Routing Area Working Group A. Atlas Internet-Draft Juniper Networks Intended status: Informational G. Enyedi Expires: April 24, 2013 A. Csaszar Ericsson A. Gopalan University of Arizona October 21, 2012 Algorithms for computing Maximally Redundant Trees for IP/LDP Fast- Reroute draft-enyedi-rtgwg-mrt-frr-algorithm-02 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 describes an algorithm that can be used to compute the necessary Maximally Redundant Trees and the associated next-hops. 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 working documents as Internet-Drafts. The list of current Internet- Drafts is at http://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any 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 April 24, 2013. Copyright Notice Copyright (c) 2012 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 (http://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents Atlas, et al. Expires April 24, 2013 [Page 1]

Internet-Draft MRT FRR Algorithm October 2012 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. Terminology and Definitions . . . . . . . . . . . . . . . . . 4 3. Algorithm Key Concepts . . . . . . . . . . . . . . . . . . . . 6 3.1. Partial Ordering for Disjoint Paths . . . . . . . . . . . 6 3.2. Finding an Ear and the Correct Direction . . . . . . . . . 8 3.3. Low-Point Values and Their Uses . . . . . . . . . . . . . 10 3.4. Blocks in a Graph . . . . . . . . . . . . . . . . . . . . 14 3.5. Determining Local-Root and Assigning Block-ID . . . . . . 15 4. Algorithm Sections . . . . . . . . . . . . . . . . . . . . . . 17 4.1. Root Selection . . . . . . . . . . . . . . . . . . . . . . 18 4.2. Initialization . . . . . . . . . . . . . . . . . . . . . . 19 4.3. Option 1: Computing GADAG using lowpoint inheritance . . . 19 4.4. Option 2: Computing GADAG using SPFs . . . . . . . . . . . 21 4.5. Option 3: Computing GADAG using a hybrid method . . . . . 27 4.6. Augmenting the GADAG by directing all links . . . . . . . 30 4.7. Compute MRT next-hops . . . . . . . . . . . . . . . . . . 32 4.7.1. MRT next-hops to all nodes partially ordered with respect to the computing node . . . . . . . . . . . . 33 4.7.2. MRT next-hops to all nodes not partially ordered with respect to the computing node . . . . . . . . . . 33 4.7.3. Computing Redundant Tree next-hops in a 2-connected Graph . . . . . . . . . . . . . . . . . . 34 4.7.4. Generalizing for graph that isn't 2-connected . . . . 36 4.7.5. Complete Algorithm to Compute MRT Next-Hops . . . . . 37 4.8. Identify MRT alternates . . . . . . . . . . . . . . . . . 39 5. Algorithm Alternatives and Evaluation . . . . . . . . . . . . 42 5.1. Algorithm Evaluation . . . . . . . . . . . . . . . . . . . 43 6. Algorithm Work to Be Done . . . . . . . . . . . . . . . . . . 44 7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 44 8. Security Considerations . . . . . . . . . . . . . . . . . . . 44 9. References . . . . . . . . . . . . . . . . . . . . . . . . . . 45 9.1. Normative References . . . . . . . . . . . . . . . . . . . 45 9.2. Informative References . . . . . . . . . . . . . . . . . . 45 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 46 Atlas, et al. Expires April 24, 2013 [Page 2]

Internet-Draft MRT FRR Algorithm October 2012 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 Blue MRT and the Red MRT. A router which experiences a local failure must also have pre-determined which alternate to use. This document describes how to compute these three things and the algorithm design decisions and rationale. The algorithms are based on those presented in [MRTLinear] and expanded in [EnyediThesis]. 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 Blue MRT and Red MRT in a consistent fashion. This is the motivation for the detail in this document. 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 Blue MRT for forwarding received traffic on the Blue MRT and primary next-hops for the Red MRT for forwarding received traffic on the Red MRT. 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 provides 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 decision- making is not captured in this document. [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 Blue MRT towards R Red MRT 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 Atlas, et al. Expires April 24, 2013 [Page 3]

Internet-Draft MRT FRR Algorithm October 2012 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 | [R] [F] [C]<--[G] | [R] [F] [C]<--[G] | ^ | ^ | | ^ | | | | V | | V [A]-->[B] [H] [A]<--[B]<--| |---[H] (b) Blue MRT towards R (c) Red MRT towards R Figure 2 2. Terminology and Definitions 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. network graph: A graph that reflects the network topology where all links connect exactly two nodes and broadcast links have been transformed into the standard pseudo-node representation. Atlas, et al. Expires April 24, 2013 [Page 4]

Internet-Draft MRT FRR Algorithm October 2012 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. 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-edge, 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 whith removing a single node (the root node). 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. Atlas, et al. Expires April 24, 2013 [Page 5]

Internet-Draft MRT FRR Algorithm October 2012 ear: A path along not-yet-included-in-the-GADAG nodes that starts at a node that is already-included-in-the-GADAG and that ends at a node that is already-included-in-the-GADAG. The starting and ending nodes may be the same node if it is a cut-vertex. X >> Y or Y << X: Indicates the relationship between X and Y in a partial order, such as found in a GADAG. X >> Y means that X is higher in the partial order than Y. Y << X means that Y is lower in the partial order than X. X > Y or Y < X: Indicates the relationship between X and Y in the total order, such as found via a topological sort. X > Y means that X is higher in the total order than Y. Y < X means that Y is lower in the total order than X. 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. 3. Algorithm Key Concepts There are five key concepts that are critical for understanding the 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-edge 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. 3.1. 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 IP loopback addresses. In a partial order, there may be some nodes for which it can't be determined whether X << Y or X >> Y. A partial order can be captured in a directed graph, as shown in Figure 3. In a graphical representation, a link directed from X to Y indicates that X is a neighbor of Y in the network graph and X << Y. Atlas, et al. Expires April 24, 2013 [Page 6]

Internet-Draft MRT FRR Algorithm October 2012 [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, it is very useful to have the root of the MRTs be at 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. Additionally, 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 Atlas, et al. Expires April 24, 2013 [Page 7]

Internet-Draft MRT FRR Algorithm October 2012 along an increasing path: it can either forward packets to C or F. 3.2. 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 algorithms given in Section 4.3 and Section 4.4. 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 ADAG (we say those nodes are *ready* in the sequel), 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 Atlas, et al. Expires April 24, 2013 [Page 8]

Internet-Draft MRT FRR Algorithm October 2012 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 were strictly greater than 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 already 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 4.7 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, we can always add ears to it: just select a non-ready neighbor N of a ready node Q, such that Q is not the root, find a path from N to the root 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 second 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. Atlas, et al. Expires April 24, 2013 [Page 9]

Internet-Draft MRT FRR Algorithm October 2012 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. There are three options being considered. The Low-Point Inheritance option is described in Section 4.3. The SPF-based option is described in Section 4.4 and the hybrid option is described in Section 4.5. 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. 3.3. 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. Atlas, et al. Expires April 24, 2013 [Page 10]

Internet-Draft MRT FRR Algorithm October 2012 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 root) dfs_number = 0 DFS_Visit(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 highest attachment point neighbouring x. What is interesting, though, is what is the highest attachment point from x and x's descendants. This is what is determined by computing the Low-Point value, as given in Algorithm Figure 9 and illustrated on a graph in Figure 8. Atlas, et al. Expires April 24, 2013 [Page 11]

Internet-Draft MRT FRR Algorithm October 2012 [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, ) (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 8 Atlas, et al. Expires April 24, 2013 [Page 12]

Internet-Draft MRT FRR Algorithm October 2012 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 x.lowpoint_parent = NONE for each 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) x.lowpoint_parent = intf.remote_node x.lowpoint_parent_intf = intf Run_Lowpoint(node root) dfs_number = 0 Lowpoint_Visit(root, NONE, NONE) Figure 9: Computing Low-Point value From the low-point value and lowpoint parent, there are two 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. This is useful because it allows identification of the cut-vertices and thus the blocks. As seen in Figure 8, 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. 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. Atlas, et al. Expires April 24, 2013 [Page 13]

Internet-Draft MRT FRR Algorithm October 2012 3.4. Blocks in a Graph A key idea for the 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. [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: 3 2-connected clusters and a cut-link [E]<--| [J]<------[I] [P]<--[O] | | | ^ | ^ V | V | V | [R] [D]<--[C] [F] [H]<---[K] [N] ^ | ^ ^ | V | | [A]------->[B] [G]---| [L]-->[M] (b) Blue MRT [E]---| [J]-------->[I] [P]-->[O] | | | V V V [R] [D]-->[C]<---[F] [H]<---[K] [N] ^ | ^ | ^ | | V | | | V [A]<-------[B] [G]<--| [L]<--[M] (c) Red MRT 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, Atlas, et al. Expires April 24, 2013 [Page 14]

Internet-Draft MRT FRR Algorithm October 2012 L, M, N, O, P, and block 4 is a cut-edge 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-edge that is block 4. No two blocks can have more than one common node, since two blocks with at least 2 common nodes would qualify as a single 2-connected cluster. 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 one in block 2 with C as a root, and finally, we need the last one 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-edge (as in block 4), that cut-edge 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. 3.5. 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 c 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(root, 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 Atlas, et al. Expires April 24, 2013 [Page 15]

Internet-Draft MRT FRR Algorithm October 2012 illustrates the concept. It is used in the second and third options for computing a GADAG using SPFs and the hybrid versions respectively. The other method for local-root computation is used in the first option for computing a GADAG using Low-Point inheritance and the essence of it is given in Figure 12. 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. The block id is useful in the ear computations involved in the SPF and hybrid based GADAG's as will be seen later. Atlas, et al. Expires April 24, 2013 [Page 16]

Internet-Draft MRT FRR Algorithm October 2012 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(root, max_block_id) Figure 13: Assigning block id to identify blocks 4. Algorithm Sections This algorithm computes one GADAG that is then used by a router to determine its blue MRT and red MRT 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, for instance, its interfaces are ordered as per Figure 14. 1. Select the root to use for the GADAG. [See Section 4.1.] 2. Initialize all interfaces to UNDIRECTED. [See Section 4.2.] 3. Compute the DFS value,e.g. D(x), and lowpoint value, L(x). [See Figure 9.] 4. Construct the GADAG. [See Section 4.3 for Option 1 using Lowpoint Inheritance, Section 4.4 for Option 2 using SPFs and Section 4.5 for Option 3 using a hybrid method.] 5. Assign directions to all interfaces that are still UNDIRECTED. [See Section 4.6.] 6. From the computing router x, compute the next-hops for the blue MRT and red MRT. [See Section 4.7.] 7. 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 4.8.] To ensure consistency in computation, it is necessary that all Atlas, et al. Expires April 24, 2013 [Page 17]

Internet-Draft MRT FRR Algorithm October 2012 routers order interfaces identically. This is necessary for the DFS, where the selection order of the interfaces to explore results in different trees, and for computing the GADAG, where the selection order of the interfaces to use to form ears can result in different GADAGs. The recommended ordering between two interfaces from the same router x is given in Figure 14. Interface_Compare(interface a, interface b) if a.metric < b.metric return A_LESS_THAN_B if b.metric < a.metric return B_LESS_THAN_A if a.neighbor.loopback_addr < b.neighbor.loopback_addr return A_LESS_THAN_B if b.neighbor.loopback_addr < a.neighbor.loopback_addr return B_LESS_THAN_A // Same metric to same node, so the order doesn't matter anymore. // To have a unique, consistent total order, // tie-break based on ifindex. if a.ifindex < b.ifindex return A_LESS_THAN_B return B_LESS_THAN_A Figure 14: Rules for ranking multiple interfaces. Order is from low to high. 4.1. Root Selection The precise mechanism by which routers advertise a priority for the GADAG root is not described in this document. Nor is the algorithm for selecting routers based upon priority described in this document. A network may be partitioned or there may be islands of routers that support MRT fast-reroute. Therefore, the root selected for use in a GADAG must be consistent only across each connected island of MRT fast-reroute support. Before beginning computation, the network graph is reduced to contain only the set of routers that support a compatible MRT fast-reroute. The selection of a GADAG root is done among only those routers in the same MRT fast-reroute island as the computing router x. Additionally, only routers that are not marked as unusable or overloaded (e.g. ISIS overload or [RFC3137]) are eligible for selection as root. Atlas, et al. Expires April 24, 2013 [Page 18]

Internet-Draft MRT FRR Algorithm October 2012 4.2. 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, whether because of configuration, overload, or due to a transient cause such as [RFC3137]. In the algorithm description, it is assumed that such links and nodes will not be explored or used and no more disussion is given of this restriction. 4.3. Option 1: Computing GADAG using lowpoint inheritance The basic idea of this is to find ears from a node x that is already in the GADAG (known as IN_GADAG). There are two methods to find ears; both are required. 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 Atlas, et al. Expires April 24, 2013 [Page 19]

Internet-Draft MRT FRR Algorithm October 2012 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 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 15. Construct_Ear(x, Stack, intf, 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 type is CHILD cur_intf = cur_node.lowpoint_parent_intf else type must be NEIGHBOR cur_intf = cur_node.dfs_parent_intf cur_node = cur_intf.remote_node else Atlas, et al. Expires April 24, 2013 [Page 20]

Internet-Draft MRT FRR Algorithm October 2012 not_done = false if (cur_node is x) //x is a cut-vertex and the local root for //the block in which the ear is computed 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, root) root.IN_GADAG = true root.localroot = root Initialize Stack to empty push root onto Stack while (Stack is not empty) x = pop(Stack) foreach 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 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, root) Figure 15: Low-point Inheritance GADAG algorithm 4.4. 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 Atlas, et al. Expires April 24, 2013 [Page 21]

Internet-Draft MRT FRR Algorithm October 2012 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. Atlas, et al. Expires April 24, 2013 [Page 22]

Internet-Draft MRT FRR Algorithm October 2012 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 is true found_in_gadag = true else foreach interface intf of min_node if ((intf.OUTGOING or intf.UNDIRECTED) and ((intf.remote_node.localroot is block_root) or (intf.remote_node is block_root)) and (intf.remote_node is not TEMP_UNUSABLE) and (intf is not TEMP_UNUSABLE)) path_metric = min_node.spf_metric + intf.metric if path_metric < intf.remote_node.spf_metric intf.remote_node.spf_metric = path_metric intf.remote_node.spf_prev_intf = intf insert_or_update(spf_heap, intf.remote_node) return min_node SPF_for_Ear(cand_intf.local_node,cand_intf.remote_node, block_root, method) Mark all interfaces between cand_intf.remote_node and cand_intf.local_node as TEMP_UNUSABLE if cand_intf.local_node is not block_root Mark cand_intf.local_node as TEMP_UNUSABLE Initialize ear_list to empty end_ear = Mod_SPF(spf_root, block_root) y = end_ear.spf_prev_hop while y.local_node is not spf_root add_to_list_start(ear_list, y) y.local_node.IN_GADAG = true y = y.local_node.spf_prev_intf if(method is not hybrid) Set_Ear_Direction(ear_list, cand_intf.local_node, end_ear,block_root) Clear TEMP_UNUSABLE from all interfaces between cand_intf.remote_node and cand_intf.local_node Clear TEMP_UNUSABLE from cand_intf.local_node return end_ear Atlas, et al. Expires April 24, 2013 [Page 23]

Internet-Draft MRT FRR Algorithm October 2012 Figure 16: 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 4.3, 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. 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. Atlas, et al. Expires April 24, 2013 [Page 24]

Internet-Draft MRT FRR Algorithm October 2012 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 17: 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 Atlas, et al. Expires April 24, 2013 [Page 25]

Internet-Draft MRT FRR Algorithm October 2012 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 will be seen in Section 4.7. 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-edges are a special case because there is no point in doing an SPF on a block of 2 nodes. The algorithm identifies cut- edges simply as links where both ends of the link are cut-vertices. Cut-edges 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 is true) 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-edges cand_intf.UNDIRECTED = false Atlas, et al. Expires April 24, 2013 [Page 26]

Internet-Draft MRT FRR Algorithm October 2012 cand_intf.remote_intf.UNDIRECTED = false cand_intf.OUTGOING = true cand_intf.INCOMING = true cand_intf.remote_intf.OUTGOING = true cand_intf.remote_intf.INCOMING = true cand_intf.remote_node.IN_GADAG = true add_eligible_interfaces_of_node( ordered_intfs_tree,cand_intf.remote_node) else if (cand_intf.remote_node.local_root == cand_intf.local_node) && check_if_block_has_ear (cand_intf.local_node, cand_intf.remote_node.block_id)) /* Skip the interface since the block root already has an incoming interface in the block */ else ear_end = SPF_for_Ear(cand_intf.local_node, cand_intf.remote_node, cand_intf.remote_node.localroot, SPF method) y = ear_end.spf_prev_hop while y.local_node is not cand_intf.local_node add_eligible_interfaces_of_node( ordered_intfs_tree, y.local_node) y = y.local_node.spf_prev_intf Figure 18: SPF-based GADAG algorithm 4.5. Option 3: Computing GADAG using a hybrid method In this option, the idea is to combine the salient features of the above two options. 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 Atlas, et al. Expires April 24, 2013 [Page 27]

Internet-Draft MRT FRR Algorithm October 2012 of an ear is pre-determined. Thus, whenever the block already has an ear computed, and we are processing an interface of the block root, we mark the block root as unusable before the SPF run that computes the ear. This ensures that the SPF terminates at some node other than the block-root. This in turn guarantees that the block-root has only one incoming interface in each block, which is necessary for correctly computing the next-hops on the GADAG. As in the SPF gadag, bridge ears are handled as a special case. The entire algorithm is shown below in Figure 19 Atlas, et al. Expires April 24, 2013 [Page 28]

Internet-Draft MRT FRR Algorithm October 2012 find_spf_stack_ear(stack, x, y, xy_intf, block_root) if L(y) == D(y) // Special case for cut-edges 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) 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) Atlas, et al. Expires April 24, 2013 [Page 29]

Internet-Draft MRT FRR Algorithm October 2012 Figure 19: Hybrid GADAG algorithm 4.6. Augmenting the GADAG by directing all links The GADAG, whether constructed via Low-Point Inheritance or with SPFs or the hybrid method, 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. Finally, all UNDIRECTED links are assigned a direction based upon the partial ordering. Any UNDIRECTED links that connect to a root of a block from within that block are assigned a direction INCOMING to that root. The exact details of this whole process are captured in Figure 20 Set_Block_Root_Incoming_Links(topo, root, mark_or_clear) foreach node x in topo if node x is a cut-vertex or root foreach interface i of x if (i.remote_node.localroot is x) if i.UNDIRECTED i.OUTGOING = true i.remote_intf.INCOMING = true i.UNDIRECTED = false i.remote_intf.UNDIRECTED = false if i.INCOMING Atlas, et al. Expires April 24, 2013 [Page 30]

Internet-Draft MRT FRR Algorithm October 2012 if mark_or_clear is mark if i.OUTGOING // a cut-edge i.STORE_INCOMING = true i.INCOMING = false i.remote_intf.STORE_OUTGOING = true i.remote_intf.OUTGOING = false i.TEMP_UNUSABLE = true i.remote_intf.TEMP_UNUSABLE = true else i.TEMP_UNUSABLE = false i.remote_intf.TEMP_UNUSABLE = false if i.STORE_INCOMING and (mark_or_clear is clear) i.INCOMING = true i.STORE_INCOMING = false i.remote_intf.OUTGOING = true i.remote_intf.STORE_OUTGOING = false Run_Topological_Sort_GADAG(topo, root) Set_Block_Root_Incoming_Links(topo, root, MARK) foreach node x set x.unvisited to the count of x's incoming interfaces that aren't marked TEMP_UNUSABLE Initialize working_list to empty Initialize topo_order_list to empty add_to_list_end(working_list, root) while working_list is not empty y = remove_start_item_from_list(working_list) add_to_list_end(topo_order_list, y) foreach interface i of y if (i.OUTGOING) and (not i.TEMP_UNUSABLE) 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 Set_Block_Root_Incoming_Links(topo, root, CLEAR) Add_Undirected_Links(topo, root) Run_Topological_Sort_GADAG(topo, root) 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 Atlas, et al. Expires April 24, 2013 [Page 31]

Internet-Draft MRT FRR Algorithm October 2012 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, root) Figure 20: Assigning direction to UNDIRECTED links Proxy-nodes are used to represent multi-homed prefixes and routers that do not support MRT Fast-Reroute. Until now, the network graph has not included proxy-nodes because the computation for a GADAG assumes that the nodes can be transited. To handle destinations that can only be reached via proxy-nodes, each proxy-node should be added into the network graph after Add_Directed_Links() has beeen run once. A proxy-node P is connected to two routers, X and Y, which have been found to offer the best cost. If X.topo_order < Y.topo_order, then the proxy-node P is added along with a link X->P and a link P->Y. Once all the proxy-nodes have been added in this fashion, Run_Topological_Sort_GADAG() should be rerun so that the topological order includes the proxy-nodes as well. This is needed for determining which MRT can offer alternates, as is explained in Section 4.8. 4.7. Compute MRT next-hops As was discussed in Section 3.1, 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 want to reuse the common GADAG and find not only one pair of redundant trees with it, but a pair rooted at each node. This is ideal, since it is faster and it results packet forwarding easier to trace and/or debug. The method for doing that is based on two basic ideas. First, if two nodes X and Y are ordered with respect to each other in the partial order, then the same SPF and reverse-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 4.7.4. The full algorithm is given in Section 4.7.5. Atlas, et al. Expires April 24, 2013 [Page 32]

Internet-Draft MRT FRR Algorithm October 2012 4.7.1. 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 21, 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) Blue MRT path: X->Cloud 2->Y Red MRT path: X->Cloud 1->R->Cloud 3->Y Figure 21: Y >> X Similar logic applies if Y << X, as shown in Figure 22. 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) Blue MRT path: X->Cloud 3->R->Cloud 1->Y Red MRT path: X->Cloud 2->Y Figure 22: Y << X 4.7.2. 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 Atlas, et al. Expires April 24, 2013 [Page 33]

Internet-Draft MRT FRR Algorithm October 2012 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 23. It is necessary to decrease and then increase for the Blue MRT and increase and then decrease for the Red MRT; if one simply increased for one and decreased for the other, then both paths would go through the root R. (Cloud 6)<---[Y]<---(Cloud 5)<------------| | | | | V | [G]--->(Cloud 4)--->[R]--->(Cloud 1)--->[H] ^ | | | | | (Cloud 3)<---[X]<---(Cloud 2)<-----------| Blue MRT path: decrease to H and increase to Y X->Cloud 2->H->Cloud 5->Y Red MRT path: increase to G and decrease to Y X->Cloud 3->G->Cloud 6->Y Figure 23: 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 Red MRT path will never reach R; it will turn at M and decrease to Y. 4.7.3. 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 4.7.1 and Section 4.7.2. 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 Atlas, et al. Expires April 24, 2013 [Page 34]

Internet-Draft MRT FRR Algorithm October 2012 running an SPF and a reverse-SPF rooted at X and not exploring any links from the ADAG root. ( Other traversal algorithms could safely be used instead where one traversal takes the links in their given directions and the other reverses the links' directions.) An 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 Blue MRT to reach Y. A reverse-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 Red MRT to reach Z. Since the root R is both greater than and less than X, after this SPF and reverse-SPF, X's next-hops on the Blue MRT and on the Red MRT to reach R are known. For every node Y >> X, X's next-hops on the Red MRT to reach Y are set to those on the Red MRT to reach R. For every node Z << X, X's next-hops on the Blue MRT to reach Z are set to those on the Blue MRT to reach R. For those nodes, which were not reached, we have the next-hops as well. The increasing Blue MRT next-hop for a node, which is not ordered, is the next-hop along the decreasing Red MRT towards R and the decreasing Red MRT next-hop is the next-hop along the increasing Blue MRT 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 specific path taken but only the appropriate next-hops to use. The identity of G and H is not determined. 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 SPF rooted at R will reach all other nodes; the Blue MRT next-hops are those found with this SPF. Similarly, since the root R is >> all other nodes, running a reverse-SPF rooted at R will reach all other nodes; the Red MRT next-hops are those found with this reverse-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 24 As an example consider the situation depicted in Figure 24. There Atlas, et al. Expires April 24, 2013 [Page 35]

Internet-Draft MRT FRR Algorithm October 2012 node C runs an SPF and a reverse-SPF The SPF reaches D, E and R and the reverse SPF reaches B, A and R. So we immediately get that e.g. towards E the increasing next-hop is D (it was reached though D), and the decreasing next-hop is B (since R was reached though B). Since both D and B, A and R will compute the next hops similarly, the packets will reach E. We have the next-hops towards F as well: since F is not ordered with respect to C, the increasing next-hop is the decreasing one towards R (which is B) and the decreasing next-hop is the increasing one towards R (which is D). Since B is ordered with F, it will find a real increasing next-hop, so packet forwarded to B will get to F on path C-B-F. Similarly, D will have a real decreasing next-hop, and packet will use path C-D-F. 4.7.4. Generalizing for graph that isn't 2-connected If a graph isn't 2-connected, then the basic approach given in Section 4.7.3 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 SPF and reverse-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 increasing SPF so Y >> X, then X's Red MRT next-hops to reach Y uses X's Red MRT next-hops to reach X's local-root and if Z <<, then X's Blue MRT next- hops to reach Z uses X's Blue MRT next-hops to reach X's local-root. C. If a node W in a common block with X was not reached in the SPF or reverse-SPF, then W is unordered with respect to X. X's Blue MRT next-hops to W are X's decreasing aka Red MRT next-hops to X's local-root. X's Red MRT next-hops to W are X's increasing aka Blue MRT next-hops to X's local-root. Atlas, et al. Expires April 24, 2013 [Page 36]

Internet-Draft MRT FRR Algorithm October 2012 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 4.7.5. 4.7.5. Complete Algorithm to Compute MRT Next-Hops The complete algorithm to compute MRT Next-Hops for a particular router X is given in Figure 25. In addition to computing the Blue MRT next-hops and Red MRT 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 Blue MRT or the Red MRT can provide an acceptable alternate for a particular primary next-hop. In_Common_Block(x, y) if ((x.localroot is y.localroot) or (x is y.localroot) or (y is x.localroot)) return true return false Store_Results(y, direction, spf_root, store_nhs) if direction is FORWARD y.higher = true if store_nhs y.blue_next_hops = y.next_hops if direction is REVERSE y.lower = true if store_nhs y.red_next_hops = y.next_hops SPF_No_Traverse_Root(spf_root, block_root, direction, store_nhs) 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, spf_root, store_nhs) if ((min_node is spf_root) or ((min_node is not block_root) and (min_node is not a proxy_node))) foreach interface intf of min_node if (((direction is FORWARD) and intf.OUTGOING) or ((direction is REVERSE) and intf.INCOMING) and In_Common_Block(spf_root, intf.remote_node)) if direction is FORWARD Atlas, et al. Expires April 24, 2013 [Page 37]

Internet-Draft MRT FRR Algorithm October 2012 path_metric = min_node.spf_metric + intf.metric else path_metric = min_node.spf_metric + intf.remote_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, 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_Root(x, x.localroot, FORWARD, TRUE) SPF_No_Traverse_Root(x, x.localroot, REVERSE, TRUE) // 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 is not x) and (y.localroot is x.localroot) and ((y is x.localroot) or (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 Atlas, et al. Expires April 24, 2013 [Page 38]

Internet-Draft MRT FRR Algorithm October 2012 y.red_next_hops = x.localroot.blue_next_hops // Inherit next-hops and order_proxies to other components if x is not root root.blue_next_hops = x.localroot.blue_next_hops root.red_next_hops = x.localroot.red_next_hops root.order_proxy = x.localroot foreach node y if (y is not root) and (y is not x) SetEdge(y) max_block_id = 0 Assign_Block_ID(root, max_block_id) Compute_MRT_NextHops(x, root) Figure 25 4.8. Identify MRT alternates At this point, a computing router S knows its Blue MRT next-hops and Red MRT next-hops for each destination. 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 Blue MRT next-hops as the alternate next-hop(s) for that primary next hop or to use the Red MRT next-hops. The algorithm is given in Figure 26 and discussed afterwards. Select_Alternates(S, D, F, primary_intf) if D.order_proxy is not D D_lower = D.order_proxy.lower D_higher = D.order_proxy.higher D_topo_order = D.order_proxy.topo_order else D_lower = D.lower D_higher = D.higher D_topo_order = D.topo_order //When D==F, we can do only link protection if ((D is F) or (D.order_proxy is F)) Atlas, et al. Expires April 24, 2013 [Page 39]

Internet-Draft MRT FRR Algorithm October 2012 if an MRT doesn't use primary_intf indicate alternate is not node-protecting return that MRT color else // parallel links are cut-edge return AVOID_LINK_ON_BLUE if (D_lower and D_higher and F.lower and F.higher) if F.topo_order < D_topo_order return USE_RED else return USE_BLUE if (D_lower and D_higher) if F.higher return USE_RED else return USE_BLUE if (F.lower and F.higher) if D_lower return USE_RED else if D_higher return USE_BLUE else if primary_intf.OUTGOING and primary_intf.INCOMING return AVOID_LINK_ON_BLUE if primary_intf.OUTGOING is true return USE_BLUE if primary_intf.INCOMING is true return USE_RED if D_higher if F.higher if F.topo_order < D_topo_order return USE_RED else return USE_BLUE else if F.lower return USE_BLUE else // F and S are neighbors so either F << S or F >> S else if D_lower if F.higher return USE_RED else if F.lower if F.topo_order < D_topo_order return USE_RED else Atlas, et al. Expires April 24, 2013 [Page 40]

Internet-Draft MRT FRR Algorithm October 2012 return USE_BLUE else // F and S are neighbors so either F << S or F >> S else // D and S not ordered if F.lower return USE_RED else if F.higher return USE_BLUE else // F and S are neighbors so either F << S or F >> S Figure 26 If either D>>S>>F or D<<S<<F holds true, the situation is simple: in the first case we should choose the increasing Blue next-hop, in the second case, the decreasing Red next-hop is the right choice. However, when both D and F are greater than S the situation is not so simple, there can be three possibilities: (i) F>>D (ii) F<<D or (iii) F and D are not ordered. In the first case, we should choose the path towards D along the Blue tree. In contrast, in case (ii) the Red path towards the root and then to D would be the solution. Finally, in case (iii) both paths would be acceptable. However, observe that if e.g. F.topo_order>D.topo_order, either case (i) or case (iii) holds true, which means that selecting the Blue next-hop is safe. Similarly, if F.topo_order<D.topo_order, we should select the Red next-hop. The situation is almost the same if both F and D are less than S. Recall that we have added each link to the GADAG in some direction, so it is impossible that S and F are not ordered. But it is possible that S and D are not ordered, so we need to deal with this case as well. If F<<S, we can use the Red next-hop, because that path is first increasing until a node definitely greater than D is reached, than decreasing; this path must avoid using F. Similarly, if F>>S, we should use the Blue next-hop. Additionally, the cases where either F or D is ordered both higher and lower must be considered; this can happen when one is a block- root or inherits its order_proxy is. If D is both higher and lower than S, then the MRT to use is the one that avoids F so if F is higher, then the Red MRT should be used and if F is lower, then the Blue MRT should be used; F and S must be ordered because they are neighbors. If F is both higher and lower, then if D is lower, using the Red MRT to decrease reaches D and if D is higher, using the Blue MRT to increase reaches D; if D is unordered compared to S, then the situation is a bit more complicated. Atlas, et al. Expires April 24, 2013 [Page 41]

Internet-Draft MRT FRR Algorithm October 2012 In the case where F<<S<<F and D and S are unordered, the direction of the link in the GADAG between S and F should be examined. If the link is directed S --> F, then use the Blue MRT (decrease to avoid that link and then increase). If the link is directed S <-- F, then use the Red MRT (increase to avoid that link and then decrease). If the link is S <----> F, then the link must be a cut-link and there is no node-protecting alternate. If there are multiple links between S and F, then they can protect against each other; of course, in this situation, they are probably already ECMP. Finally, there is the case where D is also F. In this case, only link protection is possible. The MRT that doesn't use the indicated primary next-hop is used. If both MRTs use the primary next-hop, then the primary next-hop must be a cut-edge so either MRT could be used but the set of MRT next-hops must be pruned to avoid that primary next-hop. To indicate this case, Select_Alternates returns AVOID_LINK_ON_BLUE. As an example, consider the ADAG depicted in Figure 27 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) a 2-connected graph Figure 27 5. Algorithm Alternatives and Evaluation This description of the algorithm assumes a particular approach that is believed to be a reasonable compromise between complexity and computation. There are two options given for constructing the GADAG Atlas, et al. Expires April 24, 2013 [Page 42]

Internet-Draft MRT FRR Algorithm October 2012 as both are reasonable and promising. SPF-based GADAG Compute the common GADAG using Option 2 of SPF-based inheritance. This considers metrics when constructing the GADAG, which is important for path length and operational control. It has higher computational complexity than the Low-Point Inheritance GADAG. Low-Point Inheritance GADAG Compute the common GADAG using Option 1 of Low-Point Inheritance. This ignores metrics when constructing the GADAG, but its computational complexity is O(links) which is attractive. It is possible that augmenting the GADAG by assigning directions to all links in the network graph and adding them to the GADAG will make the difference between this and the SPF-based GADAG minimal. In addition, it is possible to calculate Destination-Rooted GADAG, where for each destination, a GADAG rooted at that destination is computed. The GADAG can be computed using either Low-Point Inheritance or SPF-based. Then a router would need to 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. 5.1. Algorithm Evaluation When evaluating different algorithms and methods for IP Fast Reroute [RFC5714], there are three critical points to consider. o Coverage: For every Point of Local Repair (PLR) and local failure, is there an alternate to reach every destination? Those destinations include not only routers in the IGP area, but also prefixes outside the IGP area. o Alternate Length: What is the length of the alternate path offered compared to the optimal alternate route in the network? This is computed as the total length of the alternate path divided by the length of an optimal alternate path. The optimal alternate path is computed by removing the failed node and running an SPF to find the shortest path from the PLR to the destination. o Alternate Bandwidth: What percentage of the traffic sent to the failed point can be sent on the alternates? This is computed as the sum of the bandwidths along the alternate paths divided by the bandwidth of the primary paths that go through the failure point. Simulation and modeling to evalute the MRT algorithms is underway. Atlas, et al. Expires April 24, 2013 [Page 43]

Internet-Draft MRT FRR Algorithm October 2012 The algorithms being compared are: o SPF-based GADAG o Low-Point Inheritance GADAG o Destination-Rooted SPF-based GADAG o Destination-Rooted Low-Point Inheritance GADAG o Not-Via to Next-Next Hop[I-D.ietf-rtgwg-ipfrr-notvia-addresses] o Loop-Free Alternates[RFC5286] o Remote LFAs[I-D.shand-remote-lfa] 6. Algorithm Work to Be Done Broadcast Interfaces: The algorithm assumes that broadcast interfaces are already represented as pseudo-nodes in the network graph. The exact rules for extending the set of next-hops and ensuring that the neighboring node is avoided need to be fully specified. Local SRLG Protection: The algorithmic extensions to handle local SRLGs, where each member of the SRLG shares a common router end, need to be fully specified. General SRLG Protection: Creating MRTs that consider general SRLGs is still a challenging open research problem. 7. IANA Considerations This doument includes no request to IANA. 8. Security Considerations This architecture is not currently believed to introduce new security concerns. 9. References Atlas, et al. Expires April 24, 2013 [Page 44]

Internet-Draft MRT FRR Algorithm October 2012 9.1. Normative References [I-D.ietf-rtgwg-mrt-frr-architecture] Atlas, A., Kebler, R., Envedi, G., Csaszar, A., Konstantynowicz, M., White, R., and M. Shand, "An Architecture for IP/LDP Fast-Reroute Using Maximally Redundant Trees", draft-ietf-rtgwg-mrt-frr-architecture-01 (work in progress), March 2012. 9.2. 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/ collections/phd/Villamosmernoki_es_Informatikai_Kar/2011/ Enyedi_Gabor/ertekezes.pdf>. [I-D.ietf-rtgwg-ipfrr-notvia-addresses] Bryant, S., Previdi, S., and M. Shand, "IP Fast Reroute Using Not-via Addresses", draft-ietf-rtgwg-ipfrr-notvia-addresses-09 (work in progress), June 2012. [I-D.ietf-rtgwg-lfa-applicability] Filsfils, C. and P. Francois, "LFA applicability in SP networks", draft-ietf-rtgwg-lfa-applicability-06 (work in progress), January 2012. [I-D.shand-remote-lfa] Bryant, S., Filsfils, C., Shand, M., and N. So, "Remote LFA FRR", draft-shand-remote-lfa-01 (work in progress), June 2012. [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 Atlas, et al. Expires April 24, 2013 [Page 45]

Internet-Draft MRT FRR Algorithm October 2012 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. [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. Authors' Addresses Alia Atlas Juniper Networks 10 Technology Park Drive Westford, MA 01886 USA Email: akatlas@juniper.net Gabor Sandor Enyedi Ericsson Konyves Kalman krt 11 Budapest 1097 Hungary Email: Gabor.Sandor.Enyedi@ericsson.com Atlas, et al. Expires April 24, 2013 [Page 46]

Internet-Draft MRT FRR Algorithm October 2012 Andras Csaszar Ericsson Konyves Kalman krt 11 Budapest 1097 Hungary Email: Andras.Csaszar@ericsson.com Abishek Gopalan University of Arizona 1230 E Speedway Blvd. Tucson, AZ 85721 USA Email: abishek@ece.arizona.edu Atlas, et al. Expires April 24, 2013 [Page 47]