One document matched: draft-enyedi-rtgwg-mrt-frr-algorithm-00.xml
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<rfc category="info" docName="draft-enyedi-rtgwg-mrt-frr-algorithm-00" ipr="trust200902">
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<!-- ***** FRONT MATTER ***** -->
<front>
<!-- The abbreviated title is used in the page header - it is only necessary if the
full title is longer than 39 characters -->
<title abbrev="MRT FRR Algorithm">Algorithms for computing Maximally Redundant Trees for IP/LDP Fast-Reroute</title>
<!-- add 'role="editor"' below for the editors if appropriate -->
<!-- Another author who claims to be an editor -->
<author fullname="Alia Atlas" initials="A.K.A." surname="Atlas">
<organization>Juniper Networks</organization>
<address>
<postal>
<street>10 Technology Park Drive</street>
<city>Westford</city>
<region>MA</region>
<code>01886</code>
<country>USA</country>
</postal>
<email>akatlas@juniper.net</email>
</address>
</author>
<author fullname="Gábor Sándor Enyedi" initials="G.S.E." surname="Enyedi">
<organization>Ericsson</organization>
<address>
<postal>
<street>Konyves Kalman krt 11</street>
<city>Budapest</city>
<country>Hungary</country>
<code>1097</code>
</postal>
<email>Gabor.Sandor.Enyedi@ericsson.com</email>
</address>
</author>
<author fullname="András Császár" initials="A.C." surname="Császár">
<organization>Ericsson</organization>
<address>
<postal>
<street>Konyves Kalman krt 11</street>
<city>Budapest</city>
<country>Hungary</country>
<code>1097</code>
</postal>
<email>Andras.Csaszar@ericsson.com</email>
</address>
</author>
<date year="2011" />
<area>Routing</area>
<workgroup>Routing Area Working Group</workgroup>
<abstract>
<t>A complete solution for IP and LDP Fast-Reroute using Maximally
Redundant Trees is presented in <xref
target="I-D.atlas-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.</t>
</abstract>
</front>
<middle>
<section title="Introduction" >
<t>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 <xref
target="MRTLinear"/> and expanded in <xref
target="EnyediThesis"/>.</t>
<t>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.</t>
<t>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.</t>
<t>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.</t>
<t>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.</t>
<figure anchor="2_connected_graph_and_mrts" align="center">
<artwork align="center"><![CDATA[
[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
]]></artwork>
</figure>
<t>Algorithms for computing MRTs can handle arbitrary network
topologies where the whole network graph is not 2-connected, as in
<xref target="non-2-connected_graph_and_mrts"/>, as well as the
easier case where the network graph is 2-connected (<xref
target="2_connected_graph_and_mrts"/>). 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.</t>
<figure anchor="non-2-connected_graph_and_mrts" align="center">
<artwork align="center"><![CDATA[
[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
]]></artwork>
</figure>
</section>
<section title="Terminology and Definitions" >
<t><list style="hanging">
<t hangText="Redundant Trees (RT): ">A pair of trees where the
path from any node X to the root R along 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.</t>
<t hangText="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.</t>
<t hangText="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.</t>
<t hangText="cut-vertex: ">A vertex whose removal partitions the
network.</t>
<t hangText="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. </t>
<t hangText="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.</t>
<t hangText="spanning tree: ">A tree containing links that
connects all nodes in the network graph.</t>
<t hangText="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.</t>
<t hangText="DAG: ">Directed Acyclic Graph - a graph where all
links are directed and there are no cycles in it.</t>
<t hangText="ADAG: ">Almost Directed Acyclic Graph - a graph
that, if all links incoming to the root were removed, would be a
DAG.</t>
<t hangText="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.</t>
<t hangText="block: ">Either a 2-connected cluster, a cut-edge,
or an isolated vertex.</t>
<t hangText="GADAG: ">Generalized ADAG - a graph that is
the combination of the ADAGs of all blocks.</t>
<t hangText="DFS: ">Depth-First Search</t>
<t hangText="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.</t>
<t hangText="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.</t>
<t hangText="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.</t>
<t hangText="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.</t>
<t hangText="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.</t>
</list></t>
</section>
<section title="Algorithm Key Concepts" >
<t>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.</t>
<section anchor="sec_partial_order" title="Partial Ordering for Disjoint Paths" >
<t>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 <xref
target="partial_order_graph"/>. 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.</t>
<figure anchor="partial_order_graph" align="center"
title="Directed Graph showing a Partial Order">
<artwork align="center"><![CDATA[
[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]
]]></artwork>
</figure>
<t>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
<xref target="adag_graph"/>, 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.</t>
<figure anchor="adag_graph" align="center"
title="ADAG showing a Partial Order with R lowest and highest">
<artwork align="center"><![CDATA[
[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]
]]></artwork>
</figure>
<t>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.</t>
<t>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
along an increasing path: it can either forward packets to C or F.</t>
</section>
<section title="Finding an Ear and the Correct Direction" >
<t>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 <xref
target="sec_gadag_lowpoint"/> and <xref target="sec_gadag_spf"/>.</t>
<t>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.</t>
<t>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 <xref
target="add_ear_figure"/></t>
<figure anchor="add_ear_figure" align="center">
<artwork align="center"><![CDATA[
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
]]></artwork>
</figure>
<t>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 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.</t>
<t>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.</t>
<t>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 greater than Y. The logic for this decision is
explained in <xref target="sec_compute_mrt_next-hops"/></t>
<t>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.</t>
<t>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.</t>
<t>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.</t>
<figure anchor="alg_generic_ear" align="center"
title="Generic Algorithm to find ears and their direction in 2-connected graph">
<artwork align="center"><![CDATA[
Generic_Find_Ears_ADAG(root)
Create an empty ADAG. Add root to the ADAG.
Mark root as IN_GADAG.
Select the shortest cycle containing root.
Add the shortest 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 (X is root) or (Y<<X)
add the ear towards X to the ADAG
else // (a) Y is root or (b) Y>>X or (c)X, Y not ordered
Add the ear towards Y to the ADAG
]]></artwork>
</figure>
<t>Algorithm <xref target="alg_generic_ear"/> 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 two options being considered.
The Low-Point Inheritance option is described in <xref
target="sec_gadag_lowpoint"/>. The SPF-based option is described in
<xref target="sec_gadag_spf"/>.</t>
<t>As an example, consider <xref target="add_ear_figure"/>
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 <xref target="add_ear_figure"/> (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.</t>
</section>
<section title="Low-Point Values and Their Uses" >
<t>A basic way of computing a spanning tree on a network graph is to
run a depth-first-search, such as given in <xref
target="DFS_algorithm"/>. 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.</t>
<figure anchor="DFS_algorithm" align="center"
title="Basic Depth-First Search algorithm">
<artwork align="center">
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)
</artwork>
</figure>
<t>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 <xref target="low-point_algorithm"/> and illustrated on a
graph in <xref target="fig_lowpoint_value_example"/>.</t>
<figure anchor="fig_lowpoint_value_example" align="center">
<artwork align="center"><![CDATA[
[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))
]]></artwork>
</figure>
<figure anchor="low-point_algorithm" align="center"
title="Computing Low-Point value">
<artwork align="center"><![CDATA[
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)
]]></artwork>
</figure>
<t>From the low-point value and lowpoint parent, there are two very
useful things which motivate our computation.</t>
<t>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 <xref
target="fig_lowpoint_value_example"/>, 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.</t>
<t>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.</t>
</section>
<section title="Blocks in a Graph" >
<t> 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.</t>
<figure anchor="fig_next_hops_mrt_example" align="center">
<artwork align="center"><![CDATA[
[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
]]></artwork>
</figure>
<t>Consider the example depicted in <xref
target="fig_next_hops_mrt_example"/> (a). In this figure, a special
graph is presented, showing us all the ways 2-connected clusters can
be connected. It has four blocks: block 1 contains R, A, B, C, D, E,
block 2 contains C, F, G, H, I, J, block 3 contains K, L, M, N, O, P,
and block 4 is a cut-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.</t>
<t>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 <xref target="fig_next_hops_mrt_example"/> (b) and
(c).</t>
<t>Similarly, to create a GADAG it is sufficient to compute ADAGs in
each block and connect them.</t>
<t>It is necessary, therefore, to identify the cut-vertices, the
blocks and identify the appropriate local-root to use for each
block.</t>
</section>
<section title="Determining Local-Root" >
<t>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. </t>
<figure anchor="local-root_computation" align="center"
title="A method for computing local-roots">
<artwork align="center"><![CDATA[
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)
]]></artwork>
</figure>
<t>There are two different ways of computing the local-root for each
node. The stand-alone method is given in <xref
target="local-root_computation"/> and better illustrates the concept.
It is used in the second option for computing a GADAG using SPFs. The
other method is used in the first option for computing a GADAG using
Low-Point inheritance and the essence of it is given in <xref
target="ear-based_local-root"/>.
</t>
<figure anchor="ear-based_local-root" align="center"
title="Ear-based method for computing local-roots">
<artwork align="center"><![CDATA[
Get the current node, s.
Compute an ear from s to a child c
and then via lowpoint inheritance, e.g.
( n = c
while n is not ready:
n = n.lowpoint_parent
e = n
)
to a ready node e.
if s is e
s is a cut-vertex
x.localroot = s
else
for each node x in the ear that is not s or e
x.localroot = s.localroot
]]></artwork>
</figure>
<t>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.</t>
<t><list style="symbols">
<t>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.</t>
<t>Y's local-root is X: X is the cut-vertex closest to the root for
Y's block</t>
<t>Y is X's local-root: Y is the cut-vertex closest to the root for
X's block</t>
</list></t>
</section>
</section>
<section title="Algorithm Sections" >
<t>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 <xref target="interface_ordering"/>.</t>
<t><list style="numbers">
<t>Select the root to use for the GADAG. [See <xref target="sec_root_selection"/>.]</t>
<t>Initialize all interfaces to UNDIRECTED. [See <xref target="sec_initialize"/>.]</t>
<t>Compute the DFS value,e.g. D(x), and lowpoint value, L(x). [See
<xref target="low-point_algorithm"/>.]</t>
<t>Construct the GADAG. [See <xref target="sec_gadag_lowpoint"/> for
Option 1 using Lowpoint Inheritance and <xref target="sec_gadag_spf"/>
for Option 2 using SPFs.]</t>
<t>Assign directions to all interfaces that are still UNDIRECTED. [See
<xref target="sec_gadag_direct_links"/>.]</t>
<t>From the computing router x, compute the next-hops for the blue MRT
and red MRT. [See <xref target="sec_compute_mrt_next-hops"/>.]</t>
<t>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 <xref target="sec_mrt_alternates"/>.]</t>
</list></t>
<t>To ensure consistency in computation, it is necessary that all
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 <xref target="interface_ordering"/>.</t>
<figure anchor="interface_ordering" align="center"
title="Rules for ranking multiple interfaces. Order is from low to high.">
<artwork align="center"><![CDATA[
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
]]></artwork>
</figure>
<section anchor="sec_root_selection" title="Root Selection" >
<t>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.</t>
<t>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.</t>
<t>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 <xref target="RFC3137"/>) are
eligible for selection as root.</t>
</section>
<section anchor="sec_initialize" title="Initialization" >
<t>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.</t>
<t>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.</t>
<t>Initially, all interfaces must be initialized to UNDIRECTED.
Whether they are OUTGOING, INCOMING or both is determined when the
GADAG is constructed and augmented.</t>
<t>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 <xref target="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.</t>
</section>
<section anchor="sec_gadag_lowpoint"
title="Option 1: Computing GADAG using lowpoint inheritance" >
<t>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.</t>
<t>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.</t>
<t>Each ear represents a partial order (see <xref
target="adag_graph"/>) and processing the nodes in order along each
ear ensures that all ears connecting to a node are found before a node
higher in the partial order has its ears explored. This means that
the direction of the links in the ear is always from the node x being
processed towards the other end of the ear. Additionally, by using a
stack of ears, this means that any unprocessed nodes in previous ears
can only be ordered higher than nodes in the ears below it on the
stack.</t>
<t>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 will 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 to a
cut-vertex to start and end at the same node.</t>
<t>Finally, the algorithm simultaneously computes each node's
local-root, as described in <xref target="ear-based_local-root"/>.
The local-root can be inherited from the node x being processed to the
nodes in the ear unless the child of x is a cut-vertex in which case
the rest of the nodes in the ear are in a different block than x and
have the child of x as their local-root.</t>
<figure anchor="lowpoint_inheritance_gadag" align="center"
title="Low-point Inheritance GADAG algorithm">
<artwork align="center"><![CDATA[
Construct_GADAG_via_Lowpoint(topology, root)
root.IN_GADAG = true
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_Ear(x, Stack, intf, type)
ear_list = empty
cur_node = intf.remote_node
cur_intf = intf
while cur_node.IN_GADAG is false
cur_intf.UNDIRECTED = false
cur_intf.OUTGOING = true
cur_intf.remote_intf.UNDIRECTED = false
cur_intf.remote_intf.INCOMING = true
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
if (type is CHILD) and (cur_node is x)
localroot = x
else
localroot = x.localroot
while ear_list is not empty
y = remove_end_item_from_list(ear_list)
y.localroot = localroot
push(Stack, y)
]]></artwork>
</figure>
</section>
<section anchor="sec_gadag_spf" title="Option 2: Computing GADAG using SPFs" >
<t>The basic idea in this option is to use slightly-modified SPF
computations to find ADAGs in each block. In each block, an SPF
computation is first done to find a cycle from the local root and then
SPF computations 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.</t>
<t>To do this, first all cut-vertices must be identified and
local-roots assigned as specified in <xref
target="local-root_computation"/></t>
<t>The slight modifications to the SPF are as follows. The root of the
block is referred to as the block-root; it is either the GADAG root or
a cut-vertex.</t>
<t><list style="letters">
<t>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.</t>
<t>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.</t>
<t>Do not explore links to nodes whose local-root is not the block-root.
This keeps the SPF confined to the particular block.</t>
<t>Terminate when the first IN_GADAG node z is minimized.</t>
<t>Respect the existing directions (e.g. INCOMING, OUTGOING,
UNDIRECTED) already specified for each interface.</t>
</list></t>
<figure anchor="mod_spf_alg" align="center"
title="Modified SPF for GADAG computation">
<artwork align="center"><![CDATA[
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) and
(intf.remote_node 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(spf_root, block_root, ear_list, cut_vertex_list)
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(cut_vertex_list, y)
if y.local_node is a cut-vertex
add_to_list_end(cut_vertex_list, y.local_node)
y = y.local_node.spf_prev_intf
]]></artwork>
</figure>
<t>In <xref target="mod_spf_alg"/>, while the path is determined, any
non-end node in the path that is a cut-vertex is added to the list of
cut-vertices. This ensures that there is a path from the GADAG root
to that cut-vertex before adding it to the list of nodes. All such
cut-vertices will be treated as the root of a block and
the ADAG in that block will be computed.</t>
<t>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 <xref target="sec_gadag_lowpoint"/>, 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.</t>
<t>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.</t>
<figure anchor="ear_direction_alg" align="center"
title="Algorithm to assign links of an ear direction">
<artwork align="center"><![CDATA[
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_a 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
]]></artwork>
</figure>
<t>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
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 <xref
target="interface_ordering"/>.</t>
<t>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.</t>
<figure anchor="spf_gadag" align="center"
title="SPF-based GADAG algorithm">
<artwork align="center"><![CDATA[
Construct_GADAG_via_SPF(topology, root)
Compute_Localroot(root, root)
if root has multiple DFS-children
mark root as a cut-vertex
Initialize cut_vertex_list to empty
Initialize ordered_intfs_tree to empty
add_to_list_end(cut_vertex_list, root)
while cut_vertex_list is not empty
v = remove_start_item_from_list(cut_vertex_list)
foreach interface intf of v
if intf.remote_node is a cut-vertex
// Special case for cut-edges
intf.UNDIRECTED = false
intf.remote_intf.UNDIRECTED = false
intf.OUTGOING = true
intf.INCOMING = true
intf.remote_intf.OUTGOING = true
intf.remote_intf.INCOMING = true
else if intf.remote_node.localroot is v
insert(ordered_intfs_tree, intf)
v.IN_GADAG = true
while ordered_intfs_trees is not empty
cand_intf = remove_lowest(ordered_intfs_tree)
if cand_intf.remote_node.IN_GADAG is false
Mark all interfaces between cand_intf.remote_node
and cand_intf.local_node as TEMP_UNUSABLE
if cand_intf.local_node is not v
Mark cand_intf.local_node as TEMP_UNUSABLE
Initialize ear_list to empty
ear_end = SPF_for_Ear(cand_intf.remote_node, v, ear_list,
cut_vertex_list)
add_to_list_start(ear_list, cand_intf)
Set_Ear_Direction(ear_list, cand_intf.remote, ear_end, v)
Clear TEMP_UNUSABLE from all interfaces between
cand_intf.remote_node and cand_intf.local_node
Clear TEMP_UNUSABLE from cand_intf.local_node
]]></artwork>
</figure>
</section>
<section anchor="sec_gadag_direct_links"
title="Augmenting the GADAG by directing all links" >
<t>The GADAG, whether constructed via Low-Point Inheritance or with
SPFs, 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.</t>
<t>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.</t>
<t>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<xref target="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.</t>
<t>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 <xref target="topo_sort_links"/></t>
<figure anchor="topo_sort_links" align="center"
title="Assigning direction to UNDIRECTED links">
<artwork align="center"><![CDATA[
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.INCOMING = true
i.remote_intf.OUTGOING = true
i.UNDIRECTED = false
i.remote_intf.UNDIRECTED = false
if i.INCOMING
if mark_or_clear is mark
i.TEMP_UNUSABLE = true
i.remote_intf.TEMP_UNUSABLE = true
Run_Topological_Sort(topo, root)
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
Add_Undirected_Links(topo, root)
Set_Block_Root_Incoming_Links(topo, root, MARK)
Run_Topological_Sort(topo, root)
Set_Block_Root_Incoming_Links(topo, root, CLEAR)
foreach node x in topo
foreach interface i of x
if i.UNDIRECTED
if x.topo_order < i.remote_node.topo_order
i.OUTGOING = true
i.UNDIRECTED = false
i.remote_intf.INCOMING = true
i.remote_intf.UNDIRECTED = false
else
i.INCOMING = true
i.UNDIRECTED = false
i.remote_intf.OUTGOING = true
i.remote_intf.UNDIRECTED = false
Add_Undirected_Links(topo, root)
]]></artwork>
</figure>
</section>
<section anchor="sec_compute_mrt_next-hops"
title="Compute MRT next-hops" >
<t>As was discussed in <xref target="sec_partial_order"/>, 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.</t>
<t>As usual, the two basic ideas will be discussed assuming the
network is two-connected. The generalization to multiple blocks is
discussed in <xref target="sec_compute_mrt_next-hops_gadag"/>. The
full algorithm is given in <xref
target="sec_compute_mrt_next-hops_alg"/>.</t>
<section anchor="sec_next_hops_ordered" title="MRT next-hops to all
nodes partially ordered with respect to the computing node" >
<t>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 <xref target="fig_ordered_yx"/>, 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.</t>
<figure anchor="fig_ordered_yx" title="Y >> X" align="center">
<artwork align="center"><![CDATA[
[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
]]></artwork>
</figure>
<t>Similar logic applies if Y << X, as shown in <xref
target="fig_ordered_xy"/>. 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].</t>
<figure anchor="fig_ordered_xy" title="Y << X" align="center">
<artwork align="center"><![CDATA[
[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
]]></artwork>
</figure>
</section>
<section anchor="sec_next_hops_unordered"
title="MRT next-hops to all nodes not partially ordered with
respect to the computing node" >
<t>When X and Y are not ordered, the first path should increase until
we get to a node G, where G >> Y. At G, we need to decrease to
Y. The other path should be just the opposite: we must decrease until
we get to a node H, where H << Y, and then increase. Since R is
smaller and greater than Y, such G and H must exist. It is also easy
to see that these two paths must be node disjoint: the first path
contains nodes in interval [X,G] and [Y,G], while the second path
contains nodes in interval [H,X] and [H,Y]. This is illustrated in
<xref target="fig_unordered_xy"/>. 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.</t>
<figure anchor="fig_unordered_xy" title="X and Y unordered" align="center">
<artwork align="center"><![CDATA[
(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
]]></artwork>
</figure>
<t>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 out-going interface from the root R which is created
when the initial cycle is found. Recall from <xref
target="alg_generic_ear"/> that whenever an ear was found to have an
end that was the root R, the ear was directed towards R so that the
associated interface on R is incoming and not outgoing. Therefore,
there must be exactly one node M which is the smallest one after R, so
the Blue MRT path will never reach R; it will turn at M and increase
to Y.</t>
</section>
<section anchor="sec_next_hops_2_connect_algo"
title="Computing Redundant Tree next-hops in a 2-connected Graph" >
<t>The basic ideas for computing RT next-hops in a 2-connected graph
were given in <xref target="sec_next_hops_ordered"/> and <xref
target="sec_next_hops_unordered"/>. Given these two ideas, how can we
find the trees?</t>
<t>If some node X only wants to find the next-hops (which is usually
the case for IP networks), it is enough to find which nodes are
greater and less than X, and which are not ordered; this can be done
by running an 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.)</t>
<t>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.</t>
<t>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.</t>
<t>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.</t>
<figure anchor="fig_next_hops_example" align="center">
<artwork align="center"><![CDATA[
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
]]></artwork>
</figure>
<t>As an example consider the situation depicted in <xref
target="fig_next_hops_example"/>. There 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.</t>
<t>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.</t>
</section>
<section anchor="sec_compute_mrt_next-hops_gadag"
title="Generalizing for graph that isn't 2-connected" >
<t>If a graph isn't 2-connected, then the basic approach given in
<xref target="sec_next_hops_2_connect_algo"/> 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.</t>
<t>When computing the MRT next-hops from a router X, there are three
basic differences:</t>
<t><list style="numbers">
<t>Only nodes in a common block with X should be explored in the SPF
and reverse-SPF.</t>
<t>Instead of using the GADAG root, X's local-root should be used.
This has the following implications:
<list style="letters">
<t>The links from X's local-root should not be explored. </t>
<t>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.</t>
<t>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.</t>
</list></t>
<t>For nodes in different blocks, the next-hops must be inherited
via the relevant cut-vertex.</t>
</list></t>
<t>These are all captured in the detailed algorithm given in <xref
target="sec_compute_mrt_next-hops_alg"/>.</t>
</section>
<section anchor="sec_compute_mrt_next-hops_alg"
title="Complete Algorithm to Compute MRT Next-Hops" >
<t>The complete algorithm to compute MRT Next-Hops for a particular
router X is given in <xref target="fig_mrt_next_hops_alg"/>. 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.</t>
<figure anchor="fig_mrt_next_hops_alg" align="center">
<artwork align="center"><![CDATA[
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)
if direction is FORWARD
y.higher = true
y.blue_next_hops = y.next_hops
if direction is REVERSE
y.lower = true
y.red_next_hops = y.next_hops
SPF_No_Traverse_Root(spf_root, block_root, direction)
Initialize spf_heap to empty
Initialize nodes' spf_metric to infinity and next_hops to empty
spf_root.spf_metric = 0
insert(spf_heap, spf_root)
while (spf_heap is not empty)
min_node = remove_lowest(spf_heap)
Store_Results(min_node, direction, spf_root)
if min_node is not block_root
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
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)
SPF_No_Traverse_Root(x, x.localroot, REVERSE)
// red and blue next-hops are stored to x.localroot as different
// paths are found via the SPF and reverse-SPF.
// Similarly any nodes whose local-root is x will have their
// red_next_hops and blue_next_hops already set.
// Handle nodes in the same block that aren't the local-root
foreach node y
if (y is not x) and (y.localroot is x.localroot)
if y.higher
y.red_next_hops = x.localroot.red_next_hops
else if y.lower
y.blue_next_hops = x.localroot.blue_next_hops
else
y.blue_next_hops = x.localroot.red_next_hops
y.red_next_hops = x.localroot.blue_next_hops
// 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)
Copute_RT_NextHops(x, root)
]]></artwork>
</figure>
</section>
</section>
<section anchor="sec_mrt_alternates" title="Identify MRT alternates" >
<t>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.</t>
<t>For each primary next-hop node F to each destination D, S can call
Select_Alternates(S, D, F) 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 <xref
target="fig_alternate_selection_alg"/> and discussed afterwards.</t>
<figure anchor="fig_alternate_selection_alg" align="center">
<artwork align="center"><![CDATA[
Select_Alternates(S, D, F)
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
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
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_BLUE
else if F.upper
return USE_RED
else
// F and S are neighbors so either F << S or F >> S
]]></artwork>
</figure>
<t>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.</t>
<t>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.</t>
<t>Recall that we have added each link to the GADAG in some direction,
so that is imposible 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.</t>
<t>As an example consider the ADAG depicted in <xref
target="ADAG-for-nh-sel"/> 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 first decrease to avoid using H, so the
Blue, first decreasing then increasing, path is selected.</t>
<figure anchor="ADAG-for-nh-sel" align="center">
<artwork align="center"><![CDATA[
[E]<-[D]<-[H]<-[J]
| ^ ^ |
V | | |
[R] [C] [G]->[I]
| ^ ^ ^
V | | |
[A]->[B]->[F]---|
(a)
a 2-connected graph
]]></artwork>
</figure>
</section>
</section>
<section title="Algorithm Alternatives and Evaluation" >
<t>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
as both are reasonable and promising.</t>
<t><list style="hanging">
<t hangText="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.</t>
<t hangText="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.</t>
</list></t>
<t>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.</t>
<section title="Algorithm Evaluation" >
<t>When evaluating different algorithms and methods for IP Fast
Reroute <xref target="RFC5714"/>, there are three critical points to consider.
<list style="symbols">
<t>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.</t>
<t>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.</t>
<t>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.</t>
</list></t>
<t>Simulation and modeling to evalute the MRT algorithms is underway.
The algorithms being compared are:
<list style="symbols">
<t>SPF-based GADAG</t>
<t>Low-Point Inheritance GADAG</t>
<t>Destination-Rooted SPF-based GADAG</t>
<t>Destination-Rooted Low-Point Inheritance GADAG</t>
<t>Not-Via to Next-Next Hop<xref target="I-D.ietf-rtgwg-ipfrr-notvia-addresses"/></t>
<t>Loop-Free Alternates<xref target="RFC5286"/></t>
<t>Remote LFAs<xref target="I-D.shand-remote-lfa"/></t>
</list></t>
</section>
<!-- Alia 21-Oct-2011: I want to add the criteria for comparisons and
comparing the length of the alternates found with those by doing
not-via F to the next-next-hop as well as by doing a not-via F to the
destination.
In draft, best to describe this comparison as on-going work; I'm
hopeful of results by Taipei, but we'll see...
-->
</section>
<section anchor="IANA" title="IANA Considerations" >
<t>This doument includes no request to IANA.</t>
</section>
<section anchor="Security" title="Security Considerations" >
<t>This architecture is not currently believed to introduce new security concerns.</t>
</section>
</middle>
<back>
<!-- Alia 21-Oct-2011: This section needs more work. I'm slightly
confused by the reverse-SPFs versus just reversing the link
direction. I have been thinking of the as the same thing. Anyhow, we
don't have to have this section for the 00 version, though I do think
it is a useful thing to capture.
<section title="Computing the full MRTs - not just next-hops">
<t>The algorithm given in <xref target="sec_compute_mrt_next_hops"/>
works to compute each node's next-hops for the Blue MRT and the Red
MRT. However, it does not compute the full trees from a node X to all
other nodes. For troubleshooting and modeling, it may be useful to
compute a pair of the two complete trees along which a network using
the algorithm given in <xref target="sec_compute_mrt_next_hops"/>
would forward a packet. There are two ways of doing so that are given
here.</t>
<t>The first way is simply by simulating the MRT next-hop computations
from every node in the network. However, if we do not need all the
trees but just a single pair of them rooted at a given node X, that
can be done faster.</t>
<t>First, compute the two trees rooted at the root of the ADAG (let
that node be R); the first is computed by running a reverse-SPF rooted
at R and gives the increasing next-hops
SPF rooted at R
and gives the decreasing next-hops from all nodes to R. The second is
computed by running a reverse-SPF rooted at R and gives the increaang
next-hops from all nodes to R. In this way, the next hops towards R
along two trees are computed for each node.</t>
<t>Second, run an SPF rooted at X and not exploring links from R; this
finds all the nodes >> X. Run a reverse-SPF rooted at X and not
exploring links from R; this finds all nodes << X. For those
nodes, the next-hops are now known( see <xref
target="sec_next_hops_2_connect_algo"/>)<t>
<t>For nodes, which are not ordered with X (neither the SPF or the
reverse-SPF reached it), we need to use the fact that the increasing
next-hop is the decreasing towards the root and vice versa.</t>
<figure anchor="fig_next_hops_example2" align="center">
<artwork align="center"><![CDATA[
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
]]></artwork>
</figure>
As an example, suppose that we want to compute the trees rooted at F
<xref target="fig_next_hops_example"/>. We first do two reverse
Dijkstras from R; the Dijkstra ran in increasing direction (normal
direction of the arcs) finds the decreasing next-hops back to R, which
are R for A, A for B, B for C and F, F or C (both of them are OK) for
D and D for E. The other Dijkstra finds the increasing next-hops,
i.e. R for E, E for D, D for C and F, C or F for B and B for A.
Then, we start two revers Dijkstra algorithms from F. They reach D, E,
R (increasing direction) and B, A, R (decreasing direction), so we
have immediately the next-hops for these nodes:
Decreasing: F for D, D for E, E for R, R for A, A for B
Increasing: F for B, B for A, A for R, R for E, E for D
(The last two in each line come from the next-hops towards the root
with respect to observation 1.)
Finally, we have C, which is not ordered with F, so the increasing
next-hop is the decreasing next hop of C towards R and the decreasing
one is the increasing next-hop towards R. So we have B for increasing
and D for decreasing. Combining these next-hops as arcs immediately
generates the two trees rooted at F.
</section>
-->
<!-- References split into informative and normative -->
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<references title="Normative References">
&I-D.atlas-rtgwg-mrt-frr-architecture;
</references>
<references title="Informative References">
&RFC3137;
&RFC5286;
&RFC5714;
&I-D.ietf-rtgwg-ipfrr-notvia-addresses;
&I-D.shand-remote-lfa;
<reference anchor="Kahn_1962_topo_sort"
target="http://dl.acm.org/citation.cfm?doid=368996.369025">
<front>
<title>Topological sorting of large networks</title>
<author fullname="A.B. Kahn" initials="A.B.K." surname="Kahn"/>
<date month="Nov" year="1962"/>
</front>
<seriesInfo name="Communications of the ACM, Volume 5, Issue 11" value=""/>
</reference>
<reference anchor="EnyediThesis"
target="http://timon.tmit.bme.hu/theses/thesis_book.pdf">
<front>
<title>Novel Algorithms for IP Fast Reroute</title>
<author fullname="Gábor Sándor Enyedi" initials="G.S.E." surname="Enyedi"/>
<date month="February" year="2011"/>
</front>
<seriesInfo name="Department of Telecommunications and Media Informatics, Budapest University of Technology and Economics" value="Ph.D. Thesis"/>
<format type='PDF' target="http://timon.tmit.bme.hu/theses/thesis_book.pdf" />
</reference>
&I-D.ietf-rtgwg-lfa-applicability;
<reference anchor="LightweightNotVia"
target="http://mycite.omikk.bme.hu/doc/71691.pdf">
<front>
<title>IP Fast ReRoute: Lightweight Not-Via without Additional Addresses</title>
<author fullname="Gábor Sándor Enyedi" initials="G.S.E." surname="Enyedi"/>
<author fullname="Gabor Retvari" initials="G.R." surname="Retvari"/>
<author fullname="Peter Szilagyi" initials="P.S." surname="Szilagyi"/>
<author fullname="András Császár" initials="A.C." surname="Császár"/>
<date year="2009" />
</front>
<seriesInfo name="Proceedings of IEEE INFOCOM" value=""/>
<format type='PDF' target="http://mycite.omikk.bme.hu/doc/71691.pdf"/>
</reference>
<reference anchor="LFARevisited"
target="http://opti.tmit.bme.hu/~tapolcai/papers/retvari2011lfa_infocom.pdf">
<front>
<title>IP Fast ReRoute: Loop Free Alternates Revisited</title>
<author fullname="Gabor Retvari" initials="G.R." surname="Retvari"/>
<author fullname="Janos Tapolcai" initials="J.T." surname="Tapolcai"/>
<author fullname="Gábor Sándor Enyedi" initials="G.S.E." surname="Enyedi"/>
<author fullname="András Császár" initials="A.C." surname="Császár"/>
<date year="2011" />
</front>
<seriesInfo name="Proceedings of IEEE INFOCOM" value=""/>
<format type='PDF' target="http://opti.tmit.bme.hu/~tapolcai/papers/retvari2011lfa_infocom.pdf"/>
</reference>
<reference anchor="MRTLinear"
target="http://opti.tmit.bme.hu/~enyedi/ipfrr/distMaxRedTree.pdf">
<front>
<title>On Finding Maximally Redundant Trees in Strictly Linear Time</title>
<author fullname="Gábor Sándor Enyedi" initials="G.S.E." surname="Enyedi"/>
<author fullname="Gabor Retvari" initials="G.R." surname="Retvari"/>
<author fullname="András Császár" initials="A.C." surname="Császár"/>
<date year="2009"/>
</front>
<seriesInfo name="IEEE Symposium on Computers and Comunications (ISCC)" value=""/>
<format type='PDF' target="http://opti.tmit.bme.hu/~enyedi/ipfrr/distMaxRedTree.pdf"/>
</reference>
</references>
</back>
<!-- Change Log
v00a 2011-10-20 AKA First pass based on Gabor's initial write-up
v00b 2011-10-21 AKA Second pass
v00c 2011-10-22 Andras first pass, with minor corrections and comments
v00d 2011-10-23 Gabor alt selection added, cluster finding corrected, minor changes and comments
v003 2011-10-24 Alia changes based on comments, adding in Maciek's
comments, added references, extended comparison section,
changed terminology to block from inclusive 2-connected cluster.
-->
</rfc>
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