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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, Ed.
Internet-Draft                                                A. Csaszar
Intended status: Standards Track                                Ericsson
Expires: January 3, 2016                                   A. Atlas, Ed.
                                                               C. Bowers
                                                        Juniper Networks
                                                              A. Gopalan
                                                   University of Arizona
                                                            July 2, 2015


  Algorithms for computing Maximally Redundant Trees for IP/LDP Fast-
                                Reroute
                 draft-ietf-rtgwg-mrt-frr-algorithm-05

Abstract

   A complete solution for IP and LDP Fast-Reroute using Maximally
   Redundant Trees is presented in [I-D.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
   Task Force (IETF).  Note that other groups may also distribute
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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 January 3, 2016.

Copyright Notice

   Copyright (c) 2015 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  . . . . . . . . . . . . . . . . . . .   7
     4.1.  Partial Ordering for Disjoint Paths . . . . . . . . . . .   7
     4.2.  Finding an Ear and the Correct Direction  . . . . . . . .   9
     4.3.  Low-Point Values and Their Uses . . . . . . . . . . . . .  11
     4.4.  Blocks in a Graph . . . . . . . . . . . . . . . . . . . .  15
     4.5.  Determining Local-Root and Assigning Block-ID . . . . . .  17
   5.  Algorithm Sections  . . . . . . . . . . . . . . . . . . . . .  19
     5.1.  Interface Ordering  . . . . . . . . . . . . . . . . . . .  19
     5.2.  MRT Island Identification . . . . . . . . . . . . . . . .  22
     5.3.  GADAG Root Selection  . . . . . . . . . . . . . . . . . .  23
     5.4.  Initialization  . . . . . . . . . . . . . . . . . . . . .  23
     5.5.  MRT Lowpoint Algorithm: Computing GADAG using lowpoint
           inheritance . . . . . . . . . . . . . . . . . . . . . . .  24
     5.6.  Augmenting the GADAG by directing all links . . . . . . .  26
     5.7.  Compute MRT next-hops . . . . . . . . . . . . . . . . . .  30
       5.7.1.  MRT next-hops to all nodes partially ordered with
               respect to the computing node . . . . . . . . . . . .  30
       5.7.2.  MRT next-hops to all nodes not partially ordered with
               respect to the computing node . . . . . . . . . . . .  31
       5.7.3.  Computing Redundant Tree next-hops in a 2-connected
               Graph . . . . . . . . . . . . . . . . . . . . . . . .  32
       5.7.4.  Generalizing for a graph that isn't 2-connected . . .  34
       5.7.5.  Complete Algorithm to Compute MRT Next-Hops . . . . .  35
     5.8.  Identify MRT alternates . . . . . . . . . . . . . . . . .  37
     5.9.  Finding FRR Next-Hops for Proxy-Nodes . . . . . . . . . .  42
   6.  MRT Lowpoint Algorithm: Next-hop conformance  . . . . . . . .  45
   7.  Python Implementation of MRT Lowpoint Algorithm . . . . . . .  45
   8.  Algorithm Alternatives and Evaluation . . . . . . . . . . . .  66
     8.1.  Algorithm Evaluation  . . . . . . . . . . . . . . . . . .  66
   9.  Implementation Status . . . . . . . . . . . . . . . . . . . .  76
   10. Algorithm Work to Be Done . . . . . . . . . . . . . . . . . .  76
   11. Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  76
   12. IANA Considerations . . . . . . . . . . . . . . . . . . . . .  76
   13. Security Considerations . . . . . . . . . . . . . . . . . . .  76
   14. References  . . . . . . . . . . . . . . . . . . . . . . . . .  76



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     14.1.  Normative References . . . . . . . . . . . . . . . . . .  76
     14.2.  Informative References . . . . . . . . . . . . . . . . .  77
   Appendix A.  Option 2: Computing GADAG using SPFs . . . . . . . .  79
   Appendix B.  Option 3: Computing GADAG using a hybrid method  . .  84
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  86

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
   downstream paths, may be prefered when available and that policy-
   based alternate selection process[I-D.ietf-rtgwg-lfa-manageability]
   is not captured in this document.








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   [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

   Algorithms for computing MRTs 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.

                        [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








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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

   network graph:   A graph that reflects the network topology where all
      links connect exactly two nodes and broadcast links have been
      transformed into a pseudo-node representation (e.g. in OSPF,
      viewing a Network LSA as representing a pseudo-noe).

   Redundant Trees (RT):  A pair of trees where the path from any node X
      to the root R on the first tree is node-disjoint with the path
      from the same node X to the root along the second tree.  These can
      be computed in 2-connected graphs.

   Maximally Redundant Trees (MRT):   A pair of trees where the path
      from any node X to the root R along the first tree and the path
      from the same node X to the root along the second tree 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.
      Any RT is an MRT but many MRTs are not RTs.

   MRT Island:   From the computing router, the set of routers that
      support a particular MRT profile and are connected.

   MRT-Red:   MRT-Red is used to describe one of the two MRTs; it is
      used to describe the associated forwarding topology and MT-ID.
      Specifically, MRT-Red is the decreasing MRT where links in the
      GADAG are taken in the direction from a higher topologically
      ordered node to a lower one.

   MRT-Blue:   MRT-Blue is used to describe one of the two MRTs; it is
      used to describe the associated forwarding topology and MT-ID.
      Specifically, MRT-Blue is the increasing MRT where links in the
      GADAG are taken in the direction from a lower topologically
      ordered node to a higher one.

   cut-vertex:   A vertex whose removal partitions the network.

   cut-link:   A link whose removal partitions the network.  A cut-link
      by definition must be connected between two cut-vertices.  If
      there are multiple parallel links, then they are referred to as
      cut-links in this document if removing the set of parallel links
      would partition the network.




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   2-connected:   A graph that has no cut-vertices.  This is a graph
      that requires two nodes to be removed before the network is
      partitioned.

   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.

   2-connected cluster:   A maximal set of nodes that are 2-connected.
      In a network graph with at least one cut-vertex, there will be
      multiple 2-connected clusters.

   block:   Either a 2-connected cluster, a cut-link, or an isolated
      vertex.

   DAG:   Directed Acyclic Graph - a digraph containing no directed
      cycle.

   ADAG:   Almost Directed Acyclic Graph - a digraph that can be
      transformed into a DAG with removing a single node (the root
      node).

   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.

   GADAG:   Generalized ADAG - a digraph, which has only ADAGs as all of
      its blocks.  The root of such a block is the node closest to the
      global root (e.g. with uniform link costs).

   DFS:    Depth-First Search

   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.





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   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.

   proxy-node:   A node added to the network graph to represent a multi-
      homed prefix or routers outside the local MRT-fast-reroute-
      supporting island of routers.  The key property of proxy-nodes is
      that traffic cannot transit them.

   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 and other algorithms for computing MRTs.  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.

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



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   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.


     [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.




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   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.

           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



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   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
   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.






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   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

   Algorithm Figure 6 merely requires that a cycle or ear be selected
   without specifying how.  Regardless of the way of 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's method using Low-Point
   Inheritance is defined in Section 5.5.  Other methods are described
   in the Appendices (Appendix A and Appendix B).

   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
   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.








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                        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),
      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.












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            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 an MRT 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 destionation 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 MRT algorithms given in
   the Appendices Appendix A and Appendix B.  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.  Algorithm Sections

   This 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.  These work on a network graph after
   its interfaces have been ordered as per Figure 14.

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

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

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

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

   5.  Construct the GADAG.  [See Section 5.5]

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

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

   8.  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.]

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.

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




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      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 (or flooding protocol in the case of ISIS-PCR
   [I-D.ietf-isis-pcr]) 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
   should be considered informational.













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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.  Also note that these values are only specified in the case
   of point-to-point links.  Therefore, in the case of IS-IS for IP/LDP
   FRR, the pseudonode number (the 7th octet) will always be zero.

   In the case of IS-IS for IP/LDP FRR, this specification allows for
   the use of Multi-Topology routing.  [RFC5120] requires that
   information related to the standard/default topology (MT-ID = 0) be
   carried in the Extended IS Reachability TLV #22, while it requires
   that the Multi-Topology IS Neighbor TLV #222 only be used to carry
   topology information related to non-default topologies (with non-zero
   MT-IDs).  [RFC5120] enforces this by requiring an implementation to
   ignore TLV#222 with MT-ID = 0.  The current document also requires
   that TLV#222 with MT-ID = 0 MUST be ignored.

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 interface in next_rtr
           if interface is (not MRT-ineligible and not IGP-excluded
                            and in area)
              if ((interface.remote_node supports profile_id) and
                  (interface.remote_node.IN_MRT_ISLAND is FALSE))
                 interface.remote_node.IN_MRT_ISLAND = TRUE
                 add_to_tail(explore_list, interface.remote_node)

                   Figure 16: MRT Island Identification






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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.  In
   [I-D.ietf-ospf-mrt] and [I-D.ietf-isis-mrt], a mechanism is given for
   routers to advertise the GADAG Root Selection Priority and
   consistently select a GADAG Root inside the local MRT Island.  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 computation, the
   network graph is reduced to contain only the set of routers that
   support the specific MRT profile whose MRTs are being computed.

   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.

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 as unusable
   using standard IGP mechanisms (see section 7 of
   [I-D.ietf-rtgwg-mrt-frr-architecture]).  Due to FRR manageability
   considerations [I-D.ietf-rtgwg-lfa-manageability], it may also be
   desirable to administratively configure some interfaces as ineligible
   to carry MRT FRR traffic.  This constraint MUST be consistently
   flooded via the IGP [I-D.ietf-ospf-mrt] [I-D.ietf-isis-mrt] by the
   owner of the interface, so that links are clearly known to be MRT-
   ineligible and not explored or used in the MRT algorithm.  In the
   algorithm description, it is assumed that such links and nodes will
   not be explored or used, and no more discussion is given of this
   restriction.





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5.5.  MRT Lowpoint Algorithm: Computing 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 algorithm 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 partially 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 partially 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 considered is when the root R computes its own
   next-hops.  Since the 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 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 is 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
         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
      else if D_higher



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          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
      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
      else  //D is unordered wrt S
          if F.HIGHER and F.LOWER
              if primary_intf.OUTGOING and primary_intf.INCOMING
                  // this case should not occur
              if primary_intf.OUTGOING
                  return USE_BLUE
              if primary_intf.INCOMING
                  return USE_RED
          if F.LOWER
              return USE_RED
          if F.HIGHER
              return USE_BLUE

  Select_Alternates(D, F, primary_intf)
      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

   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



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   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 pseudocode 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
   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          |



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| 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. |      |                 | S to D or  |            |
|      | Red path:  |      |                 | 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    |            |
+------+------------+------+-----------------+------------+------------+
| 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     |            |
|      | 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    |                 |            |            |
+------+------------+------+-----------------+------------+------------+
| 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     |            |



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|      | 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    |
|      |            | and  | criteria        |            | to avoid   |
|      |            | F<<S,| not             |            | F          |
|      |            | F is | needed          |            |            |
|      |            | 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 H>>D, |      |                 | S to H.    |            |
|      | 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 G     |            |
|      | node G<<D, |      |                 |            |            |
|      | then decr. |      |                 |            |            |
|      |            +------+-----------------+------------+------------+
|      |            | 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      | Implies F is the order  |
|      |            |      | direction       | proxy for D, which has  |
|      |            |      | S<-->F          | already been handled.   |
+------+------------+------+-----------------+------------+------------+

     Figure 25: determining MRT next-hops and alternates based on the
       partial order and topological sort relationships between the



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    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.

   As an example, 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 not smaller
   than H, so we should select the decreasing path towards the root.
   If, however, the destination were instead J, we 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.  Finding FRR Next-Hops for Proxy-Nodes

   As discussed in Section 10.2 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 a named proxy-
   nodes.  An example case is for a router that is not part of that
   local MRT Island, when there is only partial MRT support in the
   domain.

   A first incorrect and naive approach to handling proxy-nodes, which
   cannot be transited, is to simply add these proxy-nodes to the graph
   of the network and connect it to the routers through which the new
   proxy-node can be reached.  Unfortunately, this can introduce some
   new ordering between the border routers connected to the new node
   which could result in routing MRT paths through the proxy-node.
   Thus, this naive approach would need to recompute GADAGs and redo
   SPTs for each proxy-node.




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   Instead of adding the proxy-node to the original network graph, each
   individual proxy-node can be individually added to the GADAG.  The
   proxy-node is connected to at most two nodes in the GADAG.
   Section 10.2 of [I-D.ietf-rtgwg-mrt-frr-architecture] defines how the
   proxy-node attachments MUST be determined.  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 attachment node;
   if there are different interfaces, then some can be assigned to MRT-
   Red and others to MRT_Blue.

   Now, consider the proxy-node that is attached to exactly two nodes in
   the GADAG.  Let the order_proxies of these nodes be A and B.  Let the
   current node, where next-hop is just being calculated, be S.  If one
   of these two nodes A and B is the local root of S, let A=S.local_root
   and the other one be B.  Otherwise, let A.topo_order < B.topo_order.

   A valid GADAG was constructed.  Instead doing an increasing-SPF and a
   decreasing-SPF to find ordering for the proxy-nodes, the following
   simple rules, providing the same result, can be used independently
   for each different proxy-node.  For the following rules, let
   X=A.local_root, and if A is the local root, let that be strictly
   lower than any other node.  Always take the first rule that matches.


   Rule   Condition     Blue NH      Red NH        Notes
    1       S=X         Blue to A    Red to B
    2       S<<A        Blue to A    Red to R
    3       S>>B        Blue to R    Red to B
    4       A<<S<<B     Red to A     Blue to B
    5       A<<S        Red to A     Blue to R     S not ordered w/ B
    6       S<<B        Red to R     Blue to B     S not ordered w/ A
    7     Otherwise     Red to R     Blue to R     S not ordered w/ A+B

   These rules are realized in the following pseudocode where P is the
   proxy-node, X and Y are the nodes that P is attached to, and S is the
   computing router:















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   Select_Proxy_Node_NHs(P, S, X, Y)
       if (X.order_proxy.topo_order < Y.order_proxy.topo_order)
           //This fits even if X.order_proxy=S.local_root
           A=X.order_proxy
           B=Y.order_proxy
       else
           A=Y.order_proxy
           B=X.order_proxy

       if (S==A.local_root)
           P.blue_next_hops = A.blue_next_hops
           P.red_next_hops  = B.red_next_hops
           return
       if (A.higher)
           P.blue_next_hops = A.blue_next_hops
           P.red_next_hops  = R.red_next_hops
           return
       if (B.lower)
           P.blue_next_hops = R.blue_next_hops
           P.red_next_hops  = B.red_next_hops
           return
       if (A.lower && B.higher)
           P.blue_next_hops = A.red_next_hops
           P.red_next_hops  = B.blue_next_hops
           return
       if (A.lower)
           P.blue_next_hops = R.red_next_hops
           P.red_next_hops  = B.blue_next_hops
           return
       if (B.higher)
           P.blue_next_hops = A.red_next_hops
           P.red_next_hops  = R.blue_next_hops
           return
       P.blue_next_hops = R.red_next_hops
       P.red_next_hops  = R.blue_next_hops
       return


   After finding the the red and the blue next-hops, it is necessary to
   know which one of these to use in the case of failure.  This can be
   done by Select_Alternates_Inner().  In order to use
   Select_Alternates_Internal(), we need to know if P is greater, less
   or unordered with S, and P.topo_order.  P.lower = B.lower, P.higher =
   A.higher, and any value is OK for P.topo_order, as long as
   A.topo_order<=P.topo_order<=B.topo_order and P.topo_order is not
   equal to the topo_order of the failed node.  So for simplicity let
   P.topo_order=A.topo_order when the next-hop is not A, and




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   P.topo_order=B.topo_order otherwise.  This gives the following
   pseudo-code:


     Select_Alternates_Proxy_Node(S, P, F, primary_intf)
        if (F is not P.neighbor_A)
           return Select_Alternates_Internal(S, P, F, primary_intf,
                                             P.neighbor_B.lower,
                                             P.neighbor_A.higher,
                                             P.neighbor_A.topo_order)
        else
           return Select_Alternates_Internal(S, P, F, primary_intf,
                                             P.neighbor_B.lower,
                                             P.neighbor_A.higher,
                                             P.neighbor_B.topo_order)

                                 Figure 27

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.  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.

 <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).



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 # 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 heapq

 # simple Class definitions allow structure-like dot notation for
 # variables and a convenient place to initialize those variables.
 class Topology:
     pass

 class Node:
     pass

 class Interface:
     pass

 class Bundle:
     pass

 class Alternate:
     def __init__(self):
         self.failed_intf = None
         self.nh_list = []
         self.fec = 'NO_ALTERNATE'
         self.prot = 'NO_PROTECTION'
         self.info = 'NONE'


 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




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

 def Initialize_Node(node):
     node.intf_list = []
     node.island_intf_list = []
     node.profile_id_list = [0]
     node.GR_sel_priority = 128
     node.IN_MRT_ISLAND = False
     node.IN_GADAG = False
     node.dfs_number = None
     node.dfs_parent = None
     node.dfs_parent_intf = None
     node.dfs_child_list = []
     node.lowpoint_number = None
     node.lowpoint_parent = None
     node.lowpoint_parent_intf = None
     node.localroot = None
     node.block_id = None
     node.IS_CUT_VERTEX = False
     node.blue_next_hops_dict = {}
     node.red_next_hops_dict = {}
     node.pnh_dict = {}
     node.alt_dict = {}

 def Initialize_Intf(intf):
     intf.metric = None
     intf.area = None
     intf.MRT_INELIGIBLE = False
     intf.IGP_EXCLUDED = False
     intf.UNDIRECTED = True
     intf.INCOMING = False
     intf.OUTGOING = False
     intf.INCOMING_STORED = False
     intf.OUTGOING_STORED = False
     intf.PROCESSED = False
     intf.IN_MRT_ISLAND = False

 def Reset_Computed_Node_and_Intf_Values(topo):
     for node in topo.node_list:
         node.IN_MRT_ISLAND = False
         node.IN_GADAG = False
         node.dfs_number = None
         node.dfs_parent = None
         node.dfs_parent_intf = None
         node.dfs_child_list = []
         node.lowpoint_number = None



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         node.lowpoint_parent = None
         node.lowpoint_parent_intf = None
         node.localroot = None
         node.block_id = None
         node.IS_CUT_VERTEX = False
         for intf in node.intf_list:
             intf.UNDIRECTED = True
             intf.INCOMING = False
             intf.OUTGOING = False
             intf.INCOMING_STORED = False
             intf.OUTGOING_STORED = False
             intf.IN_MRT_ISLAND = False


 # 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()
     topo.gadag_root = None
     topo.node_list = []
     topo.node_dict = {}
     topo.island_node_list = []
     topo.prefix_list = [] # possibly no longer needed
     node_id_set= set()
     cols_list = []
     # on first pass just create nodes
     with open(filename) 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)



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             node_id_set.add(nodeb_node_id)
     for node_id in node_id_set:
         node = Node()
         node.node_id = node_id
         Initialize_Node(node)
         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:
             reverse_metric=int(cols[3])
         nodea = topo.node_dict[nodea_node_id]
         nodeb = topo.node_dict[nodeb_node_id]
         nodea_intf = Interface()
         Initialize_Intf(nodea_intf)
         nodea_intf.metric = metric
         nodea_intf.area = 0
         nodeb_intf = Interface()
         Initialize_Intf(nodeb_intf)
         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.IGP_EXCLUDED
                  and intf.area == area ):



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                 if (profile_id in intf.remote_node.profile_id_list):
                     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 Set_Island_Intf_and_Node_Lists(topo):
     topo.island_node_list = []
     for node in topo.node_list:
         node.island_intf_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



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     Lowpoint_Visit(topo.gadag_root, None, None)

 # addresses these cases.
 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:
         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



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     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
     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()
                 bundle.UNDIRECTED = True
                 bundle.OUTGOING = False
                 bundle.INCOMING = False
                 for intf2 in node.island_intf_list:



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                     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
                 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:



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                 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
                         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:



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                     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

 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)



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     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,
                    (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:



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                                 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]
         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)



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         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,
                                 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:



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         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_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)
         if P.convert_blue_to_green:
             x.blue_to_green_nh_dict[P.node_id] = True
         if P.convert_red_to_green:
             x.red_to_green_nh_dict[P.node_id] = True

 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_Source_To_Nodes(topo,x):
     for y in topo.node_list:
         x.pnh_dict[y.node_id] = []
         Copy_List_Items(x.pnh_dict[y.node_id], y.primary_next_hops)

 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)



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 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(False)
     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'
             assert(False)
         assert(False)
     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(False)
     else: # D is unordered wrt S
         if F.HIGHER and F.LOWER:
             if primary_intf.OUTGOING and primary_intf.INCOMING:
                 assert(False)
             if primary_intf.OUTGOING:
                 # this case isn't hit it topo-9e
                 return 'USE_BLUE'
             if primary_intf.INCOMING:
                 return 'USE_RED'
             assert(False)



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         if F.LOWER:
             return 'USE_RED'
         if F.HIGHER:
             return 'USE_BLUE'
         assert(False)

 def Select_Alternates(D, F, primary_intf):
     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 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 primary_intf in D.primary_next_hops:
             alt = Alternate()
             alt.failed_intf = primary_intf
             if primary_intf in x.island_intf_list:
                 alt.info = Select_Alternates(D,
                     primary_intf.remote_node, primary_intf)
             else:
                 alt.info = 'PRIM_NH_NOT_IN_ISLAND'
                 Copy_List_Items(alt.nh_list, D.blue_next_hops)
                 alt.fec = 'BLUE'
                 alt.prot = 'NODE_PROTECTION'
             if (alt.info == 'USE_BLUE'):
                 Copy_List_Items(alt.nh_list, D.blue_next_hops)
                 alt.fec = 'BLUE'
                 alt.prot = 'NODE_PROTECTION'
             if (alt.info == 'USE_RED'):
                 Copy_List_Items(alt.nh_list, D.red_next_hops)
                 alt.fec = 'RED'
                 alt.prot = 'NODE_PROTECTION'
             if (alt.info == 'PRIM_NH_IS_D_OR_OP_FOR_D'):
                 if primary_intf.OUTGOING and primary_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:



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                         if ( intf is not primary_intf and
                              (intf.remote_node is
                              primary_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)
                 elif primary_intf in D.red_next_hops:
                     Copy_List_Items(alt.nh_list, D.blue_next_hops)
                     alt.fec = 'BLUE'
                     alt.prot = 'LINK_PROTECTION'
                 else:
                     Copy_List_Items(alt.nh_list, D.red_next_hops)
                     alt.fec = 'RED'
                     alt.prot = 'LINK_PROTECTION'
             D.alt_list.append(alt)

 def Write_GADAG_To_File(topo, file_prefix):
     gadag_edge_list = []
     for node in topo.island_node_list:
         for intf in node.island_intf_list:
             if intf.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:
         if color == 'blue':



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             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)
     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 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 = \



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                             "%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

 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()





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 def Run_MRT_for_One_Source(topo, src):
     Reset_Computed_Node_and_Intf_Values(topo)
     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)

 def Run_Prim_SPF_for_One_Source(topo,src):
     Normal_SPF(topo, src)
     Store_Primary_NHs_For_One_Source_To_Nodes(topo,src)

 def Run_MRT_for_All_Sources(topo):
     for src in topo.node_list:
         if 0 in src.profile_id_list:
             # src runs MRT if it has profile_id=0
             Run_MRT_for_One_Source(topo,src)
         else:
             # still run SPF for nodes not running MRT
             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_Example_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, '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])+','+



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                         str(item[2])+'\n')
             topo_file.write(line)

 def Generate_Example_Topology_and_Run_MRT():
     Create_Example_Topology_Input_File('example_topo_input_file.csv')
     topo = Create_Topology_From_File('example_topo_input_file.csv')
     res_file_base = 'example_topo'
     Raise_GADAG_Root_Selection_Priority(topo,3)
     Run_MRT_for_All_Sources(topo)
     Write_Output_To_Files(topo, res_file_base)

 Generate_Example_Topology_and_Run_MRT()

 <CODE ENDS>

8.  Algorithm Alternatives and Evaluation

   This specification defines the MRT Lowpoint Algorithm, which is one
   option among several possible MRT algorithms.  Other alternatives are
   described in the appendices.

   In addition, it is possible to calculate Destination-Rooted GADAG,
   where for each destination, a GADAG rooted at that destination is
   computed.  Then a router can compute the blue MRT and red MRT next-
   hops to that destination.  Building GADAGs per destination is
   computationally more expensive, but may give somewhat shorter
   alternate paths.  It may be useful for live-live multicast along
   MRTs.

8.1.  Algorithm Evaluation

   The MRT Lowpoint algorithm is the lowest computation of the MRT
   algorithms.  Two other MRT algorithms are provided in Appendix A and
   Appendix B.  When analyzed on service provider network topologies,
   they did not provide significant differences in the path lenghts for
   the alternatives.  This section does not focus on that analysis or
   the decision to use the MRT Lowpoint algorithm as the default MRT
   algorithm; it has the lowest computational and storage requirements
   and gave comparable results.

   Since this document defines the MRT Lowpoint algorithm for use in
   fast-reroute applications, it is useful to compare MRT and Remote LFA
   [RFC7490].  This section compares MRT and remote LFA for IP Fast
   Reroute in 19 service provider network topologies, focusing on
   coverage and alternate path length.  Figure 28 shows the node-
   protecting coverage provided by local LFA (LLFA), remote LFA (RLFA),
   and MRT against different failure scenarios in these topologies.  The
   coverage values are calculated as the percentage of source-



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   destination pairs protected by the given IPFRR method relative to
   those protectable by optimal routing, against the same failure modes.
   More details on alternate selection policies used for this analysis
   are described later in this section.

               +------------+-----------------------------+
               |  Topology  |    percentage of failure    |
               |            |    scenarios covered by     |
               |            |        IPFRR method         |
               |            |-----------------------------+
               |            | NP_LLFA | NP_RLFA |   MRT   |
               +------------+---------+---------+---------+
               |    T201    |   37    |   90    |   100   |
               |    T202    |   73    |   83    |   100   |
               |    T203    |   51    |   80    |   100   |
               |    T204    |   55    |   81    |   100   |
               |    T205    |   92    |   93    |   100   |
               |    T206    |   71    |   74    |   100   |
               |    T207    |   57    |   74    |   100   |
               |    T208    |   66    |   81    |   100   |
               |    T209    |   79    |   79    |   100   |
               |    T210    |   95    |   98    |   100   |
               |    T211    |   68    |   71    |   100   |
               |    T212    |   59    |   63    |   100   |
               |    T213    |   84    |   84    |   100   |
               |    T214    |   68    |   78    |   100   |
               |    T215    |   84    |   88    |   100   |
               |    T216    |   43    |   59    |   100   |
               |    T217    |   78    |   88    |   100   |
               |    T218    |   72    |   75    |   100   |
               |    T219    |   78    |   84    |   100   |
               +------------+---------+---------+---------+

                                 Figure 28

   For the topologies analyzed here, LLFA is able to provide node-
   protecting coverage ranging from 37% to 95% of the source-destination
   pairs, as seen in the column labeled NP_LLFA.  The use of RLFA in
   addition to LLFA is generally able to increase the node-protecting
   coverage.  The percentage of node-protecting coverage with RLFA is
   provided in the column labeled NP_RLFA, ranges from 59% to 98% for
   these topologies.  The node-protecting coverage provided by MRT is
   100% since MRT is able to provide protection for any source-
   destination pair for which a path still exists after the failure.

   We would also like to measure the quality of the alternate paths
   produced by these different IPFRR methods.  An obvious approach is to
   take an average of the alternate path costs over all source-



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   destination pairs and failure modes.  However, this presents a
   problem, which we will illustrate by presenting an example of results
   for one topology using this approach ( Figure 29).  In this table,
   the average relative path length is the alternate path length for the
   IPFRR method divided by the optimal alternate path length, averaged
   over all source-destination pairs and failure modes.  The first three
   columns of data in the table give the path length calculated from the
   sum of IGP metrics of the links in the path.  The results for
   topology T208 show that the metric-based path lengths for NP_LLFA and
   NP_RLFA alternates are on average 78 and 66 times longer than the
   path lengths for optimal alternates.  The metric-based path lengths
   for MRT alternates are on average 14 times longer than for optimal
   alternates.

        +--------+------------------------------------------------+
        |        |     average relative alternate path length     |
        |        |-----------------------+------------------------+
        |Topology|      IGP metric       |       hopcount         |
        |        |-----------------------+------------------------+
        |        |NP_LLFA |NP_RLFA | MRT |NP_LLFA |NP_RLFA | MRT  |
        +--------+--------+--------+-----+--------+--------+------+
        |  T208  |  78.2  |   66.0 | 13.6|  0.99  |  1.01  | 1.32 |
        +--------+--------+--------+-----+--------+--------+------+

                                 Figure 29

   The network topology represented by T208 uses values of 10, 100, and
   1000 as IGP costs, so small deviations from the optimal alternate
   path can result in large differences in relative path length.  LLFA,
   RLFA, and MRT all allow for at least one hop in the alterate path to
   be chosen independent of the cost of the link.  This can easily
   result in an alternate using a link with cost 1000, which introduces
   noise into the path length measurement.  In the case of T208, the
   adverse effects of using metric-based path lengths is obvious.
   However, we have observed that the metric-based path length
   introduces noise into alternate path length measurements in several
   other topologies as well.  For this reason, we have opted to measure
   the alternate path length using hopcount.  While IGP metrics may be
   adjusted by the network operator for a number of reasons (e.g.
   traffic engineering), the hopcount is a fairly stable measurement of
   path length.  As shown in the last three columns of Figure 29, the
   hopcount-based alternate path lengths for topology T208 are fairly
   well-behaved.

   Figure 30, Figure 31, Figure 32, and Figure 33 present the hopcount-
   based path length results for the 19 topologies examined.  The
   topologies in the four tables are grouped based on the size of the
   topologies, as measured by the number of nodes, with Figure 30 having



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   the smallest topologies and Figure 33 having the largest topologies.
   Instead of trying to represent the path lengths of a large set of
   alternates with a single number, we have chosen to present a
   histogram of the path lengths for each IPFRR method and alternate
   selection policy studied.  The first eight colums of data represent
   the percentage of failure scenarios protected by an alternate N hops
   longer than the primary path, with the first column representing an
   alternate 0 or 1 hops longer than the primary path, all the way up
   through the eighth column respresenting an alternate 14 or 15 hops
   longer than the primary path.  The last column in the table gives the
   percentage of failure scenarios for which there is no alternate less
   than 16 hops longer than the primary path.  In the case of LLFA and
   RLFA, this category includes failure scenarios for which no alternate
   was found.

   For each topology, the first row (labeled OPTIMAL) is the
   distribution of the number of hops in excess of the primary path
   hopcount for optimally routed alternates.  (The optimal routing was
   done with respect to IGP metrics, as opposed to hopcount.)  The
   second row(labeled NP_LLFA) is the distribution of the extra hops for
   node-protecting LLFA.  The third row (labeled NP_LLFA_THEN_NP_RLFA)
   is the hopcount distribution when one adds node-protecting RLFA to
   increase the coverage.  The alternate selection policy used here
   first tries to find a node-protecting LLFA.  If that does not exist,
   then it tries to find an RLFA, and checks if it is node-protecting.
   Comparing the hopcount distribution for RLFA and LLFA across these
   topologies, one can see how the coverage is increased at the expense
   of using longer alternates.  It is also worth noting that while
   superficially LLFA and RLFA appear to have better hopcount
   distributions than OPTIMAL, the presence of entries in the last
   column (no alternate < 16) mainly represent failure scenarios that
   are not protected, for which the hopcount is effectively infinite.

   The fourth and fifth rows of each topology show the hopcount
   distributions for two alternate selection policies using MRT
   alternates.  The policy represented by the label
   NP_LLFA_THEN_MRT_LOWPOINT will first use a node-protecting LLFA.  If
   a node-protecting LLFA does not exist, then it will use an MRT
   alternate.  The policy represented by the label MRT_LOWPOINT instead
   will use the MRT alternate even if a node-protecting LLFA exists.
   One can see from the data that combining node-protecting LLFA with
   MRT results in a significant shortening of the alternate hopcount
   distribution.








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   +-------------------------------------------------------------------+
   |                              |   percentage of failure scenarios  |
   |        Topology name         |  protected by an alternate N hops  |
   |             and              |   longer than the primary path     |
   |     alternate selection      +------------------------------------+
   |       policy evaluated       |   |   |   |   |   |   |   |   | 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)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 37| 37| 20|  3|  3|   |   |   |    |
   |            NP_LLFA           | 37|   |   |   |   |   |   |   |  63|
   |     NP_LLFA_THEN_NP_RLFA     | 37| 34| 19|   |   |   |   |   |  10|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 37| 33| 21|  6|  3|   |   |   |    |
   |         MRT_LOWPOINT         | 33| 36| 23|  6|  3|   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T202(avg primary hops=4.8)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 90|  9|   |   |   |   |   |   |    |
   |            NP_LLFA           | 71|  2|   |   |   |   |   |   |  27|
   |     NP_LLFA_THEN_NP_RLFA     | 78|  5|   |   |   |   |   |   |  17|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 80| 12|  5|  2|  1|   |   |   |    |
   |       MRT_LOWPOINT_ONLY      | 48| 29| 13|  7|  2|  1|   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T203(avg primary hops=4.1)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 36| 37| 21|  4|  2|   |   |   |    |
   |            NP_LLFA           | 34| 15|  3|   |   |   |   |   |  49|
   |     NP_LLFA_THEN_NP_RLFA     | 35| 19| 22|  4|   |   |   |   |  20|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 36| 35| 22|  5|  2|   |   |   |    |
   |       MRT_LOWPOINT_ONLY      | 31| 35| 26|  7|  2|   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T204(avg primary hops=3.7)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 76| 20|  3|  1|   |   |   |   |    |
   |            NP_LLFA           | 54|  1|   |   |   |   |   |   |  45|
   |     NP_LLFA_THEN_NP_RLFA     | 67| 10|  4|   |   |   |   |   |  19|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 70| 18|  8|  3|  1|   |   |   |    |
   |       MRT_LOWPOINT_ONLY      | 58| 27| 11|  3|  1|   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T205(avg primary hops=3.4)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 92|  8|   |   |   |   |   |   |    |
   |            NP_LLFA           | 89|  3|   |   |   |   |   |   |   8|
   |     NP_LLFA_THEN_NP_RLFA     | 90|  4|   |   |   |   |   |   |   7|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 91|  9|   |   |   |   |   |   |    |
   |       MRT_LOWPOINT_ONLY      | 62| 33|  5|  1|   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+

                                 Figure 30





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   +-------------------------------------------------------------------+
   |                              |   percentage of failure scenarios  |
   |        Topology name         |  protected by an alternate N hops  |
   |             and              |   longer than the primary path     |
   |     alternate selection      +------------------------------------+
   |       policy evaluated       |   |   |   |   |   |   |   |   | no |
   |                              |   |   |   |   |   |10 |12 |14 | alt|
   |                              |0-1|2-3|4-5|6-7|8-9|-11|-13|-15| <16|
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T206(avg primary hops=3.7)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 63| 30|  7|   |   |   |   |   |    |
   |            NP_LLFA           | 60|  9|  1|   |   |   |   |   |  29|
   |     NP_LLFA_THEN_NP_RLFA     | 60| 13|  1|   |   |   |   |   |  26|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 64| 29|  7|   |   |   |   |   |    |
   |         MRT_LOWPOINT         | 55| 32| 13|   |   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T207(avg primary hops=3.9)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 71| 24|  5|  1|   |   |   |   |    |
   |            NP_LLFA           | 55|  2|   |   |   |   |   |   |  43|
   |     NP_LLFA_THEN_NP_RLFA     | 63| 10|   |   |   |   |   |   |  26|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 70| 20|  7|  2|  1|   |   |   |    |
   |       MRT_LOWPOINT_ONLY      | 57| 29| 11|  3|  1|   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T208(avg primary hops=4.6)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 58| 28| 12|  2|  1|   |   |   |    |
   |            NP_LLFA           | 53| 11|  3|   |   |   |   |   |  34|
   |     NP_LLFA_THEN_NP_RLFA     | 56| 17|  7|  1|   |   |   |   |  19|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 58| 19| 10|  7|  3|  1|   |   |    |
   |       MRT_LOWPOINT_ONLY      | 34| 24| 21| 13|  6|  2|  1|   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T209(avg primary hops=3.6)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 85| 14|  1|   |   |   |   |   |    |
   |            NP_LLFA           | 79|   |   |   |   |   |   |   |  21|
   |     NP_LLFA_THEN_NP_RLFA     | 79|   |   |   |   |   |   |   |  21|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 82| 15|  2|   |   |   |   |   |    |
   |       MRT_LOWPOINT_ONLY      | 63| 29|  8|   |   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T210(avg primary hops=2.5)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 95|  4|  1|   |   |   |   |   |    |
   |            NP_LLFA           | 94|  1|   |   |   |   |   |   |   5|
   |     NP_LLFA_THEN_NP_RLFA     | 94|  3|  1|   |   |   |   |   |   2|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 95|  4|  1|   |   |   |   |   |    |
   |       MRT_LOWPOINT_ONLY      | 91|  6|  2|   |   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+

                                 Figure 31





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   +-------------------------------------------------------------------+
   |                              |   percentage of failure scenarios  |
   |        Topology name         |  protected by an alternate N hops  |
   |             and              |   longer than the primary path     |
   |     alternate selection      +------------------------------------+
   |       policy evaluated       |   |   |   |   |   |   |   |   | no |
   |                              |   |   |   |   |   |10 |12 |14 | alt|
   |                              |0-1|2-3|4-5|6-7|8-9|-11|-13|-15| <16|
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T211(avg primary hops=3.3)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 88| 11|   |   |   |   |   |   |    |
   |            NP_LLFA           | 66|  1|   |   |   |   |   |   |  32|
   |     NP_LLFA_THEN_NP_RLFA     | 68|  3|   |   |   |   |   |   |  29|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 88| 12|   |   |   |   |   |   |    |
   |         MRT_LOWPOINT         | 85| 15|  1|   |   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T212(avg primary hops=3.5)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 76| 23|  1|   |   |   |   |   |    |
   |            NP_LLFA           | 59|   |   |   |   |   |   |   |  41|
   |     NP_LLFA_THEN_NP_RLFA     | 61|  1|  1|   |   |   |   |   |  37|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 75| 24|  1|   |   |   |   |   |    |
   |       MRT_LOWPOINT_ONLY      | 66| 31|  3|   |   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T213(avg primary hops=4.3)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 91|  9|   |   |   |   |   |   |    |
   |            NP_LLFA           | 84|   |   |   |   |   |   |   |  16|
   |     NP_LLFA_THEN_NP_RLFA     | 84|   |   |   |   |   |   |   |  16|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 89| 10|  1|   |   |   |   |   |    |
   |       MRT_LOWPOINT_ONLY      | 75| 24|  1|   |   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T214(avg primary hops=5.8)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 71| 22|  5|  2|   |   |   |   |    |
   |            NP_LLFA           | 58|  8|  1|  1|   |   |   |   |  32|
   |     NP_LLFA_THEN_NP_RLFA     | 61| 13|  3|  1|   |   |   |   |  22|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 66| 14|  7|  5|  3|  2|  1|  1|   1|
   |       MRT_LOWPOINT_ONLY      | 30| 20| 18| 12|  8|  4|  3|  2|   3|
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T215(avg primary hops=4.8)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 73| 27|   |   |   |   |   |   |    |
   |            NP_LLFA           | 73| 11|   |   |   |   |   |   |  16|
   |     NP_LLFA_THEN_NP_RLFA     | 73| 13|  2|   |   |   |   |   |  12|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 74| 19|  3|  2|  1|  1|  1|   |    |
   |       MRT_LOWPOINT_ONLY      | 32| 31| 16| 12|  4|  3|  1|   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+

                                 Figure 32





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   +-------------------------------------------------------------------+
   |                              |   percentage of failure scenarios  |
   |        Topology name         |  protected by an alternate N hops  |
   |             and              |   longer than the primary path     |
   |     alternate selection      +------------------------------------+
   |       policy evaluated       |   |   |   |   |   |   |   |   | no |
   |                              |   |   |   |   |   |10 |12 |14 | alt|
   |                              |0-1|2-3|4-5|6-7|8-9|-11|-13|-15| <16|
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T216(avg primary hops=5.2)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 60| 32|  7|  1|   |   |   |   |    |
   |            NP_LLFA           | 39|  4|   |   |   |   |   |   |  57|
   |     NP_LLFA_THEN_NP_RLFA     | 46| 12|  2|   |   |   |   |   |  41|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 48| 20| 12|  7|  5|  4|  2|  1|   1|
   |         MRT_LOWPOINT         | 28| 25| 18| 11|  7|  6|  3|  2|   1|
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T217(avg primary hops=8.0)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 81| 13|  5|  1|   |   |   |   |    |
   |            NP_LLFA           | 74|  3|  1|   |   |   |   |   |  22|
   |     NP_LLFA_THEN_NP_RLFA     | 76|  8|  3|  1|   |   |   |   |  12|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 77|  7|  5|  4|  3|  2|  1|  1|    |
   |       MRT_LOWPOINT_ONLY      | 25| 18| 18| 16| 12|  6|  3|  1|    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T218(avg primary hops=5.5)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 85| 14|  1|   |   |   |   |   |    |
   |            NP_LLFA           | 68|  3|   |   |   |   |   |   |  28|
   |     NP_LLFA_THEN_NP_RLFA     | 71|  4|   |   |   |   |   |   |  25|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 77| 12|  7|  4|  1|   |   |   |    |
   |       MRT_LOWPOINT_ONLY      | 37| 29| 21| 10|  3|  1|   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T219(avg primary hops=7.7)  |   |   |   |   |   |   |   |   |    |
   |            OPTIMAL           | 77| 15|  5|  1|  1|   |   |   |    |
   |            NP_LLFA           | 72|  5|   |   |   |   |   |   |  22|
   |     NP_LLFA_THEN_NP_RLFA     | 73|  8|  2|   |   |   |   |   |  16|
   |   NP_LLFA_THEN_MRT_LOWPOINT  | 74|  8|  3|  3|  2|  2|  2|  2|   4|
   |       MRT_LOWPOINT_ONLY      | 19| 14| 15| 12| 10|  8|  7|  6|  10|
   +------------------------------+---+---+---+---+---+---+---+---+----+

                                 Figure 33

   In the preceding analysis, the following procedure for selecting an
   RLFA was used.  Nodes were ordered with respect to distance from the
   source and checked for membership in Q and P-space.  The first node
   to satisfy this condition was selected as the RLFA.  More
   sophisticated methods to select node-protecting RLFAs is an area of
   active research.





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   The analysis presented above uses the MRT Lowpoint Algorithm defined
   in this specification with a common GADAG root.  The particular
   choice of a common GADAG root is expected to affect the quality of
   the MRT alternate paths, with a more central common GADAG root
   resulting in shorter MRT alternate path lengths.  For the analysis
   above, the GADAG root was chosen for each topology by calculating
   node centrality as the sum of costs of all shortest paths to and from
   a given node.  The node with the lowest sum was chosen as the common
   GADAG root.  In actual deployments, the common GADAG root would be
   chosen based on the GADAG Root Selection Priority advertised by each
   router, the values of which would be determined off-line.

   In order to measure how sensitive the MRT alternate path lengths are
   to the choice of common GADAG root, we performed the same analysis
   using different choices of GADAG root.  All of the nodes in the
   network were ordered with respect to the node centrality as computed
   above.  Nodes were chosen at the 0th, 25th, and 50th percentile with
   respect to the centrality ordering, with 0th percentile being the
   most central node.  The distribution of alternate path lengths for
   those three choices of GADAG root are shown in Figure 34 for a subset
   of the 19 topologies (chosen arbitrarily).  The third row for each
   topology (labeled MRT_LOWPOINT ( 0 percentile) ) reproduces the
   results presented above for MRT_LOWPOINT_ONLY.  The fourth and fifth
   rows show the alternate path length distibution for the 25th and 50th
   percentile choice for GADAG root.  One can see some impact on the
   path length distribution with the less central choice of GADAG root
   resulting in longer path lenghths.

   We also looked at the impact of MRT algorithm variant on the
   alternate path lengths.  The first two rows for each topology present
   results of the same alternate path length distribution analysis for
   the SPF and Hybrid methods for computing the GADAG.  These two
   methods are described in Appendix A and Appendix B.  For three of the
   topologies in this subset (T201, T206, and T211), the use of SPF or
   Hybrid methods does not appear to provide a significant advantage
   over the Lowpoint method with respect to path length.  Instead, the
   choice of GADAG root appears to have more impact on the path length.
   However, for two of the topologies in this subset(T216 and T219) and
   for this particular choice of GAGAG root, the use of the SPF method
   results in noticeably shorter alternate path lengths than the use of
   the Lowpoint or Hybrid methods.  It remains to be determined if this
   effect applies generally across more topologies or is sensitive to
   choice of GADAG root.








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   +-------------------------------------------------------------------+
   |        Topology name         |   percentage of failure scenarios  |
   |                              |  protected by an alternate N hops  |
   |     MRT algorithm variant    |   longer than the primary path     |
   |                              +------------------------------------+
   |          (GADAG root         |   |   |   |   |   |   |   |   | no |
   |     centrality percentile)   |   |   |   |   |   |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 ( 0 percentile) | 33| 26| 23|  6|  3|   |   |   |    |
   |      MRT_SPF ( 0 percentile) | 33| 36| 23|  6|  3|   |   |   |    |
   | MRT_LOWPOINT ( 0 percentile) | 33| 36| 23|  6|  3|   |   |   |    |
   | MRT_LOWPOINT (25 percentile) | 27| 29| 23| 11| 10|   |   |   |    |
   | MRT_LOWPOINT (50 percentile) | 27| 29| 23| 11| 10|   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T206(avg primary hops=3.7)  |   |   |   |   |   |   |   |   |    |
   |   MRT_HYBRID ( 0 percentile) | 50| 35| 13|  2|   |   |   |   |    |
   |      MRT_SPF ( 0 percentile) | 50| 35| 13|  2|   |   |   |   |    |
   | MRT_LOWPOINT ( 0 percentile) | 55| 32| 13|   |   |   |   |   |    |
   | MRT_LOWPOINT (25 percentile) | 47| 25| 22|  6|   |   |   |   |    |
   | MRT_LOWPOINT (50 percentile) | 38| 38| 14| 11|   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T211(avg primary hops=3.3)  |   |   |   |   |   |   |   |   |    |
   |   MRT_HYBRID ( 0 percentile) | 86| 14|   |   |   |   |   |   |    |
   |      MRT_SPF ( 0 percentile) | 86| 14|   |   |   |   |   |   |    |
   | MRT_LOWPOINT ( 0 percentile) | 85| 15|  1|   |   |   |   |   |    |
   | MRT_LOWPOINT (25 percentile) | 70| 25|  5|  1|   |   |   |   |    |
   | MRT_LOWPOINT (50 percentile) | 80| 18|  2|   |   |   |   |   |    |
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T216(avg primary hops=5.2)  |   |   |   |   |   |   |   |   |    |
   |   MRT_HYBRID ( 0 percentile) | 23| 22| 18| 13| 10|  7|  4|  2|   2|
   |      MRT_SPF ( 0 percentile) | 35| 32| 19|  9|  3|  1|   |   |    |
   | MRT_LOWPOINT ( 0 percentile) | 28| 25| 18| 11|  7|  6|  3|  2|   1|
   | MRT_LOWPOINT (25 percentile) | 24| 20| 19| 16| 10|  6|  3|  1|    |
   | MRT_LOWPOINT (50 percentile) | 19| 14| 13| 10|  8|  6|  5|  5|  10|
   +------------------------------+---+---+---+---+---+---+---+---+----+
   |  T219(avg primary hops=7.7)  |   |   |   |   |   |   |   |   |    |
   |   MRT_HYBRID ( 0 percentile) | 20| 16| 13| 10|  7|  5|  5|  5|   3|
   |      MRT_SPF ( 0 percentile) | 31| 23| 19| 12|  7|  4|  2|  1|    |
   | MRT_LOWPOINT ( 0 percentile) | 19| 14| 15| 12| 10|  8|  7|  6|  10|
   | MRT_LOWPOINT (25 percentile) | 19| 14| 15| 13| 12| 10|  6|  5|   7|
   | MRT_LOWPOINT (50 percentile) | 19| 14| 14| 12| 11|  8|  6|  6|  10|
   +------------------------------+---+---+---+---+---+---+---+---+----+


                                 Figure 34




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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.

10.  Algorithm Work to Be Done

   Broadcast Interfaces:   The algorithm assumes that broadcast
      interfaces are already represented as pseudo-nodes in the network
      graph.  Given maximal redundancy, one of the MRT will try to avoid
      both the pseudo-node and the next hop.  The exact rules need to be
      fully specified.

11.  Acknowledgements

   The authors would like to thank Shraddha Hegde for her suggestions
   and review.  We would also like to thank Anil Kumar SN for his
   assistance in clarifying the algorithm description and pseudocode.

12.  IANA Considerations

   This document includes no request to IANA.

13.  Security Considerations

   This architecture is not currently believed to introduce new security
   concerns.

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-05 (work in progress),
              January 2015.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.








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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>.

   [I-D.ietf-isis-mrt]
              Li, Z., Wu, N., Zhao, Q., Atlas, A., Bowers, C., and J.
              Tantsura, "Intermediate System to Intermediate System (IS-
              IS) Extensions for Maximally Redundant Trees (MRT)",
              draft-ietf-isis-mrt-00 (work in progress), February 2015.

   [I-D.ietf-isis-pcr]
              Farkas, J., Bragg, N., Unbehagen, P., Parsons, G.,
              Ashwood-Smith, P., and C. Bowers, "IS-IS Path Computation
              and Reservation", draft-ietf-isis-pcr-00 (work in
              progress), April 2015.

   [I-D.ietf-mpls-ldp-mrt]
              Atlas, A., Tiruveedhula, K., Bowers, C., Tantsura, J., and
              I. Wijnands, "LDP Extensions to Support Maximally
              Redundant Trees", draft-ietf-mpls-ldp-mrt-00 (work in
              progress), January 2015.

   [I-D.ietf-ospf-mrt]
              Atlas, A., Hegde, S., Bowers, C., Tantsura, J., and Z. Li,
              "OSPF Extensions to Support Maximally Redundant Trees",
              draft-ietf-ospf-mrt-00 (work in progress), January 2015.

   [I-D.ietf-rtgwg-ipfrr-notvia-addresses]
              Bryant, S., Previdi, S., and M. Shand, "A Framework for IP
              and MPLS Fast Reroute Using Not-via Addresses", draft-
              ietf-rtgwg-ipfrr-notvia-addresses-11 (work in progress),
              May 2013.

   [I-D.ietf-rtgwg-lfa-manageability]
              Litkowski, S., Decraene, B., Filsfils, C., Raza, K.,
              Horneffer, M., and P. Sarkar, "Operational management of
              Loop Free Alternates", draft-ietf-rtgwg-lfa-
              manageability-11 (work in progress), June 2015.







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   [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>.

   [LFARevisited]
              Retvari, G., Tapolcai, J., Enyedi, G., and A. Csaszar, "IP
              Fast ReRoute: Loop Free Alternates Revisited", Proceedings
              of IEEE INFOCOM , 2011,
              <http://opti.tmit.bme.hu/~tapolcai/papers/
              retvari2011lfa_infocom.pdf>.

   [LightweightNotVia]
              Enyedi, G., Retvari, G., Szilagyi, P., and A. Csaszar, "IP
              Fast ReRoute: Lightweight Not-Via without Additional
              Addresses", Proceedings of IEEE INFOCOM , 2009,
              <http://mycite.omikk.bme.hu/doc/71691.pdf>.

   [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>.

   [RFC3137]  Retana, A., Nguyen, L., White, R., Zinin, A., and D.
              McPherson, "OSPF Stub Router Advertisement", RFC 3137,
              June 2001.

   [RFC5120]  Przygienda, T., Shen, N., and N. Sheth, "M-ISIS: Multi
              Topology (MT) Routing in Intermediate System to
              Intermediate Systems (IS-ISs)", RFC 5120, February 2008.

   [RFC5286]  Atlas, A. and A. Zinin, "Basic Specification for IP Fast
              Reroute: Loop-Free Alternates", RFC 5286, September 2008.

   [RFC5714]  Shand, M. and S. Bryant, "IP Fast Reroute Framework", RFC
              5714, January 2010.

   [RFC6571]  Filsfils, C., Francois, P., Shand, M., Decraene, B.,
              Uttaro, J., Leymann, N., and M. Horneffer, "Loop-Free
              Alternate (LFA) Applicability in Service Provider (SP)
              Networks", RFC 6571, June 2012.

   [RFC7490]  Bryant, S., Filsfils, C., Previdi, S., Shand, M., and N.
              So, "Remote Loop-Free Alternate (LFA) Fast Reroute (FRR)",
              RFC 7490, April 2015.




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Appendix A.  Option 2: Computing GADAG using SPFs

   The basic idea in this option 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.

   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



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                 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
       Clear TEMP_UNUSABLE from cand_intf.local_node
    return end_ear


               Figure 35: Modified SPF for GADAG computation

   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 algorithm, 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.



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   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 algorithm.













































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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 36: Algorithm to assign links of an ear direction

   A goal of the algorithm 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, since it is computed via



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   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,
   as in the algorithm previously described in Figure 14.

   The algorithm ignores interfaces picked from the ordered tree 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
                cand_intf.UNDIRECTED = false
                cand_intf.remote_intf.UNDIRECTED = false



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                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 37: SPF-based GADAG algorithm

Appendix B.  Option 3: Computing GADAG using a hybrid method

   In this option, the idea 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
   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,



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   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 38

      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)
         Compute_Localroot (root,root)
         Assign_Block_ID(root,0)



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         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 38: Hybrid GADAG algorithm

Authors' Addresses

   Gabor Sandor Enyedi (editor)
   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 (editor)
   Juniper Networks
   10 Technology Park Drive
   Westford, MA  01886
   USA

   Email: akatlas@juniper.net


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

   Email: cbowers@juniper.net



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

   Email: abishek@ece.arizona.edu












































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