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
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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
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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
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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.
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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.
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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.
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[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
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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<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
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partial order we can say that 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.
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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. 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<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.
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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 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.
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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, ) (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
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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
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.
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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,
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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
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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.
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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
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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.
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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
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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
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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
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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.
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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
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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.
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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 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
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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
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
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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
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of an ear is pre-determined. Thus, whenever the block already has an
ear computed, and we are processing an interface of the block root,
we mark the block root as unusable before the SPF run that computes
the ear. This ensures that the SPF terminates at some node other
than the block-root. This in turn guarantees that the block-root has
only one incoming interface in each block, which is necessary for
correctly computing the next-hops on the GADAG.
As in the SPF gadag, bridge ears are handled as a special case.
The entire algorithm is shown below in Figure 19
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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)
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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
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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
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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.
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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
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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
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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
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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.
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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
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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
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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))
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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
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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<~~>D (ii) F<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>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.
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In the case where F<~~~~ 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
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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.
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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
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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, .
[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,
.
[LFARevisited]
Retvari, G., Tapolcai, J., Enyedi, G., and A. Csaszar, "IP
Fast ReRoute: Loop Free Alternates Revisited", Proceedings
of IEEE INFOCOM , 2011, .
[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,
.
[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,
.
[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]
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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]
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