TIE-BREAKING IN SHORTEST PATH DETERMINATION
FIELD OF THE INVENTION
This invention relates to consistently selecting paths among multiple
possibilities, such as equal-cost shortest paths, in a packet-forwarding communications
network, such as an Ethernet network.
BACKGROUND TO THE INVENTION
In packet-forwarding communications networks, a node can learn about the
topology of the network and can decide, on the basis of the knowledge it acquires of the
topology, how it will route traffic to each of the other network nodes. The main basis for
selecting a path is path cost, which can be specified in terms of a number of hops between
nodes, or by some other metric such as bandwidth of links connecting nodes, or both. Open
Shortest Path First (OSPF) and Intermediate System-to-intermediate System (IS-IS) are
widely used link-state protocols which establish shortest paths based on each node's
advertisements of path cost These protocols typically do not attempt to tie-break between
multiple, equal-cost, paths. Instead, they typically spread traffic across several equal-cost
paths. The spreading algorithms are not specified and can vary from router to router.
Alternatively, they may make a local selection of a single path, but without consideration of
consistency with the selection made by other routers. Consequently, in either case the
reverse direction of a flow is not guaranteed to use the path used by the forward direction.
Multicast routing protocols such as Multicast Open Shortest Path First (MOSPF)
depend on each router in a network constructing the same shortest path tree. For this
reason, MOSPF implements a tie-breaking scheme based on link type, LAN vs. point-to-
point, and router identifier to ensure that identical trees are produced. However, basing the
tie-breaking decision on the parent with the largest identifier implies that, in general, the
paths used by the reverse flows will not be the same as the paths used by the forward flows.
Spanning Tree Protocols (Spanning Tree Protocol (STP), Rapid Spanning Tree
Protocol (RSTP), Multiple Spanning Tree Protocol (MSTP) are ways of creating loop-free
spanning trees in an arbitrary topology. The Spanning Tree Protocol is performed by each
node in the network. All of the Spanning Tree Protocols use a local tie-breaking decision
based on (bridge identifier, port identifier) to select between equal-cost paths. In Spanning
tree a root node is elected first, and then the tree is constructed with respect to that root by
all nodes. So, although all paths are symmetrical for go and return traffic (by definition, a
simple tree makes this the only possible construct), the election process is slow and the
simple tree structure cannot use any redundant capacity. Similarly, Radia Perlman's
Rbridges proposal uses the identifier of the parent node as tie-breaker.
Mick Seaman in his Shortest Path Bridging proposal to the IEEE 802.1 Working
Group (http://www.ieee802.org/1/files/public/docs2(X)5/new-seaman-shortest-path-0305-
02.pdf) describes a simple protocol enhancement to the Rapid Spanning Tree Protocol
which enforces consistent tie-breaking decisions, by adding a 'cut vector'. The proposal
uses a VID per node, to identify a Spanning Tree per node. In order to fit all the
information that needs to be transmitted by a bridge in a single legal Ethernet frame, this
technique currently limits the size of the Ethernet network to 32 bridges.
Figure 1 illustrates how, even for a trivial network example, a tie-breaking method
based on the parent node identifier fails to produce symmetric paths. In this example, the
links are considered as having equal-cost and so the determination of path cost simply
considers the number of hops. Consider first computing the path from A to B. When the
computation reaches node 2, the existence of equal-cost paths will be discovered. There is a
first path (A-1-3-6) and a second path (A-1-4-5). If the tie-breaking algorithm selects a path
based on the parent node with the smallest identifier, it will select the second path (A-1-4-5)
because node identifier 5 is smaller than node identifier 6. However, now consider
computing the path from B to A When the computation reaches node 1, the existence of
equal-cost paths will be discovered. There is a first path (B-2-6-3) and a second path (B-2-
5-4). Using the same tie-breaking criterion, the tie-breaking algorithm selects the first path
(B-2-6-3) because node identifier 3 is smaller than node identifier 4. So, it can be seen that
the shortest path computations made by nodes A and B provide inconsistent results.
There is a requirement in some emerging protocols, such as Provider Link State
Bridging (PLSB), a proposal to IEEE 802.1aq, to preserve congruency of forwarding across
the network for both unicast and unknown/multicast traffic and to use a common path in
both forward and reverse directions of flow. Accordingly, it is important that nodes can
consistently arrive at the same decision when tie-breaking between equal-cost paths.
Furthermore, it is desirable that a node can perform the tie-breaking with the minimum
amount of processing effort.
SUMMARY OF THE INVENTION
A first aspect of the invention provides a method of determining forwarding
information for use in forwarding packets at a first node of a packet-forwarding
network. The method determines the shortest paths between the first node and a
second node of the network and determines when a plurality of shortest paths have
substantially equal-cost. The method forms, for each substantially equal-cost path, a
set of node identifiers which define the set of nodes in the path and then orders each set
of node identifiers using a first ordering criterion to form a path identifier. The first
ordering criterion is independent of the order in which node identifiers appear in the
path. The method then selects between the plurality of equal-cost paths by comparing
the path identifiers. Each node of the network has a unique node identifier.
Advantageously, the first ordering criterion is increasing lexicographic order or
decreasing lexicographic order, although any ordering criterion can be used which
creates a totally ordered set of node identifiers.
Preferably, the method further comprises ordering the plurality of path
identifiers into an ordered list using a second ordering criterion. Similarly, the second
ordering criterion can be increasing lexicographic order, decreasing lexicographic
order or any ordering criterion which creates a totally ordered set of path identifiers.
Another aspect of the invention provides a method of determining forwarding
information for use in forwarding packets at a first node of a packet-forwarding
network. The method comprises determining shortest paths between the first node and
a second node of the network by iteratively forming a shortest path tree and
determines, while forming the shortest path tree, when a plurality of paths have equal-
cost, each equal-cost path comprising a branch which diverges from a divergence node
common to the equal-cost paths. The method identifies, in each diverging branch, a
node identifier using a first selection criterion to form a branch identifier and selects
between the plurality of branches by comparing the branch identifiers.
Advantageously, the method uses a total ordering criterion to compare and
select a node identifier in each branch, such as lexicographic order.
Advantageously, the method records the node identifier which meets the first
selection criterion in each of the diverging branches while backtracking to the
divergence node. This has an advantage in further simplifying computation and
reducing storage requirements.
Both aspects of the invention can be used to select two equal-cost paths by
using different first ordering/selection criteria and a common second ordering/selection
criterion or by using a common first ordering criterion/selection and different second
ordering/selection criteria. Three or four equal-cost paths can be selected in a similar
manner by consistently applying the first and second ordering/selection criteria at
nodes and selecting identifiers at a particular position in the ordered lists.
The invention can be used as a tie-breaker to select between equal-cost paths by
comparing an ordered set of node identifiers for each of a plurality of end-to-end paths.
Alternatively, it has been found that the same results can be achieved, on-the-fly, as a
shortest path tree is constructed, by making a selection of an equal-cost path using the
node identifiers of the diverging branches of the tree, local to where the selection
decision needs to be made. This has advantages of reducing the amount of
computation, and reducing the amount of data which needs to be stored. Branches can
be compared on a pair-wise basis to further reduce the amount of computation. This
becomes particularly important as the size and complexity of the network increases.
Both variants of the invention have the important property of allowing a consistent
selection to be made of equal-cost paths, regardless of where in the network the
shortest paths are calculated. This ensures that traffic flow between any two nodes, in
both the forward and reverse directions, will always follow the same path through the
network.
The invention is not intended to be restricted to any particular way of
determining a shortest path: Dijkstra's algorithm, Floyd's algorithm, or any other
suitable alternative can be used.
The invention can be used as a tie-breaker between equal-cost paths having
exactly the same value, or paths which are within a desired offset of one another both
in terms of link metric or number of hops. This may be desirable in real life situation
to increase the diversity between the set of eligible paths. For example, it may not
always be cost-effective to deploy nodes and links in the symmetrical fashion in
general required to achieve exactly equal-cost between any two end-points. By
relaxing the constraint to requiring that the hop count on different routes be within one
hop of each other, modest asymmetry can still result in eligible routes, and loop-free
topology is still guaranteed because a difference of two hops is the minimum necessary
to achieve a looping path.
It will be understood that the term "shortest path" is not limited to determining
paths based only on distance, and is intended to encompass any metric, or combination
of metrics, which can be used to specify the "cost" of a link, A non-exhaustive list of
metrics is: distance, number of hops, capacity, speed, usage, availability.
The method is stable in the sense that the selection of an equal-cost shortest
path is not affected by the removal of parts of the network that are not on the selected
paths, such as failed nodes or links.
Advantageously, the network is an Ethernet network although the invention can
be applied to other types of packet-forwarding networks, especially those that have a
requirement for symmetrical traffic-routing paths.
The functionality described here can be implemented in software, hardware or a
combination of these. The invention can be implemented by means of a suitably
programmed computer or any form of processing apparatus. Accordingly, another
aspect of the invention provides software for implementing any of the described
methods. The software may be stored on an electronic memory device, hard disk,
optical disk or other machine-readable storage medium. The software may be
delivered as a computer program product on a machine-readable carrier or it may be
downloaded to a node via a network connection.
A further aspect of the invention provides a network node comprising a
processor which is configured to perform any of the described methods.
A further aspect of the invention provides a network of nodes which each
consistently apply the described methods to select between equal-cost paths.
BRIEF DESCRIPTION OF THE DRAWINGS
Embodiments of the invention will be described, by way of example only, with
reference to the accompanying drawings in which:
Figure 1 shows a network topology having equal-cost paths;
Figure 2 shows an example of a packet-forwarding network in which the
invention can be implemented;
Figure 3 schematically shows apparatus at one of the bridging nodes of Figure
2;
Figure 4 shows the locality of tie-breaking decisions;
Figures 5 to 7 show example network topologies for illustrating calculation of
shortest paths;
Figure 8 shows a further example network topology for illustrating calculation
of shortest paths;
Figures 9 to 11 show tie-breaking steps of a shortest path calculation of the
network topology shown in Figure 8;
Figure 12 shows an example of nodes dual-homed onto a mesh network;
Figures 13A and 13B illustrate properties of the tie-breaking method of the
invention.
DESCRIPTION OF PREFERRED EMBODIMENTS
Figure 2 shows an example of a link state protocol controlled Ethernet network
10 in which the invention can be implemented and Figure 3 schematically shows
apparatus at one of the nodes 41-48. Nodes (also called bridges, or bridging nodes) 41-
48 forming the mesh network exchange link state advertisements 56 with one another.
This is achieved via the well understood mechanism of a link state routing system. A
routing system module 51 exchanges information 56 with peer nodes in the network
regarding the network topology using a link state routing protocol. This exchange of
information allows the nodes to generate a synchronized view of the network topology.
At each node, a Shortest Path Determination module 52 calculates a shortest path tree,
which determines the shortest path to each other node. The shortest paths determined
by module 52 are used to populate a Forwarding Information Base 54 with entries for
directing traffic through the network. As will be described in greater detail below,
situations will arise when module 52 will encounter multiple equal-cost paths. A tie-
breaking module 53 selects one (or more) of the equal-cost paths in a consistent
manner. In normal operation, packets are received 57 at the node and a destination
lookup module 55 determines, using the FIB 54, the port (or multiple ports in the case
of multicast distribution) over which the received packet should be forwarded 58. If
there is not a valid entry in the FIB 54 then the packet may then be discarded. It will
be appreciated that the modules shown in Figure 3 are for illustrative purposes only
and may be implemented by combining or distributing functions among the modules of
a node as would be understood by a person of skill in the art.
Various shortest path algorithms can be used to determine if a given node is on
the shortest path between a given pair of bridges. An all-pairs shortest path algorithm
such as Floyd's algorithm [R. Floyd: Algorithm 97 (shortest path), Communications of
the ACM, 7:345, 1962] or Dijkstra's single-source shortest path algorithm [E. W.
Dijkstra: A note on two problems in connexion with graphs, Numerical Mathematics,
1:269-271, 1959] can be implemented in the node 41-48 to compute the shortest path
between pairs of nodes. It should be understood that any suitable shortest path
algorithm could also be utilized. The link metric used by the shortest path algorithm
can be static or dynamically modified to take into account traffic engineering
information. For example, the link metric can include a measure of cost such as
capacity, speed, usage and availability.
By way of introduction to the problem, the requirements of a tie-breaking
algorithm which can make consistent decisions between equal-cost paths will firstly be
described. The list of requirements is set out in Table 1 below:
The essence of a tie-breaking algorithm is to always 'work'. No matter what set of paths
the algorithm is presented with, the algorithm should always be able to choose one and only
one path. First and foremost, the tie-breaking algorithm should therefore be complete (1).
For consistent tie-breaking, the algorithm must produce the same results regardless of the
order in which equal-cost paths are discovered and tie-breaking is performed. That is, the
tie-breaking algorithm should be commutative (2) and associative (3). The requirement that
tie-breaking between three paths must produce the same results regardless of the order in
which pairs of paths are considered (3) is not so obvious and yet it is absolutely necessary
for consistent results as equal-cost paths are discovered in a different order depending on
the direction of the computation through the network. The tie-breaking algorithm must be
symmetric (4), i.e. the tie-breaking algorithm must produce the same result regardless of the
direction of the path: the shortest path between two nodes A and B must be the reverse of
the shortest path between B and A. Finally, locality is a very important property of shortest
paths that is exploited by routing systems (5). The locality property simply says that: a sub-
path of a shortest path is also a shortest path. This seemingly trivial property of shortest
paths has an important application in packet networks that use destination-based
forwarding. In these networks, the forwarding decision at intermediate nodes along a path
is based solely on the destination address of the packet, not its source address.
Consequently, in order to generate its forwarding information, a node needs only compute
the shortest path from itself to all the other nodes and the amount of forwarding information
produced grows linearly, not quadratically, with the number of nodes in the network. In
order to enable destination-based forwarding, the tie-breaking algorithm must therefore
preserve the locality property of shortest paths: a sub-path of the shortest path selected by
the tie-breaking algorithm must be the shortest path selected by the tie-breaking algorithm.
Considerations of computational efficiency put another seemingly different
requirement on the tie-breaking algorithm: the algorithm should be able to make a tie-
breaking decision as soon as equal-cost paths are discovered. Figure 4 illustrates this point.
An intermediate node I is connected by two equal-cost paths, p and q, to node A and by
another pair of equal-cost paths, r and s, to node B. There are therefore four equal-cost
paths between nodes A and B, all going through node I: p+r, p+s, q+r, q+s. As the
computation of the shortest path from A to B progresses, the existence of equal-cost sub-
paths between A and I will be discovered first To avoid having to carry forward
knowledge of these two paths, the tie-breaking algorithm should be able to choose between
them as soon as the existence of the second equal-cost shortest sub-path is discovered. The
tie-breaking decisions made at intermediate nodes will ultimately affect the outcome of the
computation. By eliminating one of the two sub-paths, p and q, between nodes A and I, the
algorithm removes two of the four shortest paths between nodes A and B from further
consideration. Similarly, in the reverse direction, the tie-breaking algorithm will choose
between sub-paths r and s before making a final determination. These local decisions must
be consistent with one another and, in particular, the choice between two equal-cost paths
should remain the same if the paths were to be extended in the same way. For instance, in
the case depicted in Figure 3, the tie-breaking algorithm should verify the following four
identities:
tiebreak(concat(p, r), concat(q, r)) = concat(tiebreak(p, q), r)
tiebreak(concat(p, s), concat(q, s)) = concat(tiebreak(p, q), s)
coneat(p, tiebreak(r, s)) = tiebreak(concat(p, r), concat(p, s))
concat(q, tiebreak(r, s)) = tiebreak(concat(q, r), concat(q, s))
It turns out that the symmetry (4) and locality (5) conditions are both necessary and
sufficient to guarantee that the tie-breaking algorithm will make consistent local decisions, a
fact mat can be exploited to produce very efficient implementations of the single-source
shortest path algorithm in the presence of equal-cost shortest paths.
The list of requirements set out in Table 1 is not intended to be exhaustive, and
there are other properties of shortest paths that could have been included in Table 1. For
example, if a link whieh is not part of a shortest path is removed from the graph, the shortest
path is not affected. Likewise, the tie-breaking algorithm's selection between multiple
equal-cost paths should not be affected if a link which is not part of the selected path is
removed from the graph, and that even if this link is part of some of the equal-cost paths
that were rejected by the algorithm.
A first embodiment of a consistent tie-breaking algorithm will now be described.
This algorithm begins by forming a path identifier for each path. The path identifier is an
ordered list of the identifiers of each node traversed by the path through the network. The
node identifiers are sorted in lexicographic order. The path identifier is the resulting
concatenation of the ordered node identifiers. Figure 5 shows an example network, with
end nodes A, B and intermediate nodes 0-9. A first path (along the top of Figure 5)
between nodes A and B traverses nodes having the node identifiers A-0-5-6-1-4-8-B. After
ordering the list of node identifiers in ascending lexicographic order, the path can be
represented by the path identifier 014S68AB. This construction ensures that a path and its
reverse will have the same path identifier. Furthermore, because the algorithm is only
dealing with shortest paths or nearly shortest paths, only two paths - the direct path and the
corresponding reverse path - can share an identifier. Finally, the tie-breaking algorithm
simply selects the path with the smallest (or largest) path identifier. The algorithm can be
summarised as:
1) Sort the set of identifiers of the nodes in the path according to a first ordering
criterion which achieves a total ordering of the set of node identifiers. A preferred first
ordering criterion is increasing or decreasing lexicographic order,
2) Concatenate the set of ordered node identifiers to create a path identifier,
3) Sort the path identifiers according to a second ordering criterion which achieves a
total ordering of the set of path identifiers. A preferred second ordering criterion is
increasing or decreasing lexicographic order,
4) Select the path whose path identifier appears at one end (first or last) of the sorted
set of path identifiers. Advantageously, this step selects the path identifier appearing first in
the ordered set of path identifiers.
Each node in the network that performs this algorithm consistently uses the same
ordering criteria and selects a path at the same agreed position in the set of path identifiers,
in order to select the same path.
The term "lexicographic order" means the set of node identifiers are arranged in
order of size of identifier. So, if node identifiers are alphabetic, the set of node identifiers
are arranged in alphabetic order A, B, C, D...etc.; if node identifiers are numerical, the set
of node identifiers are arranged in numerical order. Clearly, this scheme can accommodate
any way of labelling nodes, and any combination of types of identifier. For example, a mix
of numbers and letters could be ordered by agreeing an order for numbers with respect to
letters (e.g. order numbers first, then letters). Alternatively, each character can be given it's
American Standard Code for Information Interchange (ASCII) code and the ASCII codes
can be sorted in increasing (decreasing) order. Each node uses the same convention to
order the node identifiers of paths in the same manner. This algorithm will produce
consistent results because: there is a one-to-one mapping between a path (strictly speaking
between the pair made up of a path and its reverse) and its identifier, and there is a total
ordering of the path identifiers.
Referring again to Figure 5, the top path between nodes A and B is represented,
after ordering, by the path identifier 014568AB. Similarly, a second path between nodes A
and B traverses nodes A-0-7-9-1-4-8-B and this can be represented, after ordering, by the
path identifier 014789AB. Finally, a third path (along the bottom of Figure 5) between
nodes A and B traverses nodes A-0-7-9-2-3-8-B and this can be represented, after ordering,
by the path identifier 023789AB. The tie-breaking algorithm compares each element of the
ordered path identifier, in an agreed direction. In this example, the convention that will be
used is that each node selects the lowest of the ordered path identifiers, when the path
identifiers are compared in a particular direction (e.g. left-to-right). The ordered path
identifiers, for the three equal-cost paths are:
014568AB
014789AB
023789AB
Starting with the left-hand element of the identifiers, all three path identifiers begin with '0'.
The next elements are '1' or '2', so only the top two identifiers need to be considered any
further. Reaching the fourth element, "0145..." is smaller than "0147..." and so the top
path is selected. Real node identifiers in IS-IS and Ethernet are composed of six 8-bit bytes
and are usually written as a hexadecimal string such as: 00-e0-7b-cl-a8-c2. Nicknames of
nodes can also be used, providing they are used consistently.
Figure 6 shows a simple network topology to illustrate the effects of different
ordering criteria. Two nodes, X, Y, are connected by four equal-cost paths having the node
identifiers 1-8. Four possible options will now be described:
• Sort node IDs by ascending order, sort path IDs by ascending order; select first
(smallest) path ID. If the node identifiers in each path are ordered in ascending
order of size (e.g. the top path with nodes 1, 7 becomes 17), that gives the path
identifiers 17, 28, 35, 46. Arranging these path identifiers in ascending order of
size, and selecting the first path identifier in the ordered list, has the result of
selecting the first (top) path, with the nodes 1 and 7.
• Sort node IDs by ascending order, sort path IDs by ascending order; select last
(largest) path ID. This option has the result of selecting the last (bottom) path, with
the nodes 4 and 6.
• Sort node IDs by descending order, sort path IDs by ascending order, select first
(smallest) path ID. Sorting the node identifiers in each path in descending order of
size gives path identifiers (71, 82, 53, 64). Arranging these path identifiers in
ascending order of size gives (53, 64, 71, 82) and selecting the first (smallest) path
identifier in the ordered list, has the result of selecting the third path, with the nodes
3 and 5.
• Sort node IDs by descending order, sort path IDs by ascending order; select last
(largest) path ID. This option has the result of selecting the second path, with the
nodes 8 and 2.
As will be described in more detail below, there are situations in which it is
desirable for nodes to apply multiple, different, ordering and/or selection criteria to select
multiple equal-cost paths.
So far this description assumes that the algorithm is non-local and that tie-breaking
is performed after all the equal-cost paths have been found. However, it has been found
mat a local version of this algorithm can produce the same results by considering only the
nodes on the diverging branches. Indeed, the tie-breaking result depends only on the
relative positions of the smallest node identifier in fee diverging branches. A second
embodiment of a consistent tie-breaking algorithm can be summarised as:
1) Find the node identifier in the diverging branch of the first path which meets a
first selection criterion. This can be considered a branch identifier for the first path;
2) Find the node identifier in the diverging branch of the second path which meets
the first selection criterion. This can be considered a branch identifier for the second path;
3) Select one of the paths using a second selection criterion, which operates on the
branch identifiers selected by steps (1) and (2).
Preferred options for the first selection criterion are to find the node identifier which
is the first (or last) when the node identifiers are arranged using a total ordering scheme,
such as lexicographic order (increasing or decreasing lexicographic order). As will be
explained below, it is not necessary for the scheme to compile the total set of node
identifiers in a branch and then order the set Instead, the scheme can iteratively compare
pairs of node identifiers using an awareness of lexicographic order. Similarly, preferred
options for the second selection criterion are to find the branch identifier which is the first
(or last) when the branch identifiers are arranged using a total ordering scheme, such as
lexicographic order (increasing or decreasing lexicographic order).
Referring again to the topology of Figure 6, the four equal-cost paths between nodes
X and Y can represent four equal-cost diverging branches from a parent node X. The tie-
breaking algorithm needs to select one of the four branches. There are four possible
options:
• Identify the smallest node ID in each branch. This results in (1, 2, 3, 4) as the
branch identifiers. Then, identify the smallest of the branch identifiers. This has the
result of selecting the first (top) path, with the nodes 1 and 7.
• Identify the smallest node ID in each branch. Then, identify the largest of the
branch identifiers. This option has the result of selecting the last (bottom) path, with
the nodes 4 and 6.
• Identify the largest node ID in each branch. This results in (5, 6, 7, 8) as the branch
identifiers. Then, identify the smallest of the branch identifiers. This has the result
of selecting the path with the nodes 3 and 5.
• Identify the largest node ID in each branch. Then, identify the largest of the branch
identifiers. This option has the result of selecting the path with the nodes 2 and 8.
As will be described in more detail below, there are situations in which it is
desirable for nodes to apply multiple, different, ordering and/or selection criteria to select
multiple equal-cost paths.
This algorithm can be implemented very easily and efficiently with simple
comparisons. Figure 7 shows another network topology. The local version of the method,
will start at node 13, and proceed to find two diverging branches leading from node 15.
The method explores the two separate paths as far as node 16, where the two paths
converge again. At this point, the method examines the node identifiers for each of the two
branches. For the first branch, the node identifiers are: 10, 14, 17, 21 and for the second
branch the node identifiers are: 11, 12, 19, 20. The branch with the lowest identifier (10) is
part of the top path. The method can simply backtrack from node 16 towards node IS,
keeping track of the lowest node identifier found in each branch. At each backward step,
the method compares the lowest node identifier found so far, with the new node identifier
encountered at that step. The lowest node identifier is stored. When the method has
backtracked as far as node 15, the two lowest values (10 in the top branch, 11 in the lower
branch) can simply be compared to one another to find the branch having the lowest node
identifier. Accordingly, the top branch, which forms part of the top path, is selected. The
part of the path common to both of the diverging branches is ignored when performing this
tie-breaking.
One of the most common algorithms for finding shortest cost paths in a network is
Dijkstra's algorithm [Dijkstra 59]. It solves the problem of finding the shortest paths from a
point in a graph (the source or root node) to all possible destinations when the length of a
path is defined as the sum of the positive hop-by-hop link costs. This problem is sometimes
called the single-source shortest paths problem. For a graph, G = (N, L) where N is a set of
nodes and L is a set of links connecting them, Dijkstra's algorithm uses a priority queue,
usually called TENT, to visit the nodes in order of increasing distance from the source node.
The other data structures needed to implement Dijkstra's algorithm are:
Distance: an array of best estimates of the shortest distance from the source node to
each node
Parent: an array of predecessors for each node
The following text describes the known Dijkstra's algorithm, and describes how it can be
modified to perform a tie-break when multiple equal-cost paths are discovered. Dijkstra's
algorithm is described here because it is one of the most commonly used shortest path
finding algorithms. However, it will be appreciated that other algorithms could equally be
used. The initialization phase sets the Distance of each node, except the source node itself,
to Infinity. The Distance of the source node is set to zero and its Parent is set to Null as it is
the root of the tree. At the start of the computation, the priority queue contains only the
source node. As the algorithm progresses, nodes are added to the priority queue when a
path from the source node to them is found. Nodes are pulled out of the priority queue in
order of increasing distance from the source node, after the shortest path between them and
the source node has been found. The algorithm stops when all the nodes reachable from the
source node have been cycled through the priority queue. While the priority queue TENT
is not empty, the algorithm performs the following steps:
1) Find the node N in TENT which is closest to the source node and remove it
from TENT
2) For each node connected to N, if the node's distance to the source would be
reduced by making N its parent, then change the node's parent to N, set the node's distance
to the new distance, and add the node to TENT.
Upon completion of the algorithm, Distance(node) contains the shortest distance from the
source node to the node (or Infinity if the node is not reachable from the source node) and
Parent(node) contains the predecessor of the node in the spanning tree (except for the
source node and the nodes which are not reachable from the source node). The parent of a
node is updated only if changing parents actually reduces the node's distance. This means
that, if multiple equal-cost shortest paths exist between the source node and some other
node, only the first one encountered during the execution of the algorithm will be
considered.
The above steps are conventional steps of Dijkstra's algorithm. At this point Dijkstra is
modified to add a consistent tie-breaking step. Step 2 above is modified as follows:
2) For each node connected to node N do the following:
2a) if the node's distance to the source would be reduced by making N its
parent, then change the node's parent to N, set the node's distance to the new distance, and
add the node to TENT.
2b) if the node's distance to the source node would remain the same after
making N its parent, then invoke the tie-breaking algorithm to determine if the node's
parent should be changed
The tie-breaking algorithm is invoked when a convergence point of two diverging branches
is reached. For example, considering the topology shown in Figure 7, if Dijkstra's
algorithm is started from node 13, diverging branches are discovered leading from node 15
(an upper branch with nodes 10, 14, 17, 21 and a lower branch with nodes 11, 12, 19, 20)
and these diverging branches converge at node 16. It is at node 16 that the tie-breaking
algorithm would be invoked to select between the two branches.
The pseudo-code below shows an implementation of the modified Dikstra's
algorithm with consistent tie-breaking using a priority queue implementation of the TENT
set. The Enqueue operation takes two arguments, a queue and a node, and puts the node in
the proper queue position according to its distance from the source node. The Dequeue
operation removes from the queue the node at the head of the queue i.e. the node with the
smallest distance from the source node.

The tie-breaking algorithm operates by back-tracking the two equal-cost paths, starting
from the current parent and the new candidate parent of the node respectively, all the way
back to the divergence point The fact that the two diverging paths may have a different
number of hops complicates matters slightly as the two paths must be backtracked by an
unknown, un-equal number of hops. This problem can be resolved by always back-
tracking the longer of the two paths first or both simultaneously when they have equal-cost.
Alternatively, this difficulty can be eliminated altogether by ensuring that two paths will
only be considered to be of equal-cost if, and only if, they have the same number of hops.
This is easily accomplished by either incorporating a hop count in the path cost or by using
the hop count as a first order tie-breaker.
The following pseudo-code shows an implementation of the tie-breaking algorithm
that assumes that the two paths have the same number of hops (and therefore so do their
diverging branches). The tie-breaking function takes the two nodes at the end of two equal
paths and returns one of them to indicate which of the two paths it selected.

The frequency with which the algorithm needs to be performed depends on the application.
PLSB essentially needs to compute the all-pairs shortest paths (sometimes a subset thereof).
In this case Dijkstra's algorithm needs to be run for all the nodes in the network (all but one
to be precise). Floyd's algorithm computes the all-pairs shortest paths so it would need to
be run only once. Other applications may only require the computation of a smaller number
of paths (e.g. if only one shortest path is required then Dijkstra's algorithm would have to
be run only once with one of the path's endpoints as the source).
Figure 8 shows an example network of nodes A-H, J interconnected by links. For
each link, a metric associated with that link is shown as an integer value on the link. There
are six different, equal-cost, shortest paths between node A and node B in this network.
These are shown in the table below with their respective length and path identifier:

All of these six paths have the same length, 10. The non-local version of the tie-
breaking algorithm will select the one with the smallest path identifier (ABCFH), i.e. path
AFCHB. The remainder of this section shows how the local version of the tie-breaking
algorithm arrives at the same result by making only local tie-breaking decisions as equal-
cost paths and sub-paths are discovered during the execution of Dijkstra's algorithm.
Dijsktra's algorithm initializes a table of distances and parents (or predecessors) for the
nodes in the network. All the distances are initially set to infinity except for the source node
whose distance is set to zero. The parents are undefined at this stage:

Dijkstra's algorithm also initializes its priority queue to contain only the source node, A:
TENT = [(A,0)].
The first iteration of the Dijkstra loop selects the first and only node in TENT, node A.
Then for each of node A's neighbours, namely nodes F and G, it updates their distance to
the source and makes node A their parent. Finally these two nodes are added to the TENT
priority queue.
During this first iteration of Dijkstra's algorithm the table of distances and parents becomes:

At the end of this first iteration the priority queue is: TENT = [(G, 1), (F, 2)].
The second iteration of the Dijkstra loop removes the node with the smallest distance, node
G, from the priority queue. It updates two of G's neighbours that have not been processed
yet, nodes C and D, and adds mem to the priority queue:

At the end of the second iteration, the priority queue is: TENT = [(F, 2), (D, 4), (C, 5)].
The third iteration of the Dijkstra loop removes node F from the priority queue. It updates
two neighbours of node F, nodes C and E, and adds node E to the priority queue (node C is
there already). The distance of node C does not change but there is a new candidate equal
path between node A and node C going through node F. The tie-breaking algorithm must
therefore be invoked to choose between this new path going through node F and the old one
going through node G. This is shown in Figure 9. The tie-breaking algorithm is invoked
with the new candidate parent of node C, node F, and its old parent, node G. oldMin is set
to the identifier of the old parent, G, and newMin is set to the identifier of the new parent, F.
Because nodes F and G share the same parent (node A), the backtracking loop is not
executed. The tiebreakmg simply compares oldMin and newMin and because newMin = F
< G=oldMin, node F is selected as the new parent of node C:

At the end of the third iteration, the priority queue is: TENT = [(D, 4), (E, 4), (C, 5)].
The fourth iteration of the Dijkstra loop removes one of the two nodes with distance 4, node
D for instance, from the priority queue. Of D's two neighbours only one, node H, is
updated and added to the priority queue:

At the end of the fourth iteration, the priority queue is: TENT = [(E, 4), (C, 5), (H, 6)].
The fifth iteration of the Dijkstra loop removes node E from the priority queue. Of E's two
neighbours only one, node J, is updated and added to the priority queue.

At the end of the fifth iteration, the priority queue is: TENT = [(C, 5), (H, 6), (J, 6)].
The sixth iteration of the Dijkstra's loop removes node C from the priority queue. Two of
C's neighbours, nodes J and H, have equal-cost paths to node A through node C. The tie-
breaking algorithm must therefore be invoked twice for nodes J and H respectively.
For node J, the tie-breaking algorithm is invoked with the new potential parent, node C, and
the old parent, node E. oldMin is set to the identifier of the old parent, E, and newMin is set
to the identifier of the new parent, C. Because these two nodes, E and C, share the same
parent (node F), the backtracking loop is not executed. The tiebreaking simply compares
oldMin and newMin and because newMin = C < E = oldMin, the new parent is selected.
Node J's parent is therefore replaced by node C. This is shown in Figure 10.
For node H, the tie-breaking algorithm is invoked with the new potential parent, node C,
and the old parent, node D. oldMin is set to the identifier of the old parent, D, and newMin
is set to the identifier of the new parent, C. Because these two nodes have different parents,
both paths must be backtracked one hop further. D's parent is G and because G > oldMin
(=D), oldMin does not change. C's parent is F and because F > newMin (=C), newMin
does not change either. Because F and G share the same parent, node A, the backtracking
loop stops. The tiebreaking algorithm then compares oldMin and newMin and because
newMin=C < D = oldMin, node C is selected to become node H's new parent. This is
shown in Figure 11.

At the end of the sixth iteration, the priority queue is: TENT = [(H, 6), (J, 6)].
The seventh iteration of the Dijkstra's loop removes one of the two nodes with distance 6,
node H for instance, from the priority queue. Only one of H's neighbours, node B, is
updated and added to the priority queue:

At the end of the seventh iteration, the priority queue is: TENT = [(J, 6), (B, 10)].
The eighth iteration of the Dijkstra's loop removes node J from the priority queue. Of J's
neighbours, only node B needs to be updated. Its distance does not change but there is a
new candidate equal path between node A and node B going through node J.
The tie-breaking algorithm is invoked with the new potential parent of node B, node J, and
the old parent, node H. oldMin is set to the identifier of the old parent, H, and newMin is set
to the identifier of the new parent, J. Because these two nodes, H and J, share the same
parent (node C), the backtracking loop is not executed. The tiebreaking simply compares
oldMin and newMin and because oldMin = H < J = newMin, the old parent is selected and
node B's parent remains the same.

At the end of the eighth iteration, the priority queue is: TENT = [(B, 10)].
Finally the last iteration of the Dijkstra's loop removes node B from the queue and the
algorithm terminates because none of B's neighbours can be updated (node B is the node
that is the furthest away from the source node A).
The reverse of the shortest path from node A to node B can be read directly from the parent
table starting at node B and following the parents until node A is reached: BHCFA. The
shortest path from node A to node B selected by the local tie-breaking algorithm is therefore
its reverse path: AFCHB.
Although there are 6 equal-cost paths between nodes A and B, the local tie-breaking was
only invoked a total of 4 times during the execution of Dijkstra's algorithm. At its first
invocation, the tie-breaking algorithm had to choose between sub-paths AFC and AGC. It
selected sub-path AFC, thereby elirninating two paths, AGCJB and AGCHB, from further
consideration At its second invocation, the tie-breaking algorithm had to choose between
sub-paths AFCJ and AFEJ. It selected sub-path AFCJ, thereby eliminating a third path,
AFEJB, from further consideration. At its third invocation, the tie-breaking algorithm had
to choose between sub-paths AGDH and AGCH. It selected sub-path AGCH, thereby
eliminating a fourth path, AGDHB, from further consideration. Finally, at its fourth
invocation, the tie-breaking algorithm had to choose between paths AFCHB and AFCJB. It
eliminated a fifth path, AFCJB, and selected path AFCHB as the final solution.
Selection of Equal-Cost Multi-Paths for Load Spreading
In many networking applications it is often advantageous to use several equal-cost
paths, especially if this can be achieved in a consistent fashion. By using two variants of the
tie-breaking algorithm, it is possible to use two equal-cost paths between a pair of nodes
when they exist. Figure 12 shows a common networking scenario in which edge nodes X
and Y are each dual-homed on a full mesh of core nodes A, B, C, D. For redundancy, each
edge node is connected to two core nodes, with node X connected to core nodes A and B
and node Y connected to nodes C and D. Each core node is connected to all of the other
core nodes, e.g. node A is connected to B, C, and D. The problem with this topology is that
if only one shortest path is used between a pair of nodes, a lot of access capacity will be
wasted under normal circumstances. When multiple equal-cost shortest paths exist between
two nodes, two variants of the tie-breaking algorithm can be used to consistently select
exactly two paths. Any convention, agreed by all nodes, can be used to make the selection
between equal-cost paths. One particularly convenient convention is to select a first path
having the smallest identifier and a second path having the largest identifier. In Figure 12,
since the core nodes are fully meshed, four equal-cost paths exist between the edge nodes X
and Y: (X, A, C, Y), (X, A, D, Y), (X, B, C, Y), (X, B, D, Y). The two variants of the tie-
breaking algorithm will select these two paths:
(X, min(A, B), min(C, D), Y) and,
(X, max(A, B), max(C, D), Y).
Because me node identifiers are unique, min(A, B) != max(A, B) and min(C, D) != max(C,
D): these two paths are maximally diverse: they have only their endpoints in common. In
Figure 12, the two selected paths are path (X, A, C, Y) and path (X, B, D, Y).
One of the important properties of the tie-breaking method described above is
that a change to the network which does not affect one of the set of paths for which the
tie-break needs to decide between has no impact on the outcome of the tie-break. Such
changes may involve removal of parts of the network that are not on the selected paths,
such as failed nodes or links. Another important property is that when multiple paths
equal-cost paths are used, a failure in one path does not affect the stability of the
others. Similarly, the addition of a link will only affect one of the equal cost paths, not
both. This is important for stability of the network.
Figures 13A and 13B illustrate other important properties of the tie-breaking
method of the present invention:
• a single failure in the presence of equal-cost paths cannot force a loop;
• a failure cannot both close the loop and shift the point of attachment of the
root;
• a failure cannot produce a shorter path;
• the tie-breaking algorithm prevents ranking of equal-cost paths from
changing the shortest path.
Figures 13 A and 13B illustrate these properties with a simple network topology having
nodes A, B, C, D and R. Considering Figure 13A, the shortest path between R and a set of
nodes A-D uses a link R-A There is a choice of two equal-cost branches to reach node C
from Node A Using one of the tie-breaking methods described above, the branch A-B-C is
consistently selected rather than the branch A-D-C. Similarly, in the reverse direction, the
link C-B-A is consistently selected instead of C-D-A. Figure 13B shows a situation, at a
later point in time, when the link R-A has failed. Node R now connects to the set of nodes
A-D via the next best link, R-C. There is a choice of two equal-cost branches to reach node
A from Node C. Again, the branch C-B-A is consistently selected rather than the branch
C-D-A. Without the use of this consistent tie-breaking algorithm, a loop A-B-C-D-A could
arise following the failure in link R-A with nodes A and B being slow and promiscuous in
their behaviour and nodes C and D being agile. This property is particularly useful to
guarantee loop freeness for multicast forwarding.
The invention is not limited to the embodiments described herein, which may
be modified or varied without departing from the scope of the invention.
We claim:
1. A method of determining forwarding information for use in forwarding packets
at a first node of a packet-forwarding network, each node of the network having a
unique node identifier, the method comprising:
determining shortest paths between the first node and a second node of the
network;
determining when a plurality of shortest paths have substantially equal-cost;
forming, for each substantially equal-cost path, a set of node identifiers which
define the set of nodes in the path;
ordering each set of node identifiers using a first ordering criterion to form a
path identifier, wherein the first ordering criterion is independent of the order in which
node identifiers appear in the path;
selecting between the plurality of equal-cost paths by comparing the path
identifiers.
2. A method according to claim 1 where the step of determining when a plurality
of shortest paths have substantially equal-cost is replaced with determining when a
plurality of shortest paths have exactly equal-cost.
3. A method according to claim 1 wherein the first ordering criterion creates a
totally ordered set of node identifiers.
4. A method according to claim 1 wherein the first ordering criterion is one of:
increasing lexicographic order, decreasing lexicographic order.
5. A method according to claim 1 further comprising ordering the plurality of path
identifiers into an ordered list using a second ordering criterion.
6. A method according to claim 5 wherein the second ordering criterion creates a
totally ordered set of path identifiers; and, the step of selecting between the plurality of
equal-cost paths comprises selecting the equal-cost path that appears at one end of the
ordered list of path identifiers.
7. A method according to claim 6 wherein the step of selecting between a plurality
of equal-cost paths comprises selecting the equal-cost path that appears one of: first in
the ordered list of path identifiers; last in the ordered list of path identifiers.
8. A method according to claim 5 wherein the second ordering criterion is one of:
increasing lexicographic order, decreasing lexicographic order.
9. A method according to claim 5 further comprising selecting two of the
substantially equal-cost paths by at least one of: using different first ordering criteria
and a common second ordering criterion; using a common first ordering criterion and
different second ordering criteria.
10. A method according to claim 5 further comprising selecting four of the
substantially equal-cost paths by: using two different first ordering criteria and a
common second ordering criterion; using the same two first ordering criteria and a
different second ordering criterion.
11. A method according to claim 9 wherein the first ordering criteria are: increasing
lexicographic order, decreasing lexicographic order; and the second ordering criteria
are: increasing lexicographic order, decreasing lexicographic order.
12. A method of determining forwarding information for use in forwarding packets
at a first node of a packet-forwarding network, each node of the network having a
unique node identifier, the method comprising:
determining shortest paths between the first node and a second node of the
network by iteratively forming a shortest path tree;
determining, while forming the shortest path tree, when a plurality of paths
have equal-cost, each equal-cost path comprising a branch which diverges from a
divergence node common to the equal-cost paths;
identifying, in each diverging branch, a node identifier using a first selection
criterion to form a branch identifier;
selecting between the plurality of branches by comparing the branch identifiers.
13. A method according to claim 12 wherein the first selection criterion uses a total
ordering criterion to compare and select a node identifier in each branch.
14. A method according to claim 12 wherein the first selection criterion uses
lexicographic order to compare and select a node identifier in each branch.
15. A method according to claim 14 wherein the first selection criterion uses
lexicographic order to select one of: a node identifier appearing first in lexicographic
order, a node identifier appearing last in lexicographic order.
16. A method according to claim 12 further comprising recording the node
identifier which meets the first selection criterion in each of the diverging branches
while backtracking to the divergence node.
17. A method according to claim 16 further comprising, at each backwards step,
comparing the recorded node identifier with a new node identifier encountered at that
step and recording the node identifier which meets the first selection criterion.
18. A method according to claim 12 further comprising selecting between the
plurality of branches by selecting a branch identifier using a second selection criterion.
19. A method according to claim 18 wherein the second selection criterion uses a
total ordering criterion to compare and select the branch identifiers.
20. A method according to claim 19 wherein the second selection criterion uses
lexicographic order to select a branch identifier.
21. A method according to claim 20 wherein the second selection criterion uses
lexicographic order to select one of: a node identifier appearing first in lexicographic
order; a node identifier appearing last in lexicographic order.
22. A method according to claim 12 which selects between the plurality of
branches on a pair-wise basis.
23. A method according to claim 18 further comprising selecting two of the equal-
cost paths by at least one of: using different first selection criteria and a common
second selection criterion; using a common first selection criterion and different
second selection criteria.
24. A method according to claim 18 further comprising selecting four of the equal-
cost paths by: using two different first selection criteria and a common second
selection criterion; using the same two first selection criteria and a different second
selection criterion.
25. A method according to claim 23 wherein the first selection criteria are: largest
node identifier, smallest node identifier, and the second selection criteria are largest
branch identifier, smallest branch identifier.
26. A method according to claim 12 comprising using Dijkstra's algorithm to
iteratively form a shortest path tree.
27. A computer program product comprising a machine-readable medium bearing
instructions which, when executed by a processor, cause the processor to implement
the method of claim I.
28. A network node comprising a processor which is configured to perform the
method of claim 1.
29. A network of nodes which each consistently apply the method according to
claim 1 to select between equal-cost paths.
30. A computer program product comprising a machine-readable medium bearing
instructions which, when executed by a processor, cause the processor to implement
the method of claim 12.
31. A network node comprising a processor which is configured to perform the
method of claim 12.
32. A network of nodes which each consistently apply the method according to
claim 12 to select between equal-cost paths.

A consistent tie-breaking decision between equal-cost shortest (lowest cost) paths is achieved by comparing an ordered
set of node identifiers for each of a plurality of end-to-end paths. Alternatively, the same results can be achieved, on-the-fly,
as a shortest path tree is constructed, by making a selection of an equal-cost path using the node identifiers of the diverging
branches of the tree. Both variants allow a consistent selection to be made of equal-cost paths, regardless of where in the network
the shortest paths are calculated. This ensures that traffic flow between any two nodes, in both the forward and reverse directions,
will always follow the same path through the network.