links in
   OSPFv3 Router-LSAs given in Figure 15.



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   Pseudonodes MUST NOT be considered as candidates for selection as
   GADAG root.  This rule is intended to result in a more stable
   network- wide selection of GADAG root by removing the possibility
   that the change of Designated Router or Designated Intermediate
   System on a broadcast network can result in a change of GADAG root.

7.1.  Computing MRT next-hops on broadcast networks

   The pseudonode does not correspond to an real node, so it is not
   actually involved in forwarding.  A real node on a broadcast network
   cannot simply forward traffic to the broadcast network.  It must
   specify another real node on the broadcast network as the next-hop.
   On a network graph where a broadcast network is represented by a
   pseudonode, this means that if a real node determines that the next-
   hop to reach a given destination is a pseudonode, it must also
   determine the next-next-hop for that destination in the network
   graph, which corresponds to a real node attached to the broadcast
   network.

   It is interesting to note that this issue is not unique to the MRT
   algorithm, but is also encountered in normal SPF computations for
   IGPs.  Section 16.1.1 of [RFC2328] describes how this is done for
   OSPF.  As OSPF runs Dijkstra's algorithm, whenever a shorter path is
   found reach a real destination node, and the shorter path is one hop
   from the computing routing, and that one hop is a pseudonode, then
   the next-hop for that destination is taken from the interface IP
   address in the Router-LSA correspond to the link to the real
   destination node

   For IS-IS, in the example pseudo-code implementation of Dijkstra's
   algorithm in Annex C of [ISO10589-Second-Edition] whenever the
   algorithm encounters an adjacency from a real node to a pseudonode,
   it gets converted to a set of adjacencies from the real node to the
   neighbors of the pseudonode.  In this way, the computed next-hops
   point all the way to the real node, and not the pseudonode.

   We could avoid the problem of determining next-hops across
   pseudonodes in MRT by converting the pseudonode representation of
   broadcast networks to a full mesh of links between real nodes on the
   same network.  However, if we make that conversion before computing
   the GADAG, we lose information about which links actually correspond
   to a single physical interface into the broadcast network.  This
   could result computing red and blue next-hops that use the same
   broadcast interface, in which case neither the red nor the blue next-
   hop would be usable as an alternate on failure of the broadcast
   interface.





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   Instead, we take the following approach, which maintains the property
   that either the red and blue next-hop will avoid the broadcast
   network, if topologically allowed.  We run the MRT algorithm treating
   the pseudonodes as equivalent to real nodes in the network graph,
   with the exceptions noted above.  In addition to running the MRT
   algorithm from the point of view of itself, a computing router
   connected to a pseudonode MUST also run the MRT algorithm from the
   point of view of each of its pseudonode neighbors.  For example, if a
   computing router S determines that its MRT red next-hop to reach a
   destination D is a pseudonode P, S looks at its MRT algorithm
   computation from P's point of view to determine P's red next-hop to
   reach D, say interface 1 on node X.  S now knows that its real red
   next-hop to reach D is interface 1 on node X on the broadcast network
   represented by P, and can install the corresponding entry in its FIB.

7.2.  Using MRT next-hops as alternates in the event of failures on
      broadcast networks

   In the previous section, we specified how to compute MRT next-hops
   when broadcast networks are involved.  In this section, we discuss
   how a PLR can use those MRT next-hops in the event of failures
   involving broadcast networks.

   A PLR attached to a broadcast network running only OSPF or IS-IS with
   large Hello intervals has limited ability to quickly detect failures
   on a broadcast network.  The only failure mode that can be quickly
   detected is the failure of the physical interface connecting the PLR
   to the broadcast network.  For the failure of the interface
   connecting the PLR to the broadcast network, the alternate that
   avoids the broadcast network can be computed by using the broadcast
   network pseudonode as F, the primary next-hop node, in
   Select_Alternates().  This will choose an alternate path that avoids
   the broadcast network.  However, the alternate path will not
   necessarily avoid all of the real nodes connected to the broadcast
   network.  This is because we have used the pseudonode to represent
   the broadcast network.  And we have enforced the node-protecting
   property of MRT on the pseudonode to provide protection against
   failure of the broadcast network, not the real next-hop nodes on the
   broadcast network.  This is the best that we can hope to do if
   failure of the broadcast interface is the only failure mode that the
   PLR can respond to.

   We can improve on this if the PLR also has the ability to quickly
   detect a lack of connectivity across the broadcast network to a given
   IP-layer node.  This can be accomplished by running BFD between all
   pairs of IGP neighbors on the broadcast network.  Note that in the
   case of OSPF, this would require establishing BFD sessions between
   all pairs of neighbors in the 2-WAY state.  When the PLR can quickly



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   detect the failure of a particular next-hop across a broadcast
   network, then the PLR can be more selective in its choice of
   alternates.  For example, when the PLR observes that connectivity to
   an IP-layer node on a broadcast network has failed, the PLR may
   choose to still use the broadcast network to reach other IP-layer
   nodes which are still reachable.  Or if the PLR observes that
   connectivity has failed to several IP-layer nodes on the same
   broadcast network, it may choose to treat the entire broadcast
   network as failed.  The choice of MRT alternates by a PLR for a
   particular set of failure conditions is a local decision, since it
   does not require coordination with other nodes.

8.  Evaluation of Alternative Methods for Constructing GADAGs

   This document specifies the MRT Lowpoint algorithm.  One component of
   the algorithm involves constructing a common GADAG based on the
   network topology.  The MRT Lowpoint algorithm computes the GADAG
   using the method described in Section 5.5.  This method aims to
   minimize the amount of computation required to compute the GADAG.  In
   the process of developing the MRT Lowpoint algorithm, two alternative
   methods for constructing GADAGs were also considered.  These
   alternative methods are described in Appendix B and Appendix C.  In
   general, these other two methods require more computation to compute
   the GADAG.  The analysis below was performed to determine if the
   alternative GADAG construction methods produce shorter MRT alternate
   paths in real network topologies, and if so, to what extent.

   Figure 30 compares results obtained using the three different methods
   for constructing GADAGs on five different service provider network
   topologies.  MRT_LOWPOINT indicates the method specified in
   Section 5.5, while MRT_SPF and MRT_HYBRID indicate the methods
   specified in Appendix B and Appendix C, respectively.  The columns on
   the right present the distribution of alternate path lengths for each
   GADAG construction method.  Each MRT computation was performed using
   a same GADAG root chosen based on centrality.

   For three of the topologies analyzed (T201, T206, and T211), the use
   of MRT_SPF or MRT_HYBRID methods does not appear to provide a
   significantly shorter alternate path lengths compared to the
   MRT_LOWPOINT method.  However, for two of the topologies (T216 and
   T219), the use of the MRT_SPF method resulted in noticeably shorter
   alternate path lengths than the use of the MRT_LOWPOINT or MRT_HYBRID
   methods.

   It was decided to use the MRT_LOWPOINT method to construct the GADAG
   in the algorithm specified in this draft, in order to initially offer
   an algorithm with lower computational requirements.  These results
   indicate that in the future it may be useful to evaluate and



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   potentially specify other MRT algorithm variants that use different
   GADAG construction methods.

   +-------------------------------------------------------------------+
   |        Topology name         |   percentage of failure scenarios  |
   |                              |  protected by an alternate N hops  |
   |      GADAG construction      |   longer than the primary path     |
   |            method            +------------------------------------+
   |                              |   |   |   |   |   |   |   |   | no |
   |                              |   |   |   |   |   |10 |12 |14 | alt|
   |                              |0-1|2-3|4-5|6-7|8-9|-11|-13|-15| <16|
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T201(avg primary hops=3.5)  |   |   |   |   |   |   |   |   |    |
   |  MRT_HYBRID                  | 33| 26| 23|  6|  3|   |   |   |    |
   |  MRT_SPF                     | 33| 36| 23|  6|  3|   |   |   |    |
   |  MRT_LOWPOINT                | 33| 36| 23|  6|  3|   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T206(avg primary hops=3.7)  |   |   |   |   |   |   |   |   |    |
   |  MRT_HYBRID                  | 50| 35| 13|  2|   |   |   |   |    |
   |  MRT_SPF                     | 50| 35| 13|  2|   |   |   |   |    |
   |  MRT_LOWPOINT                | 55| 32| 13|   |   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T211(avg primary hops=3.3)  |   |   |   |   |   |   |   |   |    |
   |  MRT_HYBRID                  | 86| 14|   |   |   |   |   |   |    |
   |  MRT_SPF                     | 86| 14|   |   |   |   |   |   |    |
   |  MRT_LOWPOINT                | 85| 15|  1|   |   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T216(avg primary hops=5.2)  |   |   |   |   |   |   |   |   |    |
   |  MRT_HYBRID                  | 23| 22| 18| 13| 10|  7|  4|  2|   2|
   |  MRT_SPF                     | 35| 32| 19|  9|  3|  1|   |   |    |
   |  MRT_LOWPOINT                | 28| 25| 18| 11|  7|  6|  3|  2|   1|
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T219(avg primary hops=7.7)  |   |   |   |   |   |   |   |   |    |
   |  MRT_HYBRID                  | 20| 16| 13| 10|  7|  5|  5|  5|   3|
   |  MRT_SPF                     | 31| 23| 19| 12|  7|  4|  2|  1|    |
   |  MRT_LOWPOINT                | 19| 14| 15| 12| 10|  8|  7|  6|  10|
   +------------------------------+---+---+---+---+---+---+---+---+----+


                                 Figure 30

9.  Implementation Status

   [RFC Editor: please remove this section prior to publication.]

   Please see [I-D.ietf-rtgwg-mrt-frr-architecture] for details on
   implementation status.




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10.  Operational Considerations

   This sections discusses operational considerations related to the the
   MRT Lowpoint algorithm and other potential MRT algorithm variants.
   For a discussion of operational considerations related to MRT-FRR in
   general, see the Operational Considerations section of
   [I-D.ietf-rtgwg-mrt-frr-architecture].

10.1.  GADAG Root Selection

   The Default MRT Profile uses the GADAG Root Selection Priority
   advertised by routers as the primary criterion for selecting the
   GADAG root.  It is RECOMMENDED that an operator designate a set of
   routers as good choices for selection as GADAG root by setting the
   GADAG Root Selection Priority for that set of routers to lower (more
   preferred) numerical values.  Criteria for making this designation
   are discussed below.

   Analysis has shown that the centrality of a router can have a
   significant impact on the lengths of the alternate paths computed.
   Therefore, it is RECOMMENDED that off-line analysis that considers
   the centrality of a router be used to help determine how good a
   choice a particular router is for the role of GADAG root.

   If the router currently selected as GADAG root becomes unreachable in
   the IGP topology, then a new GADAG root will be selected.  Changing
   the GADAG root can change the overall structure of the GADAG as well
   the paths of the red and blue MRT trees built using that GADAG.  In
   order to minimize change in the associated red and blue MRT
   forwarding entries that can result from changing the GADAG root, it
   is RECOMMENDED that operators prioritize for selection as GADAG root
   those routers that are expected to consistently remain part of the
   IGP topology.

10.2.  Destination-rooted GADAGs

   The MRT Lowpoint algorithm constructs a single GADAG rooted at a
   single node selected as the GADAG root.  It is also possible to
   construct a different GADAG for each destination, with the GADAG
   rooted at the destination.  A router can compute the MRT-Red and MRT-
   Blue next-hops for that destination based on the GADAG rooted at that
   destination.  Building a different GADAG for each destination is
   computationally more expensive, but it may give somewhat shorter
   alternate paths.  Using destination-rooted GADAGs would require a new
   MRT profile to be created with a new MRT algorithm specification,
   since all routers in the MRT Island would need to use destination-
   rooted GADAGs.




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11.  Acknowledgements

   The authors would like to thank Shraddha Hegde, Eric Wu, Janos
   Farkas, Stewart Bryant, and Alvaro Retana for their suggestions and
   review.  We would also like to thank Anil Kumar SN for his assistance
   in clarifying the algorithm description and pseudo-code.

12.  IANA Considerations

   This document includes no request to IANA.

13.  Security Considerations

   The algorithm described in this document does not introduce new
   security concerns beyond those already discussed in the document
   describing the MRT FRR architecture
   [I-D.ietf-rtgwg-mrt-frr-architecture].

14.  References

14.1.  Normative References

   [I-D.ietf-rtgwg-mrt-frr-architecture]
              Atlas, A., Kebler, R., Bowers, C., Envedi, G., Csaszar,
              A., Tantsura, J., and R. White, "An Architecture for IP/
              LDP Fast-Reroute Using Maximally Redundant Trees", draft-
              ietf-rtgwg-mrt-frr-architecture-07 (work in progress),
              October 2015.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <http://www.rfc-editor.org/info/rfc2119>.

14.2.  Informative References

   [EnyediThesis]
              Enyedi, G., "Novel Algorithms for IP Fast Reroute",
              Department of Telecommunications and Media Informatics,
              Budapest University of Technology and Economics Ph.D.
              Thesis, February 2011, <http://www.omikk.bme.hu/collection
              s/phd/Villamosmernoki_es_Informatikai_Kar/2011/
              Enyedi_Gabor/ertekezes.pdf>.








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   [IEEE8021Qca]
              IEEE 802.1, "IEEE 802.1Qca Bridges and Bridged Networks -
              Amendment: Path Control and Reservation - Draft 2.1",
              (work in progress), June 24, 2015,
              <http://www.ieee802.org/1/pages/802.1ca.html>.

   [ISO10589-Second-Edition]
              International Organization for Standardization,
              "Intermediate system to Intermediate system intra-domain
              routeing information exchange protocol for use in
              conjunction with the protocol for providing the
              connectionless-mode Network Service (ISO 8473)", ISO/
              IEC 10589:2002, Second Edition, Nov. 2002.

   [Kahn_1962_topo_sort]
              Kahn, A., "Topological sorting of large networks",
              Communications of the ACM, Volume 5, Issue 11 , Nov 1962,
              <http://dl.acm.org/citation.cfm?doid=368996.369025>.

   [MRTLinear]
              Enyedi, G., Retvari, G., and A. Csaszar, "On Finding
              Maximally Redundant Trees in Strictly Linear Time", IEEE
              Symposium on Computers and Comunications (ISCC) , 2009,
              <http://opti.tmit.bme.hu/~enyedi/ipfrr/
              distMaxRedTree.pdf>.

   [RFC2328]  Moy, J., "OSPF Version 2", STD 54, RFC 2328,
              DOI 10.17487/RFC2328, April 1998,
              <http://www.rfc-editor.org/info/rfc2328>.

   [RFC5120]  Przygienda, T., Shen, N., and N. Sheth, "M-ISIS: Multi
              Topology (MT) Routing in Intermediate System to
              Intermediate Systems (IS-ISs)", RFC 5120,
              DOI 10.17487/RFC5120, February 2008,
              <http://www.rfc-editor.org/info/rfc5120>.

   [RFC7490]  Bryant, S., Filsfils, C., Previdi, S., Shand, M., and N.
              So, "Remote Loop-Free Alternate (LFA) Fast Reroute (FRR)",
              RFC 7490, DOI 10.17487/RFC7490, April 2015,
              <http://www.rfc-editor.org/info/rfc7490>.

Appendix A.  Python Implementation of MRT Lowpoint Algorithm

   Below is Python code implementing the MRT Lowpoint algorithm
   specified in this document.  In order to avoid the page breaks in the
   .txt version of the draft, one can cut and paste the Python code from
   the .xml version.  The code is also posted on Github.




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   While this Python code is believed to correctly implement the pseudo-
   code description of the algorithm, in the event of a difference, the
   pseudo-code description should be considered normative.

<CODE BEGINS>
# This program has been tested to run on Python 2.6 and 2.7
# (specifically Python 2.6.6 and 2.7.8 were tested).
# The program has known incompatibilities with Python 3.X.

# When executed, this program will generate a text file describing
# an example topology.  It then reads that text file back in as input
# to create the example topology, and runs the MRT algorithm.This
# was done to simplify the inclusion of the program as a single text
# file that can be extracted from the IETF draft.

# The output of the program is four text files containing a description
# of the GADAG, the blue and red MRTs for all destinations, and the
# MRT alternates for all failures.

import random
import os.path
import heapq

# simple Class definitions allow structure-like dot notation for
# variables and a convenient place to initialize those variables.
class Topology:
    def __init__(self):
        self.gadag_root = None
        self.node_list = []
        self.node_dict = {}
        self.test_gr = None
        self.island_node_list_for_test_gr = []
        self.stored_named_proxy_dict = {}
        self.init_new_computing_router()
    def init_new_computing_router(self):
        self.island_node_list = []
        self.named_proxy_dict = {}

class Node:
    def __init__(self):
        self.node_id = None
        self.intf_list = []
        self.profile_id_list = [0]
        self.GR_sel_priority = 128
        self.blue_next_hops_dict = {}
        self.red_next_hops_dict = {}
        self.blue_to_green_nh_dict = {}
        self.red_to_green_nh_dict = {}



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        self.prefix_cost_dict = {}
        self.pnh_dict = {}
        self.alt_dict = {}
        self.init_new_computing_router()
    def init_new_computing_router(self):
        self.island_intf_list = []
        self.IN_MRT_ISLAND = False
        self.IN_GADAG = False
        self.dfs_number = None
        self.dfs_parent = None
        self.dfs_parent_intf = None
        self.dfs_child_list = []
        self.lowpoint_number = None
        self.lowpoint_parent = None
        self.lowpoint_parent_intf = None
        self.localroot = None
        self.block_id = None
        self.IS_CUT_VERTEX = False
        self.blue_next_hops = []
        self.red_next_hops = []
        self.primary_next_hops = []
        self.alt_list = []

class Interface:
    def __init__(self):
        self.metric = None
        self.area = None
        self.MRT_INELIGIBLE = False
        self.IGP_EXCLUDED = False
        self.SIMULATION_OUTGOING = False
        self.init_new_computing_router()
    def init_new_computing_router(self):
        self.UNDIRECTED = True
        self.INCOMING = False
        self.OUTGOING = False
        self.INCOMING_STORED = False
        self.OUTGOING_STORED = False
        self.IN_MRT_ISLAND = False
        self.PROCESSED = False

class Bundle:
    def __init__(self):
        self.UNDIRECTED = True
        self.OUTGOING = False
        self.INCOMING = False

class Alternate:
    def __init__(self):



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        self.failed_intf = None
        self.red_or_blue = None
        self.nh_list = []
        self.fec = 'NO_ALTERNATE'
        self.prot = 'NO_PROTECTION'
        self.info = 'NONE'

class Proxy_Node_Attachment_Router:
    def __init__(self):
        self.prefix = None
        self.node = None
        self.named_proxy_cost = None
        self.min_lfin = None
        self.nh_intf_list = []

class Named_Proxy_Node:
    def __init__(self):
        self.node_id = None  #this is the prefix_id
        self.node_prefix_cost_list = []
        self.lfin_list = []
        self.pnar1 = None
        self.pnar2 = None
        self.pnar_X = None
        self.pnar_Y = None
        self.blue_next_hops = []
        self.red_next_hops = []
        self.primary_next_hops = []
        self.blue_next_hops_dict = {}
        self.red_next_hops_dict = {}
        self.pnh_dict = {}
        self.alt_dict = {}

def Interface_Compare(intf_a, intf_b):
    if intf_a.metric < intf_b.metric:
        return -1
    if intf_b.metric < intf_a.metric:
        return 1
    if intf_a.remote_node.node_id < intf_b.remote_node.node_id:
        return -1
    if intf_b.remote_node.node_id < intf_a.remote_node.node_id:
        return 1
    return 0

def Sort_Interfaces(topo):
    for node in topo.island_node_list:
        node.island_intf_list.sort(Interface_Compare)

def Reset_Computed_Node_and_Intf_Values(topo):



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    topo.init_new_computing_router()
    for node in topo.node_list:
        node.init_new_computing_router()
        for intf in node.intf_list:
            intf.init_new_computing_router()

# This function takes a file with links represented by 2-digit
# numbers in the format:
# 01,05,10
# 05,02,30
# 02,01,15
# which represents a triangle topology with nodes 01, 05, and 02
# and symmetric metrics of 10, 30, and 15.

# Inclusion of a fourth column makes the metrics for the link
# asymmetric.  An entry of:
# 02,07,10,15
# creates a link from node 02 to 07 with metrics 10 and 15.
def Create_Topology_From_File(filename):
    topo = Topology()
    node_id_set= set()
    cols_list = []
    # on first pass just create nodes
    with open(filename + '.csv') as topo_file:
        for line in topo_file:
            line = line.rstrip('\r\n')
            cols=line.split(',')
            cols_list.append(cols)
            nodea_node_id = int(cols[0])
            nodeb_node_id = int(cols[1])
            if (nodea_node_id > 999 or nodeb_node_id > 999):
                print("node_id must be between 0 and 999.")
                print("exiting.")
                exit()
            node_id_set.add(nodea_node_id)
            node_id_set.add(nodeb_node_id)
    for node_id in node_id_set:
        node = Node()
        node.node_id = node_id
        topo.node_list.append(node)
        topo.node_dict[node_id] = node
    # on second pass create interfaces
    for cols in cols_list:
        nodea_node_id = int(cols[0])
        nodeb_node_id = int(cols[1])
        metric = int(cols[2])
        reverse_metric = int(cols[2])
        if len(cols) > 3:



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            reverse_metric=int(cols[3])
        nodea = topo.node_dict[nodea_node_id]
        nodeb = topo.node_dict[nodeb_node_id]
        nodea_intf = Interface()
        nodea_intf.metric = metric
        nodea_intf.area = 0
        nodeb_intf = Interface()
        nodeb_intf.metric = reverse_metric
        nodeb_intf.area = 0
        nodea_intf.remote_intf = nodeb_intf
        nodeb_intf.remote_intf = nodea_intf
        nodea_intf.remote_node = nodeb
        nodeb_intf.remote_node = nodea
        nodea_intf.local_node = nodea
        nodeb_intf.local_node = nodeb
        nodea_intf.link_data = len(nodea.intf_list)
        nodeb_intf.link_data = len(nodeb.intf_list)
        nodea.intf_list.append(nodea_intf)
        nodeb.intf_list.append(nodeb_intf)
    return topo

def MRT_Island_Identification(topo, computing_rtr, profile_id, area):
    if profile_id in computing_rtr.profile_id_list:
        computing_rtr.IN_MRT_ISLAND = True
        explore_list = [computing_rtr]
    else:
        return
    while explore_list != []:
        next_rtr = explore_list.pop()
        for intf in next_rtr.intf_list:
            if ( (not intf.MRT_INELIGIBLE)
                 and (not intf.remote_intf.MRT_INELIGIBLE)
                 and (not intf.IGP_EXCLUDED) and intf.area == area
                 and (profile_id in intf.remote_node.profile_id_list)):
                intf.IN_MRT_ISLAND = True
                intf.remote_intf.IN_MRT_ISLAND = True
                if (not intf.remote_node.IN_MRT_ISLAND):
                    intf.remote_node.IN_MRT_ISLAND = True
                    explore_list.append(intf.remote_node)

def Compute_Island_Node_List_For_Test_GR(topo, test_gr):
    Reset_Computed_Node_and_Intf_Values(topo)
    topo.test_gr = topo.node_dict[test_gr]
    MRT_Island_Identification(topo, topo.test_gr, 0, 0)
    for node in topo.node_list:
        if node.IN_MRT_ISLAND:
            topo.island_node_list_for_test_gr.append(node)




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def Set_Island_Intf_and_Node_Lists(topo):
    for node in topo.node_list:
        if node.IN_MRT_ISLAND:
            topo.island_node_list.append(node)
            for intf in node.intf_list:
                if intf.IN_MRT_ISLAND:
                    node.island_intf_list.append(intf)

global_dfs_number = None

def Lowpoint_Visit(x, parent, intf_p_to_x):
    global global_dfs_number
    x.dfs_number = global_dfs_number
    x.lowpoint_number = x.dfs_number
    global_dfs_number += 1
    x.dfs_parent = parent
    if intf_p_to_x == None:
        x.dfs_parent_intf = None
    else:
        x.dfs_parent_intf = intf_p_to_x.remote_intf
    x.lowpoint_parent = None
    if parent != None:
        parent.dfs_child_list.append(x)
    for intf in x.island_intf_list:
        if intf.remote_node.dfs_number == None:
            Lowpoint_Visit(intf.remote_node, x, intf)
            if intf.remote_node.lowpoint_number < x.lowpoint_number:
                x.lowpoint_number = intf.remote_node.lowpoint_number
                x.lowpoint_parent = intf.remote_node
                x.lowpoint_parent_intf = intf
        else:
            if intf.remote_node is not parent:
                if intf.remote_node.dfs_number < x.lowpoint_number:
                    x.lowpoint_number = intf.remote_node.dfs_number
                    x.lowpoint_parent = intf.remote_node
                    x.lowpoint_parent_intf = intf

def Run_Lowpoint(topo):
    global global_dfs_number
    global_dfs_number = 0
    Lowpoint_Visit(topo.gadag_root, None, None)

max_block_id = None

def Assign_Block_ID(x, cur_block_id):
    global max_block_id
    x.block_id = cur_block_id
    for c in x.dfs_child_list:



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        if (c.localroot is x):
            max_block_id += 1
            Assign_Block_ID(c, max_block_id)
        else:
            Assign_Block_ID(c, cur_block_id)

def Run_Assign_Block_ID(topo):
    global max_block_id
    max_block_id = 0
    Assign_Block_ID(topo.gadag_root, max_block_id)

def Construct_Ear(x, stack, intf, ear_type):
    ear_list = []
    cur_intf = intf
    not_done = True
    while not_done:
        cur_intf.UNDIRECTED = False
        cur_intf.OUTGOING = True
        cur_intf.remote_intf.UNDIRECTED = False
        cur_intf.remote_intf.INCOMING = True
        if cur_intf.remote_node.IN_GADAG == False:
            cur_intf.remote_node.IN_GADAG = True
            ear_list.append(cur_intf.remote_node)
            if ear_type == 'CHILD':
                cur_intf = cur_intf.remote_node.lowpoint_parent_intf
            else:
                assert ear_type == 'NEIGHBOR'
                cur_intf = cur_intf.remote_node.dfs_parent_intf
        else:
            not_done = False

    if ear_type == 'CHILD' and cur_intf.remote_node is x:
        # x is a cut-vertex and the local root for the block
        # in which the ear is computed
        x.IS_CUT_VERTEX = True
        localroot = x
    else:
        # inherit local root from the end of the ear
        localroot = cur_intf.remote_node.localroot

    while ear_list != []:
        y = ear_list.pop()
        y.localroot = localroot
        stack.append(y)

def Construct_GADAG_via_Lowpoint(topo):
    gadag_root = topo.gadag_root
    gadag_root.IN_GADAG = True



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    gadag_root.localroot = None
    stack = []
    stack.append(gadag_root)
    while stack != []:
        x = stack.pop()
        for intf in x.island_intf_list:
            if ( intf.remote_node.IN_GADAG == False
                 and intf.remote_node.dfs_parent is x ):
                Construct_Ear(x, stack, intf, 'CHILD' )
        for intf in x.island_intf_list:
            if (intf.remote_node.IN_GADAG == False
                and intf.remote_node.dfs_parent is not x):
                Construct_Ear(x, stack, intf, 'NEIGHBOR')

def Assign_Remaining_Lowpoint_Parents(topo):
    for node in topo.island_node_list:
        if ( node is not topo.gadag_root
            and node.lowpoint_parent == None ):
            node.lowpoint_parent = node.dfs_parent
            node.lowpoint_parent_intf = node.dfs_parent_intf
            node.lowpoint_number = node.dfs_parent.dfs_number

def Add_Undirected_Block_Root_Links(topo):
    for node in topo.island_node_list:
        if node.IS_CUT_VERTEX or node is topo.gadag_root:
            for intf in node.island_intf_list:
                if ( intf.remote_node.localroot is not node
                     or intf.PROCESSED ):
                    continue
                bundle_list = []
                bundle = Bundle()
                for intf2 in node.island_intf_list:
                    if intf2.remote_node is intf.remote_node:
                        bundle_list.append(intf2)
                        if not intf2.UNDIRECTED:
                            bundle.UNDIRECTED = False
                            if intf2.INCOMING:
                                bundle.INCOMING = True
                            if intf2.OUTGOING:
                                bundle.OUTGOING = True
                if bundle.UNDIRECTED:
                    for intf3 in bundle_list:
                        intf3.UNDIRECTED = False
                        intf3.remote_intf.UNDIRECTED = False
                        intf3.PROCESSED = True
                        intf3.remote_intf.PROCESSED = True
                        intf3.OUTGOING = True
                        intf3.remote_intf.INCOMING = True



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                else:
                    if (bundle.OUTGOING and bundle.INCOMING):
                        for intf3 in bundle_list:
                            intf3.UNDIRECTED = False
                            intf3.remote_intf.UNDIRECTED = False
                            intf3.PROCESSED = True
                            intf3.remote_intf.PROCESSED = True
                            intf3.OUTGOING = True
                            intf3.INCOMING = True
                            intf3.remote_intf.INCOMING = True
                            intf3.remote_intf.OUTGOING = True
                    elif bundle.OUTGOING:
                        for intf3 in bundle_list:
                            intf3.UNDIRECTED = False
                            intf3.remote_intf.UNDIRECTED = False
                            intf3.PROCESSED = True
                            intf3.remote_intf.PROCESSED = True
                            intf3.OUTGOING = True
                            intf3.remote_intf.INCOMING = True
                    elif bundle.INCOMING:
                        for intf3 in bundle_list:
                            intf3.UNDIRECTED = False
                            intf3.remote_intf.UNDIRECTED = False
                            intf3.PROCESSED = True
                            intf3.remote_intf.PROCESSED = True
                            intf3.INCOMING = True
                            intf3.remote_intf.OUTGOING = True

def Modify_Block_Root_Incoming_Links(topo):
    for node in topo.island_node_list:
        if ( node.IS_CUT_VERTEX == True or node is topo.gadag_root ):
            for intf in node.island_intf_list:
                if intf.remote_node.localroot is node:
                    if intf.INCOMING:
                        intf.INCOMING = False
                        intf.INCOMING_STORED = True
                        intf.remote_intf.OUTGOING = False
                        intf.remote_intf.OUTGOING_STORED = True

def Revert_Block_Root_Incoming_Links(topo):
    for node in topo.island_node_list:
        if ( node.IS_CUT_VERTEX == True or node is topo.gadag_root ):
            for intf in node.island_intf_list:
                if intf.remote_node.localroot is node:
                    if intf.INCOMING_STORED:
                        intf.INCOMING = True
                        intf.remote_intf.OUTGOING = True
                        intf.INCOMING_STORED = False



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                        intf.remote_intf.OUTGOING_STORED = False

def Run_Topological_Sort_GADAG(topo):
    Modify_Block_Root_Incoming_Links(topo)
    for node in topo.island_node_list:
        node.unvisited = 0
        for intf in node.island_intf_list:
            if (intf.INCOMING == True):
                node.unvisited += 1
    working_list = []
    topo_order_list = []
    working_list.append(topo.gadag_root)
    while working_list != []:
        y = working_list.pop(0)
        topo_order_list.append(y)
        for intf in y.island_intf_list:
            if ( intf.OUTGOING == True):
                intf.remote_node.unvisited -= 1
                if intf.remote_node.unvisited == 0:
                    working_list.append(intf.remote_node)
    next_topo_order = 1
    while topo_order_list != []:
        y = topo_order_list.pop(0)
        y.topo_order = next_topo_order
        next_topo_order += 1
    Revert_Block_Root_Incoming_Links(topo)

def Set_Other_Undirected_Links_Based_On_Topo_Order(topo):
    for node in topo.island_node_list:
        for intf in node.island_intf_list:
            if intf.UNDIRECTED:
                if node.topo_order < intf.remote_node.topo_order:
                    intf.OUTGOING = True
                    intf.UNDIRECTED = False
                    intf.remote_intf.INCOMING = True
                    intf.remote_intf.UNDIRECTED = False
                else:
                    intf.INCOMING = True
                    intf.UNDIRECTED = False
                    intf.remote_intf.OUTGOING = True
                    intf.remote_intf.UNDIRECTED = False

def Initialize_Temporary_Interface_Flags(topo):
    for node in topo.island_node_list:
        for intf in node.island_intf_list:
            intf.PROCESSED = False
            intf.INCOMING_STORED = False
            intf.OUTGOING_STORED = False



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def Add_Undirected_Links(topo):
    Initialize_Temporary_Interface_Flags(topo)
    Add_Undirected_Block_Root_Links(topo)
    Run_Topological_Sort_GADAG(topo)
    Set_Other_Undirected_Links_Based_On_Topo_Order(topo)

def In_Common_Block(x,y):
    if (  (x.block_id == y.block_id)
          or ( x is y.localroot) or (y is x.localroot) ):
        return True
    return False

def Copy_List_Items(target_list, source_list):
    del target_list[:] # Python idiom to remove all elements of a list
    for element in source_list:
        target_list.append(element)

def Add_Item_To_List_If_New(target_list, item):
    if item not in target_list:
        target_list.append(item)

def Store_Results(y, direction):
    if direction == 'INCREASING':
        y.HIGHER = True
        Copy_List_Items(y.blue_next_hops, y.next_hops)
    if direction == 'DECREASING':
        y.LOWER = True
        Copy_List_Items(y.red_next_hops, y.next_hops)
    if direction == 'NORMAL_SPF':
        y.primary_spf_metric = y.spf_metric
        Copy_List_Items(y.primary_next_hops, y.next_hops)
    if direction == 'MRT_ISLAND_SPF':
        Copy_List_Items(y.mrt_island_next_hops, y.next_hops)
    if direction == 'COLLAPSED_SPF':
        y.collapsed_metric = y.spf_metric
        Copy_List_Items(y.collapsed_next_hops, y.next_hops)

# Note that the Python heapq fucntion allows for duplicate items,
# so we use the 'spf_visited' property to only consider a node
# as min_node the first time it gets removed from the heap.
def SPF_No_Traverse_Block_Root(topo, spf_root, block_root, direction):
    spf_heap = []
    for y in topo.island_node_list:
        y.spf_metric = 2147483647 # 2^31-1
        y.next_hops = []
        y.spf_visited = False
    spf_root.spf_metric = 0
    heapq.heappush(spf_heap,



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                   (spf_root.spf_metric, spf_root.node_id,  spf_root) )
    while spf_heap != []:
        #extract third element of tuple popped from heap
        min_node = heapq.heappop(spf_heap)[2]
        if min_node.spf_visited:
            continue
        min_node.spf_visited = True
        Store_Results(min_node, direction)
        if ( (min_node is spf_root) or (min_node is not block_root) ):
            for intf in min_node.island_intf_list:
                if ( ( (direction == 'INCREASING' and intf.OUTGOING )
                    or (direction == 'DECREASING' and intf.INCOMING ) )
                    and In_Common_Block(spf_root, intf.remote_node) ) :
                    path_metric = min_node.spf_metric + intf.metric
                    if path_metric < intf.remote_node.spf_metric:
                        intf.remote_node.spf_metric = path_metric
                        if min_node is spf_root:
                            intf.remote_node.next_hops = [intf]
                        else:
                            Copy_List_Items(intf.remote_node.next_hops,
                                            min_node.next_hops)
                        heapq.heappush(spf_heap,
                                       ( intf.remote_node.spf_metric,
                                         intf.remote_node.node_id,
                                         intf.remote_node ) )
                    elif path_metric == intf.remote_node.spf_metric:
                        if min_node is spf_root:
                            Add_Item_To_List_If_New(
                                intf.remote_node.next_hops,intf)
                        else:
                            for nh_intf in min_node.next_hops:
                                Add_Item_To_List_If_New(
                                    intf.remote_node.next_hops,nh_intf)

def Normal_SPF(topo, spf_root):
    spf_heap = []
    for y in topo.node_list:
        y.spf_metric = 2147483647 # 2^31-1 as max metric
        y.next_hops = []
        y.primary_spf_metric = 2147483647
        y.primary_next_hops = []
        y.spf_visited = False
    spf_root.spf_metric = 0
    heapq.heappush(spf_heap,
                   (spf_root.spf_metric,spf_root.node_id,spf_root) )
    while spf_heap != []:
        #extract third element of tuple popped from heap
        min_node = heapq.heappop(spf_heap)[2]



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        if min_node.spf_visited:
            continue
        min_node.spf_visited = True
        Store_Results(min_node, 'NORMAL_SPF')
        for intf in min_node.intf_list:
            path_metric = min_node.spf_metric + intf.metric
            if path_metric < intf.remote_node.spf_metric:
                intf.remote_node.spf_metric = path_metric
                if min_node is spf_root:
                    intf.remote_node.next_hops = [intf]
                else:
                    Copy_List_Items(intf.remote_node.next_hops,
                                    min_node.next_hops)
                heapq.heappush(spf_heap,
                               ( intf.remote_node.spf_metric,
                                 intf.remote_node.node_id,
                                 intf.remote_node ) )
            elif path_metric == intf.remote_node.spf_metric:
                if min_node is spf_root:
                    Add_Item_To_List_If_New(
                        intf.remote_node.next_hops,intf)
                else:
                    for nh_intf in min_node.next_hops:
                        Add_Item_To_List_If_New(
                            intf.remote_node.next_hops,nh_intf)

def Set_Edge(y):
    if (y.blue_next_hops == [] and y.red_next_hops == []):
        Set_Edge(y.localroot)
        Copy_List_Items(y.blue_next_hops,y.localroot.blue_next_hops)
        Copy_List_Items(y.red_next_hops ,y.localroot.red_next_hops)
        y.order_proxy = y.localroot.order_proxy

def Compute_MRT_NH_For_One_Src_To_Island_Dests(topo,x):
    for y in topo.island_node_list:
        y.HIGHER = False
        y.LOWER = False
        y.red_next_hops = []
        y.blue_next_hops = []
        y.order_proxy = y
    SPF_No_Traverse_Block_Root(topo, x, x.localroot, 'INCREASING')
    SPF_No_Traverse_Block_Root(topo, x, x.localroot, 'DECREASING')
    for y in topo.island_node_list:
        if ( y is not x and (y.block_id == x.block_id) ):
            assert (not ( y is x.localroot or x is y.localroot) )
            assert(not (y.HIGHER and y.LOWER) )
            if y.HIGHER == True:
                Copy_List_Items(y.red_next_hops,



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                                x.localroot.red_next_hops)
            elif y.LOWER == True:
                Copy_List_Items(y.blue_next_hops,
                                x.localroot.blue_next_hops)
            else:
                Copy_List_Items(y.blue_next_hops,
                                x.localroot.red_next_hops)
                Copy_List_Items(y.red_next_hops,
                                x.localroot.blue_next_hops)

    # Inherit x's MRT next-hops to reach the GADAG root
    # from x's MRT next-hops to reach its local root,
    # but first check if x is the gadag_root (in which case
    # x does not have a local root) or if x's local root
    # is the gadag root (in which case we already have the
    # x's MRT next-hops to reach the gadag root)
    if x is not topo.gadag_root and x.localroot is not topo.gadag_root:
        Copy_List_Items(topo.gadag_root.blue_next_hops,
                        x.localroot.blue_next_hops)
        Copy_List_Items(topo.gadag_root.red_next_hops,
                        x.localroot.red_next_hops)
        topo.gadag_root.order_proxy = x.localroot

    # Inherit next-hops and order_proxies to other blocks
    for y in topo.island_node_list:
        if (y is not topo.gadag_root and y is not x ):
            Set_Edge(y)

def Store_MRT_Nexthops_For_One_Src_To_Island_Dests(topo,x):
    for y in topo.island_node_list:
        if y is x:
            continue
        x.blue_next_hops_dict[y.node_id] = []
        x.red_next_hops_dict[y.node_id] = []
        Copy_List_Items(x.blue_next_hops_dict[y.node_id],
                        y.blue_next_hops)
        Copy_List_Items(x.red_next_hops_dict[y.node_id],
                        y.red_next_hops)

def Store_Primary_and_Alts_For_One_Src_To_Island_Dests(topo,x):
    for y in topo.island_node_list:
        x.pnh_dict[y.node_id] = []
        Copy_List_Items(x.pnh_dict[y.node_id], y.primary_next_hops)
        x.alt_dict[y.node_id] = []
        Copy_List_Items(x.alt_dict[y.node_id], y.alt_list)

def Store_Primary_NHs_For_One_Source_To_Nodes(topo,x):
    for y in topo.node_list:



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        x.pnh_dict[y.node_id] = []
        Copy_List_Items(x.pnh_dict[y.node_id], y.primary_next_hops)

def Store_MRT_NHs_For_One_Src_To_Named_Proxy_Nodes(topo,x):
    for prefix in topo.named_proxy_dict:
        P = topo.named_proxy_dict[prefix]
        x.blue_next_hops_dict[P.node_id] = []
        x.red_next_hops_dict[P.node_id] = []
        Copy_List_Items(x.blue_next_hops_dict[P.node_id],
                        P.blue_next_hops)
        Copy_List_Items(x.red_next_hops_dict[P.node_id],
                        P.red_next_hops)

def Store_Alts_For_One_Src_To_Named_Proxy_Nodes(topo,x):
    for prefix in topo.named_proxy_dict:
        P = topo.named_proxy_dict[prefix]
        x.alt_dict[P.node_id] = []
        Copy_List_Items(x.alt_dict[P.node_id],
                        P.alt_list)

def Store_Primary_NHs_For_One_Src_To_Named_Proxy_Nodes(topo,x):
    for prefix in topo.named_proxy_dict:
        P = topo.named_proxy_dict[prefix]
        x.pnh_dict[P.node_id] = []
        Copy_List_Items(x.pnh_dict[P.node_id],
                        P.primary_next_hops)

def Select_Alternates_Internal(D, F, primary_intf,
                               D_lower, D_higher, D_topo_order):
    if D_higher and D_lower:
        if F.HIGHER and F.LOWER:
            if F.topo_order > D_topo_order:
                return 'USE_BLUE'
            else:
                return 'USE_RED'
        if F.HIGHER:
            return 'USE_RED'
        if F.LOWER:
            return 'USE_BLUE'
        assert(primary_intf.MRT_INELIGIBLE
               or primary_intf.remote_intf.MRT_INELIGIBLE)
        return 'USE_RED_OR_BLUE'
    if D_higher:
        if F.HIGHER and F.LOWER:
            return 'USE_BLUE'
        if F.LOWER:
            return 'USE_BLUE'
        if F.HIGHER:



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            if (F.topo_order > D_topo_order):
                return 'USE_BLUE'
            if (F.topo_order < D_topo_order):
                return 'USE_RED'
            assert(False)
        assert(primary_intf.MRT_INELIGIBLE
               or primary_intf.remote_intf.MRT_INELIGIBLE)
        return 'USE_RED_OR_BLUE'
    if D_lower:
        if F.HIGHER and F.LOWER:
            return 'USE_RED'
        if F.HIGHER:
            return 'USE_RED'
        if F.LOWER:
            if F.topo_order > D_topo_order:
                return 'USE_BLUE'
            if F.topo_order < D_topo_order:
                return 'USE_RED'
            assert(False)
        assert(primary_intf.MRT_INELIGIBLE
               or primary_intf.remote_intf.MRT_INELIGIBLE)
        return 'USE_RED_OR_BLUE'
    else: # D is unordered wrt S
        if F.HIGHER and F.LOWER:
            if primary_intf.OUTGOING and primary_intf.INCOMING:
                # This can happen when F and D are in different blocks
                return 'USE_RED_OR_BLUE'
            if primary_intf.OUTGOING:
                return 'USE_BLUE'
            if primary_intf.INCOMING:
                return 'USE_RED'
            #This can occur when primary_intf is MRT_INELIGIBLE.
            #This appears to be a case where the special
            #construction of the GADAG allows us to choose red,
            #whereas with an arbitrary GADAG, neither red nor blue
            #is guaranteed to work.
            assert(primary_intf.MRT_INELIGIBLE
                   or primary_intf.remote_intf.MRT_INELIGIBLE)
            return 'USE_RED'
        if F.LOWER:
            return 'USE_RED'
        if F.HIGHER:
            return 'USE_BLUE'
        assert(primary_intf.MRT_INELIGIBLE
               or primary_intf.remote_intf.MRT_INELIGIBLE)
        if F.topo_order > D_topo_order:
            return 'USE_BLUE'
        else:



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            return 'USE_RED'

def Select_Alternates(D, F, primary_intf):
    S = primary_intf.local_node
    if not In_Common_Block(F, S):
        return 'PRIM_NH_IN_DIFFERENT_BLOCK'
    if (D is F) or (D.order_proxy is F):
        return 'PRIM_NH_IS_D_OR_OP_FOR_D'
    D_lower = D.order_proxy.LOWER
    D_higher = D.order_proxy.HIGHER
    D_topo_order = D.order_proxy.topo_order
    return Select_Alternates_Internal(D, F, primary_intf,
                                      D_lower, D_higher, D_topo_order)


def Is_Remote_Node_In_NH_List(node, intf_list):
    for intf in intf_list:
        if node is intf.remote_node:
            return True
    return False

def Select_Alts_For_One_Src_To_Island_Dests(topo,x):
    Normal_SPF(topo, x)
    for D in topo.island_node_list:
        D.alt_list = []
        if D is x:
            continue
        for failed_intf in D.primary_next_hops:
            alt = Alternate()
            alt.failed_intf = failed_intf
            cand_alt_list = []
            F = failed_intf.remote_node
            #We need to test if F is in the island, as opposed
            #to just testing if failed_intf is in island_intf_list,
            #because failed_intf could be marked as MRT_INELIGIBLE.
            if F in topo.island_node_list:
                alt.info = Select_Alternates(D, F, failed_intf)
            else:
                #The primary next-hop is not in the MRT Island.
                #Either red or blue will avoid the primary next-hop,
                #because the primary next-hop is not even in the
                #GADAG.
                alt.info = 'USE_RED_OR_BLUE'

            if (alt.info == 'USE_RED_OR_BLUE'):
                alt.red_or_blue = random.choice(['USE_RED','USE_BLUE'])
            if (alt.info == 'USE_BLUE'
                or alt.red_or_blue == 'USE_BLUE'):



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                Copy_List_Items(alt.nh_list, D.blue_next_hops)
                alt.fec = 'BLUE'
                alt.prot = 'NODE_PROTECTION'
            if (alt.info == 'USE_RED' or alt.red_or_blue == 'USE_RED'):
                Copy_List_Items(alt.nh_list, D.red_next_hops)
                alt.fec = 'RED'
                alt.prot = 'NODE_PROTECTION'
            if (alt.info == 'PRIM_NH_IN_DIFFERENT_BLOCK'):
                alt.fec = 'NO_ALTERNATE'
                alt.prot = 'NO_PROTECTION'
            if (alt.info == 'PRIM_NH_IS_D_OR_OP_FOR_D'):
                if failed_intf.OUTGOING and failed_intf.INCOMING:
                    # cut-link: if there are parallel cut links, use
                    # the link(s) with lowest metric that are not
                    # primary intf or None
                    cand_alt_list = [None]
                    min_metric = 2147483647
                    for intf in x.island_intf_list:
                        if ( intf is not failed_intf and
                             (intf.remote_node is
                             failed_intf.remote_node)):
                            if intf.metric < min_metric:
                                cand_alt_list = [intf]
                                min_metric = intf.metric
                            elif intf.metric == min_metric:
                                cand_alt_list.append(intf)
                    if cand_alt_list != [None]:
                        alt.fec = 'GREEN'
                        alt.prot = 'PARALLEL_CUTLINK'
                    else:
                        alt.fec = 'NO_ALTERNATE'
                        alt.prot = 'NO_PROTECTION'
                    Copy_List_Items(alt.nh_list, cand_alt_list)

                # Is_Remote_Node_In_NH_List() is used, as opposed
                # to just checking if failed_intf is in D.red_next_hops,
                # because failed_intf could be marked as MRT_INELIGIBLE.
                elif Is_Remote_Node_In_NH_List(F, D.red_next_hops):
                    Copy_List_Items(alt.nh_list, D.blue_next_hops)
                    alt.fec = 'BLUE'
                    alt.prot = 'LINK_PROTECTION'
                elif Is_Remote_Node_In_NH_List(F, D.blue_next_hops):
                    Copy_List_Items(alt.nh_list, D.red_next_hops)
                    alt.fec = 'RED'
                    alt.prot = 'LINK_PROTECTION'
                else:
                    alt.fec = random.choice(['RED','BLUE'])
                    alt.prot = 'LINK_PROTECTION'



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            D.alt_list.append(alt)

def Write_GADAG_To_File(topo, file_prefix):
    gadag_edge_list = []
    for node in topo.node_list:
        for intf in node.intf_list:
            if intf.SIMULATION_OUTGOING:
                local_node =  "%04d" % (intf.local_node.node_id)
                remote_node = "%04d" % (intf.remote_node.node_id)
                intf_data = "%03d" % (intf.link_data)
                edge_string=(local_node+','+remote_node+','+
                             intf_data+'\n')
                gadag_edge_list.append(edge_string)
    gadag_edge_list.sort();
    filename = file_prefix + '_gadag.csv'
    with open(filename, 'w') as gadag_file:
        gadag_file.write('local_node,'\
                         'remote_node,local_intf_link_data\n')
        for edge_string in gadag_edge_list:
            gadag_file.write(edge_string);

def Write_MRTs_For_All_Dests_To_File(topo, color, file_prefix):
    edge_list = []
    for node in topo.island_node_list_for_test_gr:
        if color == 'blue':
            node_next_hops_dict = node.blue_next_hops_dict
        elif color == 'red':
            node_next_hops_dict = node.red_next_hops_dict
        for dest_node_id in node_next_hops_dict:
            for intf in node_next_hops_dict[dest_node_id]:
                gadag_root =  "%04d" % (topo.gadag_root.node_id)
                dest_node =  "%04d" % (dest_node_id)
                local_node =  "%04d" % (intf.local_node.node_id)
                remote_node = "%04d" % (intf.remote_node.node_id)
                intf_data = "%03d" % (intf.link_data)
                edge_string=(gadag_root+','+dest_node+','+local_node+
                               ','+remote_node+','+intf_data+'\n')
                edge_list.append(edge_string)
    edge_list.sort()
    filename = file_prefix + '_' + color + '_to_all.csv'
    with open(filename, 'w') as mrt_file:
        mrt_file.write('gadag_root,dest,'\
            'local_node,remote_node,link_data\n')
        for edge_string in edge_list:
            mrt_file.write(edge_string);

def Write_Both_MRTs_For_All_Dests_To_File(topo, file_prefix):
    Write_MRTs_For_All_Dests_To_File(topo, 'blue', file_prefix)



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    Write_MRTs_For_All_Dests_To_File(topo, 'red', file_prefix)

def Write_Alternates_For_All_Dests_To_File(topo, file_prefix):
    edge_list = []
    for x in topo.island_node_list_for_test_gr:
        for dest_node_id in x.alt_dict:
            alt_list = x.alt_dict[dest_node_id]
            for alt in alt_list:
                for alt_intf in alt.nh_list:
                    gadag_root =  "%04d" % (topo.gadag_root.node_id)
                    dest_node =  "%04d" % (dest_node_id)
                    prim_local_node =  \
                        "%04d" % (alt.failed_intf.local_node.node_id)
                    prim_remote_node = \
                        "%04d" % (alt.failed_intf.remote_node.node_id)
                    prim_intf_data = \
                        "%03d" % (alt.failed_intf.link_data)
                    if alt_intf == None:
                        alt_local_node = "None"
                        alt_remote_node = "None"
                        alt_intf_data = "None"
                    else:
                        alt_local_node = \
                            "%04d" % (alt_intf.local_node.node_id)
                        alt_remote_node = \
                            "%04d" % (alt_intf.remote_node.node_id)
                        alt_intf_data = \
                            "%03d" % (alt_intf.link_data)
                    edge_string = (gadag_root+','+dest_node+','+
                        prim_local_node+','+prim_remote_node+','+
                        prim_intf_data+','+alt_local_node+','+
                        alt_remote_node+','+alt_intf_data+','+
                        alt.fec +'\n')
                    edge_list.append(edge_string)
    edge_list.sort()
    filename = file_prefix + '_alts_to_all.csv'
    with open(filename, 'w') as alt_file:
        alt_file.write('gadag_root,dest,'\
            'prim_nh.local_node,prim_nh.remote_node,'\
            'prim_nh.link_data,alt_nh.local_node,'\
            'alt_nh.remote_node,alt_nh.link_data,'\
            'alt_nh.fec\n')
        for edge_string in edge_list:
            alt_file.write(edge_string);

def Raise_GADAG_Root_Selection_Priority(topo,node_id):
    node = topo.node_dict[node_id]
    node.GR_sel_priority = 255



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def Lower_GADAG_Root_Selection_Priority(topo,node_id):
    node = topo.node_dict[node_id]
    node.GR_sel_priority = 128

def GADAG_Root_Compare(node_a, node_b):
    if (node_a.GR_sel_priority > node_b.GR_sel_priority):
        return 1
    elif (node_a.GR_sel_priority < node_b.GR_sel_priority):
        return -1
    else:
        if node_a.node_id > node_b.node_id:
            return 1
        elif node_a.node_id < node_b.node_id:
            return -1

def Set_GADAG_Root(topo,computing_router):
    gadag_root_list = []
    for node in topo.island_node_list:
        gadag_root_list.append(node)
    gadag_root_list.sort(GADAG_Root_Compare)
    topo.gadag_root = gadag_root_list.pop()

def Add_Prefix_Advertisements_From_File(topo, filename):
    prefix_filename = filename + '.prefix'
    cols_list = []
    if not os.path.exists(prefix_filename):
        return
    with open(prefix_filename) as prefix_file:
        for line in prefix_file:
            line = line.rstrip('\r\n')
            cols=line.split(',')
            cols_list.append(cols)
            prefix_id = int(cols[0])
            if prefix_id < 2000 or prefix_id >2999:
                print('skipping the following line of prefix file')
                print('prefix id should be between 2000 and 2999')
                print(line)
                continue
            prefix_node_id = int(cols[1])
            prefix_cost = int(cols[2])
            advertising_node = topo.node_dict[prefix_node_id]
            advertising_node.prefix_cost_dict[prefix_id] = prefix_cost

def Add_Prefixes_for_Non_Island_Nodes(topo):
    for node in topo.node_list:
        if node.IN_MRT_ISLAND:
            continue
        prefix_id = node.node_id + 1000



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        node.prefix_cost_dict[prefix_id] = 0

def Add_Profile_IDs_from_File(topo, filename):
    profile_filename = filename + '.profile'
    for node in topo.node_list:
        node.profile_id_list = []
    cols_list = []
    if os.path.exists(profile_filename):
        with open(profile_filename) as profile_file:
            for line in profile_file:
                line = line.rstrip('\r\n')
                cols=line.split(',')
                cols_list.append(cols)
                node_id = int(cols[0])
                profile_id = int(cols[1])
                this_node = topo.node_dict[node_id]
                this_node.profile_id_list.append(profile_id)
    else:
        for node in topo.node_list:
            node.profile_id_list = [0]

def Island_Marking_SPF(topo,spf_root):
    spf_root.isl_marking_spf_dict = {}
    for y in topo.node_list:
        y.spf_metric = 2147483647 # 2^31-1 as max metric
        y.PATH_HITS_ISLAND = False
        y.next_hops = []
        y.spf_visited = False
    spf_root.spf_metric = 0
    spf_heap = []
    heapq.heappush(spf_heap,
                   (spf_root.spf_metric,spf_root.node_id,spf_root) )
    while spf_heap != []:
        #extract third element of tuple popped from heap
        min_node = heapq.heappop(spf_heap)[2]
        if min_node.spf_visited:
            continue
        min_node.spf_visited = True
        spf_root.isl_marking_spf_dict[min_node.node_id] = \
            (min_node.spf_metric, min_node.PATH_HITS_ISLAND)
        for intf in min_node.intf_list:
            path_metric = min_node.spf_metric + intf.metric
            if path_metric < intf.remote_node.spf_metric:
                intf.remote_node.spf_metric = path_metric
                if min_node is spf_root:
                    intf.remote_node.next_hops = [intf]
                else:
                    Copy_List_Items(intf.remote_node.next_hops,



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                                    min_node.next_hops)
                if (intf.remote_node.IN_MRT_ISLAND):
                    intf.remote_node.PATH_HITS_ISLAND = True
                else:
                    intf.remote_node.PATH_HITS_ISLAND = \
                        min_node.PATH_HITS_ISLAND
                heapq.heappush(spf_heap,
                               ( intf.remote_node.spf_metric,
                                 intf.remote_node.node_id,
                                 intf.remote_node ) )
            elif path_metric == intf.remote_node.spf_metric:
                if min_node is spf_root:
                    Add_Item_To_List_If_New(
                        intf.remote_node.next_hops,intf)
                else:
                    for nh_intf in min_node.next_hops:
                        Add_Item_To_List_If_New(
                            intf.remote_node.next_hops,nh_intf)
                if (intf.remote_node.IN_MRT_ISLAND):
                    intf.remote_node.PATH_HITS_ISLAND = True
                else:
                    if (intf.remote_node.PATH_HITS_ISLAND
                        or min_node.PATH_HITS_ISLAND):
                        intf.remote_node.PATH_HITS_ISLAND = True


def Create_Basic_Named_Proxy_Nodes(topo):
    for node in topo.node_list:
        for prefix in node.prefix_cost_dict:
            prefix_cost = node.prefix_cost_dict[prefix]
            if prefix in topo.named_proxy_dict:
                P = topo.named_proxy_dict[prefix]
                P.node_prefix_cost_list.append((node,prefix_cost))
            else:
                P = Named_Proxy_Node()
                topo.named_proxy_dict[prefix] = P
                P.node_id = prefix
                P.node_prefix_cost_list = [(node,prefix_cost)]

def Compute_Loop_Free_Island_Neighbors_For_Each_Prefix(topo):
    topo.island_nbr_set = set()
    topo.island_border_set = set()
    for node in topo.node_list:
        if node.IN_MRT_ISLAND:
            continue
        for intf in node.intf_list:
            if intf.remote_node.IN_MRT_ISLAND:
                topo.island_nbr_set.add(node)



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                topo.island_border_set.add(intf.remote_node)

    for island_nbr in topo.island_nbr_set:
        Island_Marking_SPF(topo,island_nbr)

    for prefix in topo.named_proxy_dict:
        P = topo.named_proxy_dict[prefix]
        P.lfin_list = []
        for island_nbr in topo.island_nbr_set:
            min_isl_nbr_to_pref_cost = 2147483647
            for (adv_node, prefix_cost) in P.node_prefix_cost_list:
                (adv_node_cost, path_hits_island) = \
                    island_nbr.isl_marking_spf_dict[adv_node.node_id]
                isl_nbr_to_pref_cost = adv_node_cost + prefix_cost
                if isl_nbr_to_pref_cost < min_isl_nbr_to_pref_cost:
                    min_isl_nbr_to_pref_cost = isl_nbr_to_pref_cost
                    min_path_hits_island = path_hits_island
                elif isl_nbr_to_pref_cost == min_isl_nbr_to_pref_cost:
                    if min_path_hits_island or path_hits_island:
                        min_path_hits_island = True
            if not min_path_hits_island:
                P.lfin_list.append( (island_nbr,
                                     min_isl_nbr_to_pref_cost) )


def Compute_Island_Border_Router_LFIN_Pairs_For_Each_Prefix(topo):
    for ibr in topo.island_border_set:
        ibr.prefix_lfin_dict = {}
        ibr.min_intf_metric_dict = {}
        ibr.min_intf_list_dict = {}
        ibr.min_intf_list_dict[None] = None
        for intf in ibr.intf_list:
            if not intf.remote_node in topo.island_nbr_set:
                continue
            if not intf.remote_node in ibr.min_intf_metric_dict:
                ibr.min_intf_metric_dict[intf.remote_node] = \
                    intf.metric
                ibr.min_intf_list_dict[intf.remote_node] = [intf]
            else:
                if (intf.metric
                    < ibr.min_intf_metric_dict[intf.remote_node]):
                    ibr.min_intf_metric_dict[intf.remote_node] = \
                         intf.metric
                    ibr.min_intf_list_dict[intf.remote_node] = [intf]
                elif (intf.metric
                      < ibr.min_intf_metric_dict[intf.remote_node]):
                    ibr.min_intf_list_dict[intf.remote_node].\
                        append(intf)



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    for prefix in topo.named_proxy_dict:
        P = topo.named_proxy_dict[prefix]
        for ibr in topo.island_border_set:
            min_ibr_lfin_pref_cost = 2147483647
            min_lfin = None
            for (lfin, lfin_to_pref_cost) in P.lfin_list:
                if not lfin in ibr.min_intf_metric_dict:
                    continue
                ibr_lfin_pref_cost = \
                    ibr.min_intf_metric_dict[lfin] + lfin_to_pref_cost
                if ibr_lfin_pref_cost < min_ibr_lfin_pref_cost:
                    min_ibr_lfin_pref_cost = ibr_lfin_pref_cost
                    min_lfin = lfin
            ibr.prefix_lfin_dict[prefix] = (min_lfin,
                min_ibr_lfin_pref_cost,
                ibr.min_intf_list_dict[min_lfin])

def Proxy_Node_Att_Router_Compare(pnar_a, pnar_b):
    if pnar_a.named_proxy_cost < pnar_b.named_proxy_cost:
        return -1
    if pnar_b.named_proxy_cost < pnar_a.named_proxy_cost:
        return 1
    if pnar_a.node.node_id < pnar_b.node.node_id:
        return -1
    if pnar_b.node.node_id < pnar_a.node.node_id:
        return 1
    if pnar_a.min_lfin == None:
        return -1
    if pnar_b.min_lfin == None:
        return 1

def Choose_Proxy_Node_Attachment_Routers(topo):
    for prefix in topo.named_proxy_dict:
        P = topo.named_proxy_dict[prefix]
        pnar_candidate_list = []
        for (node, prefix_cost) in P.node_prefix_cost_list:
            if not node.IN_MRT_ISLAND:
                continue
            pnar = Proxy_Node_Attachment_Router()
            pnar.prefix = prefix
            pnar.named_proxy_cost = prefix_cost
            pnar.node = node
            pnar_candidate_list.append(pnar)
        for ibr in topo.island_border_set:
            (min_lfin, prefix_cost, min_intf_list) = \
                ibr.prefix_lfin_dict[prefix]
            if min_lfin == None:
                continue



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            pnar = Proxy_Node_Attachment_Router()
            pnar.named_proxy_cost = prefix_cost
            pnar.node = ibr
            pnar.min_lfin = min_lfin
            pnar.nh_intf_list = min_intf_list
            pnar_candidate_list.append(pnar)
        pnar_candidate_list.sort(cmp=Proxy_Node_Att_Router_Compare)
        #pop first element from list
        first_pnar = pnar_candidate_list.pop(0)
        second_pnar = None
        for next_pnar in pnar_candidate_list:
            if next_pnar.node is first_pnar.node:
                continue
            second_pnar = next_pnar
            break

        P.pnar1 = first_pnar
        P.pnar2 = second_pnar

def Attach_Named_Proxy_Nodes(topo):
    Compute_Loop_Free_Island_Neighbors_For_Each_Prefix(topo)
    Compute_Island_Border_Router_LFIN_Pairs_For_Each_Prefix(topo)
    Choose_Proxy_Node_Attachment_Routers(topo)

def Select_Proxy_Node_NHs(P,S):
    if P.pnar1.node.node_id < P.pnar2.node.node_id:
        X = P.pnar1.node
        Y = P.pnar2.node
    else:
        X = P.pnar2.node
        Y = P.pnar1.node
    P.pnar_X = X
    P.pnar_Y = Y
    A = X.order_proxy
    B = Y.order_proxy
    if (A is S.localroot
        and B is S.localroot):
        #print("1.0")
        Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
        Copy_List_Items(P.red_next_hops, Y.red_next_hops)
        return
    if (A is S.localroot
        and B is not S.localroot):
        #print("2.0")
        if B.LOWER:
            #print("2.1")
            Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
            Copy_List_Items(P.red_next_hops, Y.red_next_hops)



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            return
        if B.HIGHER:
            #print("2.2")
            Copy_List_Items(P.blue_next_hops, X.red_next_hops)
            Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
            return
        else:
            #print("2.3")
            Copy_List_Items(P.blue_next_hops, X.red_next_hops)
            Copy_List_Items(P.red_next_hops, Y.red_next_hops)
            return
    if (A is not S.localroot
        and B is S.localroot):
        #print("3.0")
        if A.LOWER:
            #print("3.1")
            Copy_List_Items(P.blue_next_hops, X.red_next_hops)
            Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
            return
        if A.HIGHER:
            #print("3.2")
            Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
            Copy_List_Items(P.red_next_hops, Y.red_next_hops)
            return
        else:
            #print("3.3")
            Copy_List_Items(P.blue_next_hops, X.red_next_hops)
            Copy_List_Items(P.red_next_hops, Y.red_next_hops)
            return
    if (A is not S.localroot
        and B is not S.localroot):
        #print("4.0")
        if (S is A.localroot or S is B.localroot):
            #print("4.05")
            if A.topo_order < B.topo_order:
                #print("4.05.1")
                Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                return
            else:
                #print("4.05.2")
                Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                return
        if A.LOWER:
            #print("4.1")
            if B.HIGHER:
                #print("4.1.1")



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                Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                return
            if B.LOWER:
                #print("4.1.2")
                if A.topo_order < B.topo_order:
                    #print("4.1.2.1")
                    Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                    Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                    return
                else:
                    #print("4.1.2.2")
                    Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                    Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                    return
            else:
                #print("4.1.3")
                Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                return
        if A.HIGHER:
            #print("4.2")
            if B.HIGHER:
                #print("4.2.1")
                if A.topo_order < B.topo_order:
                    #print("4.2.1.1")
                    Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                    Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                    return
                else:
                    #print("4.2.1.2")
                    Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                    Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                    return
            if B.LOWER:
                #print("4.2.2")
                Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                return
            else:
                #print("4.2.3")
                Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                return
        else:
            #print("4.3")
            if B.LOWER:
                #print("4.3.1")



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                Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                return
            if B.HIGHER:
                #print("4.3.2")
                Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                return
            else:
                #print("4.3.3")
                if A.topo_order < B.topo_order:
                    #print("4.3.3.1")
                    Copy_List_Items(P.blue_next_hops, X.blue_next_hops)
                    Copy_List_Items(P.red_next_hops, Y.red_next_hops)
                    return
                else:
                    #print("4.3.3.2")
                    Copy_List_Items(P.blue_next_hops, X.red_next_hops)
                    Copy_List_Items(P.red_next_hops, Y.blue_next_hops)
                    return
    assert(False)

def Compute_MRT_NHs_For_One_Src_To_Named_Proxy_Nodes(topo,S):
    for prefix in topo.named_proxy_dict:
        P = topo.named_proxy_dict[prefix]
        if P.pnar2 == None:
            if S is P.pnar1.node:
                # set the MRT next-hops for the PNAR to
                # reach the LFIN and change FEC to green
                Copy_List_Items(P.blue_next_hops,
                                P.pnar1.nh_intf_list)
                S.blue_to_green_nh_dict[P.node_id] = True
                Copy_List_Items(P.red_next_hops,
                                P.pnar1.nh_intf_list)
                S.red_to_green_nh_dict[P.node_id] = True
            else:
                # inherit MRT NHs for P from pnar1
                Copy_List_Items(P.blue_next_hops,
                                P.pnar1.node.blue_next_hops)
                Copy_List_Items(P.red_next_hops,
                                P.pnar1.node.red_next_hops)
        else:
            Select_Proxy_Node_NHs(P,S)
            # set the MRT next-hops for the PNAR to reach the LFIN
            # and change FEC to green rely on the red or blue
            # next-hops being empty to figure out which one needs
            # to point to the LFIN.
            if S is P.pnar1.node:



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                this_pnar = P.pnar1
            elif S is P.pnar2.node:
                this_pnar = P.pnar2
            else:
                continue
            if P.blue_next_hops == []:
                Copy_List_Items(P.blue_next_hops,
                    this_pnar.nh_intf_list)
                S.blue_to_green_nh_dict[P.node_id] = True
            if P.red_next_hops == []:
                Copy_List_Items(P.red_next_hops,
                    this_pnar.nh_intf_list)
                S.red_to_green_nh_dict[P.node_id] = True

def Select_Alternates_Proxy_Node(P,F,primary_intf):
    S = primary_intf.local_node
    X = P.pnar_X
    Y = P.pnar_Y
    A = X.order_proxy
    B = Y.order_proxy
    if F is A and F is B:
        return 'PRIM_NH_IS_OP_FOR_BOTH_X_AND_Y'
    if F is A:
        return 'USE_RED'
    if F is B:
        return 'USE_BLUE'

    if not In_Common_Block(A, B):
        if In_Common_Block(F, A):
            return 'USE_RED'
        elif In_Common_Block(F, B):
            return 'USE_BLUE'
        else:
            return 'USE_RED_OR_BLUE'
    if (not In_Common_Block(F, A)
        and not In_Common_Block(F, A) ):
        return 'USE_RED_OR_BLUE'

    alt_to_X = Select_Alternates(X, F, primary_intf)
    alt_to_Y = Select_Alternates(Y, F, primary_intf)

    if (alt_to_X == 'USE_RED_OR_BLUE'
        and alt_to_Y == 'USE_RED_OR_BLUE'):
        return 'USE_RED_OR_BLUE'
    if alt_to_X == 'USE_RED_OR_BLUE':
        return 'USE_BLUE'
    if alt_to_Y == 'USE_RED_OR_BLUE':
        return 'USE_RED'



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    if (A is S.localroot
        and B is S.localroot):
        #print("1.0")
        if (alt_to_X == 'USE_BLUE' and alt_to_Y == 'USE_RED'):
            return 'USE_RED_OR_BLUE'
        if alt_to_X == 'USE_BLUE':
            return 'USE_BLUE'
        if alt_to_Y == 'USE_RED':
            return 'USE_RED'
        assert(False)
    if (A is S.localroot
        and B is not S.localroot):
        #print("2.0")
        if B.LOWER:
            #print("2.1")
            if (alt_to_X == 'USE_BLUE' and alt_to_Y == 'USE_RED'):
                return 'USE_RED_OR_BLUE'
            if alt_to_X == 'USE_BLUE':
                return 'USE_BLUE'
            if alt_to_Y == 'USE_RED':
                return 'USE_RED'
            assert(False)
        if B.HIGHER:
            #print("2.2")
            if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_BLUE'):
                return 'USE_RED_OR_BLUE'
            if alt_to_X == 'USE_RED':
                return 'USE_BLUE'
            if alt_to_Y == 'USE_BLUE':
                return 'USE_RED'
            assert(False)
        else:
            #print("2.3")
            if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_RED'):
                return 'USE_RED_OR_BLUE'
            if alt_to_X == 'USE_RED':
                return 'USE_BLUE'
            if alt_to_Y == 'USE_RED':
                return 'USE_RED'
            assert(False)
    if (A is not S.localroot
        and B is S.localroot):
        #print("3.0")
        if A.LOWER:
            #print("3.1")
            if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_BLUE'):
                return 'USE_RED_OR_BLUE'
            if alt_to_X == 'USE_RED':



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                return 'USE_BLUE'
            if alt_to_Y == 'USE_BLUE':
                return 'USE_RED'
            assert(False)
        if A.HIGHER:
            #print("3.2")
            if (alt_to_X == 'USE_BLUE' and alt_to_Y == 'USE_RED'):
                return 'USE_RED_OR_BLUE'
            if alt_to_X == 'USE_BLUE':
                return 'USE_BLUE'
            if alt_to_Y == 'USE_RED':
                return 'USE_RED'
            assert(False)
        else:
            #print("3.3")
            if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_RED'):
                return 'USE_RED_OR_BLUE'
            if alt_to_X == 'USE_RED':
                return 'USE_BLUE'
            if alt_to_Y == 'USE_RED':
                return 'USE_RED'
            assert(False)
    if (A is not S.localroot
        and B is not S.localroot):
        #print("4.0")
        if (S is A.localroot or S is B.localroot):
            #print("4.05")
            if A.topo_order < B.topo_order:
                #print("4.05.1")
                if (alt_to_X == 'USE_BLUE' and alt_to_Y == 'USE_RED'):
                    return 'USE_RED_OR_BLUE'
                if alt_to_X == 'USE_BLUE':
                    return 'USE_BLUE'
                if alt_to_Y == 'USE_RED':
                    return 'USE_RED'
                assert(False)
            else:
                #print("4.05.2")
                if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_BLUE'):
                    return 'USE_RED_OR_BLUE'
                if alt_to_X == 'USE_RED':
                    return 'USE_BLUE'
                if alt_to_Y == 'USE_BLUE':
                    return 'USE_RED'
                assert(False)
        if A.LOWER:
            #print("4.1")
            if B.HIGHER:



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                #print("4.1.1")
                if (alt_to_X == 'USE_RED' and alt_to_Y == 'USE_BLUE'):
                    return 'USE_RED_OR_BLUE'
                if alt_to_X == 'USE_RED':
                    return 'USE_BLUE'
                if alt_to_Y == 'USE_BLUE':
                    return 'USE_RED'
                assert(False)
            if B.LOWER:
                #print("4.1.2")
                if A.topo_order < B.topo_order:
                    #print("4.1.2.1")
                    if (alt_to_X == 'USE_BLUE'
                        and alt_to_Y == 'USE_RED'):
                        return 'USE_RED_OR_BLUE'
                    if alt_to_X == 'USE_BLUE':
                        return 'USE_BLUE'
                    if alt_to_Y == 'USE_RED':
                        return 'USE_RED'
                    assert(False)
                else:
                    #print("4.1.2.2")
                    if (alt_to_X == 'USE_RED'
                        and alt_to_Y == 'USE_BLUE'):
                        return 'USE_RED_OR_BLUE'
                    if alt_to_X == 'USE_RED':
                        return 'USE_BLUE'
                    if alt_to_Y == 'USE_BLUE':
                        return 'USE_RED'
                    assert(False)
            else:
                #print("4.1.3")
                if (F.LOWER and not F.HIGHER
                    and F.topo_order > A.topo_order):
                    #print("4.1.3.1")
                    return 'USE_RED'
                else:
                    #print("4.1.3.2")
                    return 'USE_BLUE'
        if A.HIGHER:
            #print("4.2")
            if B.HIGHER:
                #print("4.2.1")
                if A.topo_order < B.topo_order:
                    #print("4.2.1.1")
                    if (alt_to_X == 'USE_BLUE'
                        and alt_to_Y == 'USE_RED'):
                        return 'USE_RED_OR_BLUE'



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                    if alt_to_X == 'USE_BLUE':
                        return 'USE_BLUE'
                    if alt_to_Y == 'USE_RED':
                        return 'USE_RED'
                    assert(False)
                else:
                    #print("4.2.1.2")
                    if (alt_to_X == 'USE_RED'
                        and alt_to_Y == 'USE_BLUE'):
                        return 'USE_RED_OR_BLUE'
                    if alt_to_X == 'USE_RED':
                        return 'USE_BLUE'
                    if alt_to_Y == 'USE_BLUE':
                        return 'USE_RED'
                    assert(False)
            if B.LOWER:
                #print("4.2.2")
                if (alt_to_X == 'USE_BLUE'
                    and alt_to_Y == 'USE_RED'):
                    return 'USE_RED_OR_BLUE'
                if alt_to_X == 'USE_BLUE':
                    return 'USE_BLUE'
                if alt_to_Y == 'USE_RED':
                    return 'USE_RED'
                assert(False)
            else:
                #print("4.2.3")
                if (F.HIGHER and not F.LOWER
                    and F.topo_order < A.topo_order):
                    return 'USE_RED'
                else:
                    return 'USE_BLUE'
        else:
            #print("4.3")
            if B.LOWER:
                #print("4.3.1")
                if (F.LOWER and not F.HIGHER
                    and F.topo_order > B.topo_order):
                    return 'USE_BLUE'
                else:
                    return 'USE_RED'
            if B.HIGHER:
                #print("4.3.2")
                if (F.HIGHER and not F.LOWER
                    and F.topo_order < B.topo_order):
                    return 'USE_BLUE'
                else:
                    return 'USE_RED'



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            else:
                #print("4.3.3")
                if A.topo_order < B.topo_order:
                    #print("4.3.3.1")
                    if (alt_to_X == 'USE_BLUE'
                        and alt_to_Y == 'USE_RED'):
                        return 'USE_RED_OR_BLUE'
                    if alt_to_X == 'USE_BLUE':
                        return 'USE_BLUE'
                    if alt_to_Y == 'USE_RED':
                        return 'USE_RED'
                    assert(False)
                else:
                    #print("4.3.3.2")
                    if (alt_to_X == 'USE_RED'
                        and alt_to_Y == 'USE_BLUE'):
                        return 'USE_RED_OR_BLUE'
                    if alt_to_X == 'USE_RED':
                        return 'USE_BLUE'
                    if alt_to_Y == 'USE_BLUE':
                        return 'USE_RED'
                    assert(False)
    assert(False)

def Compute_Primary_NHs_For_One_Src_To_Named_Proxy_Nodes(topo,src):
    for prefix in topo.named_proxy_dict:
        P = topo.named_proxy_dict[prefix]
        min_total_pref_cost = 2147483647
        for (adv_node, prefix_cost) in P.node_prefix_cost_list:
            total_pref_cost = (adv_node.primary_spf_metric
                               + prefix_cost)
            if total_pref_cost < min_total_pref_cost:
                min_total_pref_cost = total_pref_cost
                Copy_List_Items(P.primary_next_hops,
                                adv_node.primary_next_hops)
            elif total_pref_cost == min_total_pref_cost:
                for nh_intf in adv_node.primary_next_hops:
                    Add_Item_To_List_If_New(P.primary_next_hops,
                                            nh_intf)

def Select_Alts_For_One_Src_To_Named_Proxy_Nodes(topo,src):
    for prefix in topo.named_proxy_dict:
        P = topo.named_proxy_dict[prefix]
        P.alt_list = []
        for failed_intf in P.primary_next_hops:
            alt = Alternate()
            alt.failed_intf = failed_intf
            if failed_intf not in src.island_intf_list:



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                alt.info = 'PRIM_NH_FOR_PROXY_NODE_NOT_IN_ISLAND'
            elif P.pnar1 is None:
                alt.info = 'NO_PNARs_EXIST_FOR_THIS_PREFIX'
            elif src is P.pnar1.node:
                alt.info = 'SRC_IS_PNAR'
            elif P.pnar2 is not None and src is P.pnar2.node:
                alt.info = 'SRC_IS_PNAR'
            elif P.pnar2 is None:
                #inherit alternates from the only pnar.
                alt.info = Select_Alternates(P.pnar1.node,
                            failed_intf.remote_node, failed_intf)
            elif failed_intf in src.island_intf_list:
                alt.info = Select_Alternates_Proxy_Node(P,
                            failed_intf.remote_node, failed_intf)

            if alt.info == 'USE_RED_OR_BLUE':
                alt.red_or_blue = \
                    random.choice(['USE_RED','USE_BLUE'])
            if (alt.info == 'USE_BLUE'
                or alt.red_or_blue == 'USE_BLUE'):
                Copy_List_Items(alt.nh_list, P.blue_next_hops)
                alt.fec = 'BLUE'
                alt.prot = 'NODE_PROTECTION'
            elif (alt.info == 'USE_RED'
                  or alt.red_or_blue == 'USE_RED'):
                Copy_List_Items(alt.nh_list, P.red_next_hops)
                alt.fec = 'RED'
                alt.prot = 'NODE_PROTECTION'
            elif (alt.info == 'PRIM_NH_IS_D_OR_OP_FOR_D'
                or alt.info == 'PRIM_NH_IS_OP_FOR_BOTH_X_AND_Y'):
                if failed_intf.OUTGOING and failed_intf.INCOMING:
                    # cut-link: if there are parallel cut links, use
                    # the link(s) with lowest metric that are not
                    # primary intf or None
                    cand_alt_list = [None]
                    min_metric = 2147483647
                    for intf in src.island_intf_list:
                        if ( intf is not failed_intf and
                             (intf.remote_node is
                             failed_intf.remote_node)):
                            if intf.metric < min_metric:
                                cand_alt_list = [intf]
                                min_metric = intf.metric
                            elif intf.metric == min_metric:
                                cand_alt_list.append(intf)
                    if cand_alt_list != [None]:
                        alt.fec = 'GREEN'
                        alt.prot = 'PARALLEL_CUTLINK'



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                    else:
                        alt.fec = 'NO_ALTERNATE'
                        alt.prot = 'NO_PROTECTION'
                    Copy_List_Items(alt.nh_list, cand_alt_list)
                else:
                    # set Z as the node to inherit blue next-hops from
                    if alt.info == 'PRIM_NH_IS_D_OR_OP_FOR_D':
                        Z = P.pnar1.node
                    else:
                        Z = P
                    if failed_intf in Z.red_next_hops:
                        Copy_List_Items(alt.nh_list, Z.blue_next_hops)
                        alt.fec = 'BLUE'
                        alt.prot = 'LINK_PROTECTION'
                    else:
                        assert(failed_intf in Z.blue_next_hops)
                        Copy_List_Items(alt.nh_list, Z.red_next_hops)
                        alt.fec = 'RED'
                        alt.prot = 'LINK_PROTECTION'

            elif alt.info == 'PRIM_NH_FOR_PROXY_NODE_NOT_IN_ISLAND':
                if (P.pnar2 == None and src is P.pnar1.node):
                    #MRT Island is singly connected to non-island dest
                    alt.fec = 'NO_ALTERNATE'
                    alt.prot = 'NO_PROTECTION'
                elif P.node_id in src.blue_to_green_nh_dict:
                    # blue to P goes to failed LFIN so use red to P
                    Copy_List_Items(alt.nh_list, P.red_next_hops)
                    alt.fec = 'RED'
                    alt.prot = 'LINK_PROTECTION'
                elif P.node_id in src.red_to_green_nh_dict:
                    # red to P goes to failed LFIN so use blue to P
                    Copy_List_Items(alt.nh_list, P.blue_next_hops)
                    alt.fec = 'BLUE'
                    alt.prot = 'LINK_PROTECTION'
                else:
                    Copy_List_Items(alt.nh_list, P.blue_next_hops)
                    alt.fec = 'BLUE'
                    alt.prot = 'LINK_PROTECTION'
            elif alt.info == 'TEMP_NO_ALTERNATE':
                alt.fec = 'NO_ALTERNATE'
                alt.prot = 'NO_PROTECTION'

            P.alt_list.append(alt)

def Run_Basic_MRT_for_One_Source(topo, src):
    MRT_Island_Identification(topo, src, 0, 0)
    Set_Island_Intf_and_Node_Lists(topo)



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    Set_GADAG_Root(topo,src)
    Sort_Interfaces(topo)
    Run_Lowpoint(topo)
    Assign_Remaining_Lowpoint_Parents(topo)
    Construct_GADAG_via_Lowpoint(topo)
    Run_Assign_Block_ID(topo)
    Add_Undirected_Links(topo)
    Compute_MRT_NH_For_One_Src_To_Island_Dests(topo,src)
    Store_MRT_Nexthops_For_One_Src_To_Island_Dests(topo,src)
    Select_Alts_For_One_Src_To_Island_Dests(topo,src)
    Store_Primary_and_Alts_For_One_Src_To_Island_Dests(topo,src)

def Store_GADAG_and_Named_Proxies_Once(topo):
    for node in topo.node_list:
        for intf in node.intf_list:
            if intf.OUTGOING:
                intf.SIMULATION_OUTGOING = True
            else:
                intf.SIMULATION_OUTGOING = False
    for prefix in topo.named_proxy_dict:
        P = topo.named_proxy_dict[prefix]
        topo.stored_named_proxy_dict[prefix] = P

def Run_Basic_MRT_for_All_Sources(topo):
    for src in topo.node_list:
        Reset_Computed_Node_and_Intf_Values(topo)
        Run_Basic_MRT_for_One_Source(topo,src)
        if src is topo.gadag_root:
            Store_GADAG_and_Named_Proxies_Once(topo)

def Run_MRT_for_One_Source(topo, src):
    MRT_Island_Identification(topo, src, 0, 0)
    Set_Island_Intf_and_Node_Lists(topo)
    Set_GADAG_Root(topo,src)
    Sort_Interfaces(topo)
    Run_Lowpoint(topo)
    Assign_Remaining_Lowpoint_Parents(topo)
    Construct_GADAG_via_Lowpoint(topo)
    Run_Assign_Block_ID(topo)
    Add_Undirected_Links(topo)
    Compute_MRT_NH_For_One_Src_To_Island_Dests(topo,src)
    Store_MRT_Nexthops_For_One_Src_To_Island_Dests(topo,src)
    Select_Alts_For_One_Src_To_Island_Dests(topo,src)
    Store_Primary_and_Alts_For_One_Src_To_Island_Dests(topo,src)
    Create_Basic_Named_Proxy_Nodes(topo)
    Attach_Named_Proxy_Nodes(topo)
    Compute_MRT_NHs_For_One_Src_To_Named_Proxy_Nodes(topo,src)
    Store_MRT_NHs_For_One_Src_To_Named_Proxy_Nodes(topo,src)



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    Compute_Primary_NHs_For_One_Src_To_Named_Proxy_Nodes(topo,src)
    Store_Primary_NHs_For_One_Src_To_Named_Proxy_Nodes(topo,src)
    Select_Alts_For_One_Src_To_Named_Proxy_Nodes(topo,src)
    Store_Alts_For_One_Src_To_Named_Proxy_Nodes(topo,src)

def Run_Prim_SPF_for_One_Source(topo,src):
    Normal_SPF(topo, src)
    Store_Primary_NHs_For_One_Source_To_Nodes(topo,src)
    Create_Basic_Named_Proxy_Nodes(topo)
    Compute_Primary_NHs_For_One_Src_To_Named_Proxy_Nodes(topo,src)
    Store_Primary_NHs_For_One_Src_To_Named_Proxy_Nodes(topo,src)

def Run_MRT_for_All_Sources(topo):
    for src in topo.node_list:
        Reset_Computed_Node_and_Intf_Values(topo)
        if src in topo.island_node_list_for_test_gr:
            # src runs MRT if it is in same MRT island as test_gr
            Run_MRT_for_One_Source(topo,src)
            if src is topo.gadag_root:
                Store_GADAG_and_Named_Proxies_Once(topo)
        else:
            # src still runs SPF if not in MRT island
            Run_Prim_SPF_for_One_Source(topo,src)

def Write_Output_To_Files(topo,file_prefix):
    Write_GADAG_To_File(topo,file_prefix)
    Write_Both_MRTs_For_All_Dests_To_File(topo,file_prefix)
    Write_Alternates_For_All_Dests_To_File(topo,file_prefix)

def Create_Basic_Topology_Input_File(filename):
    data = [[01,02,10],[02,03,10],[03,04,11],[04,05,10,20],[05,06,10],
            [06,07,10],[06,07,10],[06,07,15],[07,01,10],[07,51,10],
            [51,52,10],[52,53,10],[53,03,10],[01,55,10],[55,06,10],
            [04,12,10],[12,13,10],[13,14,10],[14,15,10],[15,16,10],
            [16,17,10],[17,04,10],[05,76,10],[76,77,10],[77,78,10],
            [78,79,10],[79,77,10]]
    with open(filename + '.csv', 'w') as topo_file:
        for item in data:
            if len(item) > 3:
                line = (str(item[0])+','+str(item[1])+','+
                        str(item[2])+','+str(item[3])+'\n')
            else:
                line = (str(item[0])+','+str(item[1])+','+
                        str(item[2])+'\n')
            topo_file.write(line)

def Create_Complex_Topology_Input_File(filename):
    data = [[01,02,10],[02,03,10],[03,04,11],[04,05,10,20],[05,06,10],



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            [06,07,10],[06,07,10],[06,07,15],[07,01,10],[07,51,10],
            [51,52,10],[52,53,10],[53,03,10],[01,55,10],[55,06,10],
            [04,12,10],[12,13,10],[13,14,10],[14,15,10],[15,16,10],
            [16,17,10],[17,04,10],[05,76,10],[76,77,10],[77,78,10],
            [78,79,10],[79,77,10]]
    with open(filename + '.csv', 'w') as topo_file:
        for item in data:
            if len(item) > 3:
                line = (str(item[0])+','+str(item[1])+','+
                        str(item[2])+','+str(item[3])+'\n')
            else:
                line = (str(item[0])+','+str(item[1])+','+
                        str(item[2])+'\n')
            topo_file.write(line)

    data = [[01,0],[02,0],[03,0],[04,0],[05,0],
            [06,0],[07,0],
            [51,0],[55,0],
            [12,0],[13,0],[14,0],[15,0],
            [16,0],[17,0],[76,0],[77,0],
            [78,0],[79,0]]
    with open(filename + '.profile', 'w') as topo_file:
        for item in data:
            line = (str(item[0])+','+str(item[1])+'\n')
            topo_file.write(line)

    data = [[2001,05,100],[2001,07,120],[2001,03,130],
            [2002,13,100],[2002,15,110],
            [2003,52,100],[2003,78,100]]
    with open(filename + '.prefix', 'w') as topo_file:
        for item in data:
            line = (str(item[0])+','+str(item[1])+','+
                    str(item[2])+'\n')
            topo_file.write(line)

def Generate_Basic_Topology_and_Run_MRT():
    this_gadag_root = 3
    Create_Basic_Topology_Input_File('basic_topo_input')
    topo = Create_Topology_From_File('basic_topo_input')
    res_file_base = 'basic_topo'
    Compute_Island_Node_List_For_Test_GR(topo, this_gadag_root)
    Raise_GADAG_Root_Selection_Priority(topo,this_gadag_root)
    Run_Basic_MRT_for_All_Sources(topo)
    Write_Output_To_Files(topo, res_file_base)

def Generate_Complex_Topology_and_Run_MRT():
    this_gadag_root = 3
    Create_Complex_Topology_Input_File('complex_topo_input')



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    topo = Create_Topology_From_File('complex_topo_input')
    Add_Profile_IDs_from_File(topo,'complex_topo_input')
    Add_Prefix_Advertisements_From_File(topo,'complex_topo_input')
    Compute_Island_Node_List_For_Test_GR(topo, this_gadag_root)
    Add_Prefixes_for_Non_Island_Nodes(topo)
    res_file_base = 'complex_topo'
    Raise_GADAG_Root_Selection_Priority(topo,this_gadag_root)
    Run_MRT_for_All_Sources(topo)
    Write_Output_To_Files(topo, res_file_base)

Generate_Basic_Topology_and_Run_MRT()

Generate_Complex_Topology_and_Run_MRT()

<CODE ENDS>

Appendix B.  Constructing a GADAG using SPFs

   The basic idea in this method for constructing a GADAG is to use
   slightly-modified SPF computations to find ears.  In every block, an
   SPF computation is first done to find a cycle from the local root and
   then SPF computations in that block find ears until there are no more
   interfaces to be explored.  The used result from the SPF computation
   is the path of interfaces indicated by following the previous hops
   from the mininized IN_GADAG node back to the SPF root.

   To do this, first all cut-vertices must be identified and local-roots
   assigned as specified in Figure 12.

   The slight modifications to the SPF are as follows.  The root of the
   block is referred to as the block-root; it is either the GADAG root
   or a cut-vertex.

   a.  The SPF is rooted at a neighbor x of an IN_GADAG node y.  All
       links between y and x are marked as TEMP_UNUSABLE.  They should
       not be used during the SPF computation.

   b.  If y is not the block-root, then it is marked TEMP_UNUSABLE.  It
       should not be used during the SPF computation.  This prevents
       ears from starting and ending at the same node and avoids cycles;
       the exception is because cycles to/from the block-root are
       acceptable and expected.

   c.  Do not explore links to nodes whose local-root is not the block-
       root.  This keeps the SPF confined to the particular block.

   d.  Terminate when the first IN_GADAG node z is minimized.




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   e.  Respect the existing directions (e.g.  INCOMING, OUTGOING,
       UNDIRECTED) already specified for each interface.


    Mod_SPF(spf_root, block_root)
       Initialize spf_heap to empty
       Initialize nodes' spf_metric to infinity
       spf_root.spf_metric = 0
       insert(spf_heap, spf_root)
       found_in_gadag = false
       while (spf_heap is not empty) and (found_in_gadag is false)
           min_node = remove_lowest(spf_heap)
           if min_node.IN_GADAG
              found_in_gadag = true
           else
              foreach interface intf of min_node
                 if ((intf.OUTGOING or intf.UNDIRECTED) and
                     ((intf.remote_node.localroot is block_root) or
                      (intf.remote_node is block_root)) and
                     (intf.remote_node is not TEMP_UNUSABLE) and
                     (intf is not TEMP_UNUSABLE))
                    path_metric = min_node.spf_metric + intf.metric
                    if path_metric < intf.remote_node.spf_metric
                       intf.remote_node.spf_metric = path_metric
                       intf.remote_node.spf_prev_intf = intf
                       insert_or_update(spf_heap, intf.remote_node)
       return min_node



    SPF_for_Ear(cand_intf.local_node,cand_intf.remote_node, block_root,
                method)
       Mark all interfaces between cand_intf.remote_node
                  and cand_intf.local_node as TEMP_UNUSABLE
       if cand_intf.local_node is not block_root
          Mark cand_intf.local_node as TEMP_UNUSABLE
       Initialize ear_list to empty
       end_ear = Mod_SPF(spf_root, block_root)
       y = end_ear.spf_prev_hop
       while y.local_node is not spf_root
         add_to_list_start(ear_list, y)
         y.local_node.IN_GADAG = true
         y = y.local_node.spf_prev_intf
       if(method is not hybrid)
          Set_Ear_Direction(ear_list, cand_intf.local_node,
                            end_ear,block_root)
       Clear TEMP_UNUSABLE from all interfaces between
             cand_intf.remote_node and cand_intf.local_node



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       Clear TEMP_UNUSABLE from cand_intf.local_node
    return end_ear


              Figure 31: Modified SPF for GADAG construction

   Assume that an ear is found by going from y to x and then running an
   SPF that terminates by minimizing z (e.g. y<->x...q<->z).  Now it is
   necessary to determine the direction of the ear; if y << z, then the
   path should be y->x...q->z but if y >> z, then the path should be y<-
   x...q<-z.  In Section 5.5, the same problem was handled by finding
   all ears that started at a node before looking at ears starting at
   nodes higher in the partial order.  In this GADAG construction
   method, using that approach could mean that new ears aren't added in
   order of their total cost since all ears connected to a node would
   need to be found before additional nodes could be found.

   The alternative is to track the order relationship of each node with
   respect to every other node.  This can be accomplished by maintaining
   two sets of nodes at each node.  The first set, Higher_Nodes,
   contains all nodes that are known to be ordered above the node.  The
   second set, Lower_Nodes, contains all nodes that are known to be
   ordered below the node.  This is the approach used in this GADAG
   construction method.



























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      Set_Ear_Direction(ear_list, end_a, end_b, block_root)
        // Default of A_TO_B for the following cases:
        // (a) end_a and end_b are the same (root)
        // or (b) end_a is in end_b's Lower Nodes
        // or (c) end_a and end_b were unordered with respect to each
        //        other
        direction = A_TO_B
        if (end_b is block_root) and (end_a is not end_b)
           direction = B_TO_A
        else if end_a is in end_b.Higher_Nodes
           direction = B_TO_A
        if direction is B_TO_A
           foreach interface i in ear_list
               i.UNDIRECTED = false
               i.INCOMING = true
               i.remote_intf.UNDIRECTED = false
               i.remote_intf.OUTGOING = true
        else
           foreach interface i in ear_list
               i.UNDIRECTED = false
               i.OUTGOING = true
               i.remote_intf.UNDIRECTED = false
               i.remote_intf.INCOMING = true
         if end_a is end_b
            return
         // Next, update all nodes' Lower_Nodes and Higher_Nodes
         if (end_a is in end_b.Higher_Nodes)
            foreach node x where x.localroot is block_root
                if end_a is in x.Lower_Nodes
                   foreach interface i in ear_list
                      add i.remote_node to x.Lower_Nodes
                if end_b is in x.Higher_Nodes
                   foreach interface i in ear_list
                      add i.local_node to x.Higher_Nodes
          else
            foreach node x where x.localroot is block_root
                if end_b is in x.Lower_Nodes
                   foreach interface i in ear_list
                      add i.local_node to x.Lower_Nodes
                if end_a is in x.Higher_Nodes
                   foreach interface i in ear_list
                      add i.remote_node to x.Higher_Nodes

         Figure 32: Algorithm to assign links of an ear direction

   A goal of this GADAG construction method is to find the shortest
   cycles and ears.  An ear is started by going to a neighbor x of an
   IN_GADAG node y.  The path from x to an IN_GADAG node is minimal,



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   since it is computed via SPF.  Since a shortest path is made of
   shortest paths, to find the shortest ears requires reaching from the
   set of IN_GADAG nodes to the closest node that isn't IN_GADAG.
   Therefore, an ordered tree is maintained of interfaces that could be
   explored from the IN_GADAG nodes.  The interfaces are ordered by
   their characteristics of metric, local loopback address, remote
   loopback address, and ifindex, based on the Interface_Compare
   function defined in Figure 14.

   This GADAG construction method ignores interfaces picked from the
   ordered list that belong to the block root if the block in which the
   interface is present already has an ear that has been computed.  This
   is necessary since we allow at most one incoming interface to a block
   root in each block.  This requirement stems from the way next-hops
   are computed as was seen in Section 5.7.  After any ear gets
   computed, we traverse the newly added nodes to the GADAG and insert
   interfaces whose far end is not yet on the GADAG to the ordered tree
   for later processing.

   Finally, cut-links are a special case because there is no point in
   doing an SPF on a block of 2 nodes.  The algorithm identifies cut-
   links simply as links where both ends of the link are cut-vertices.
   Cut-links can simply be added to the GADAG with both OUTGOING and
   INCOMING specified on their interfaces.

     add_eligible_interfaces_of_node(ordered_intfs_tree,node)
        for each interface of node
           if intf.remote_node.IN_GADAG is false
              insert(intf,ordered_intfs_tree)

     check_if_block_has_ear(x,block_id)
        block_has_ear = false
           for all interfaces of x
              if ( (intf.remote_node.block_id == block_id) &&
                    intf.remote_node.IN_GADAG )
                 block_has_ear = true
     return block_has_ear

     Construct_GADAG_via_SPF(topology, root)
       Compute_Localroot (root,root)
       Assign_Block_ID(root,0)
       root.IN_GADAG = true
          add_eligible_interfaces_of_node(ordered_intfs_tree,root)
       while ordered_intfs_tree is not empty
          cand_intf = remove_lowest(ordered_intfs_tree)
          if cand_intf.remote_node.IN_GADAG is false
             if L(cand_intf.remote_node) == D(cand_intf.remote_node)
                // Special case for cut-links



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                cand_intf.UNDIRECTED = false
                cand_intf.remote_intf.UNDIRECTED = false
                cand_intf.OUTGOING = true
                cand_intf.INCOMING = true
                cand_intf.remote_intf.OUTGOING = true
                cand_intf.remote_intf.INCOMING = true
                cand_intf.remote_node.IN_GADAG = true
             add_eligible_interfaces_of_node(
                            ordered_intfs_tree,cand_intf.remote_node)
          else
             if (cand_intf.remote_node.local_root ==
                 cand_intf.local_node) &&
                 check_if_block_has_ear(cand_intf.local_node,
                              cand_intf.remote_node.block_id))
                 /* Skip the interface since the block root
                 already has an incoming interface in the
                 block */
             else
             ear_end = SPF_for_Ear(cand_intf.local_node,
                     cand_intf.remote_node,
                     cand_intf.remote_node.localroot,
                     SPF method)
             y = ear_end.spf_prev_hop
             while y.local_node is not cand_intf.local_node
                 add_eligible_interfaces_of_node(
                     ordered_intfs_tree, y.local_node)
                 y = y.local_node.spf_prev_intf


            Figure 33: SPF-based method for GADAG construction

Appendix C.  Constructing a GADAG using a hybrid method

   The idea of this method is to combine the salient features of the
   lowpoint inheritance and SPF methods.  To this end, we process nodes
   as they get added to the GADAG just like in the lowpoint inheritance
   by maintaining a stack of nodes.  This ensures that we do not need to
   maintain lower and higher sets at each node to ascertain ear
   directions since the ears will always be directed from the node being
   processed towards the end of the ear.  To compute the ear however, we
   resort to an SPF to have the possibility of better ears (path
   lentghs) thus giving more flexibility than the restricted use of
   lowpoint/dfs parents.

   Regarding ears involving a block root, unlike the SPF method which
   ignored interfaces of the block root after the first ear, in the
   hybrid method we would have to process all interfaces of the block
   root before moving on to other nodes in the block since the direction



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   of an ear is pre-determined.  Thus, whenever the block already has an
   ear computed, and we are processing an interface of the block root,
   we mark the block root as unusable before the SPF run that computes
   the ear.  This ensures that the SPF terminates at some node other
   than the block-root.  This in turn guarantees that the block-root has
   only one incoming interface in each block, which is necessary for
   correctly computing the next-hops on the GADAG.

   As in the SPF gadag, bridge ears are handled as a special case.

   The entire algorithm is shown below in Figure 34

      find_spf_stack_ear(stack, x, y, xy_intf, block_root)
         if L(y) == D(y)
            // Special case for cut-links
            xy_intf.UNDIRECTED = false
            xy_intf.remote_intf.UNDIRECTED = false
            xy_intf.OUTGOING = true
            xy_intf.INCOMING = true
            xy_intf.remote_intf.OUTGOING = true
            xy_intf.remote_intf.INCOMING = true
            xy_intf.remote_node.IN_GADAG = true
            push y onto stack
            return
         else
            if (y.local_root == x) &&
                 check_if_block_has_ear(x,y.block_id)
               //Avoid the block root during the SPF
               Mark x as TEMP_UNUSABLE
            end_ear = SPF_for_Ear(x,y,block_root,hybrid)
            If x was set as TEMP_UNUSABLE, clear it
            cur = end_ear
            while (cur != y)
               intf = cur.spf_prev_hop
               prev = intf.local_node
               intf.UNDIRECTED = false
               intf.remote_intf.UNDIRECTED = false
               intf.OUTGOING = true
               intf.remote_intf.INCOMING = true
               push prev onto stack
            cur = prev
            xy_intf.UNDIRECTED = false
            xy_intf.remote_intf.UNDIRECTED = false
            xy_intf.OUTGOING = true
            xy_intf.remote_intf.INCOMING = true
            return

      Construct_GADAG_via_hybrid(topology,root)



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         Compute_Localroot (root,root)
         Assign_Block_ID(root,0)
         root.IN_GADAG = true
         Initialize Stack to empty
         push root onto Stack
         while (Stack is not empty)
            x = pop(Stack)
            for each interface intf of x
               y = intf.remote_node
               if y.IN_GADAG is false
                  find_spf_stack_ear(stack, x, y, intf, y.block_root)

                Figure 34: Hybrid GADAG construction method

Authors' Addresses

   Gabor Sandor Enyedi
   Ericsson
   Konyves Kalman krt 11
   Budapest  1097
   Hungary

   Email: Gabor.Sandor.Enyedi@ericsson.com


   Andras Csaszar
   Ericsson
   Konyves Kalman krt 11
   Budapest  1097
   Hungary

   Email: Andras.Csaszar@ericsson.com


   Alia Atlas
   Juniper Networks
   10 Technology Park Drive
   Westford, MA  01886
   USA

   Email: akatlas@juniper.net










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Internet-Draft              MRT FRR Algorithm              February 2016


   Chris Bowers
   Juniper Networks
   1194 N. Mathilda Ave.
   Sunnyvale, CA  94089
   USA

   Email: cbowers@juniper.net


   Abishek Gopalan
   University of Arizona
   1230 E Speedway Blvd.
   Tucson, AZ  85721
   USA

   Email: abishek@ece.arizona.edu



































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