Photon Netw Commun (2007) 14:123–133
DOI 10.1007/s11107-007-0059-0
Comparison of p-cycles and p-trees in a unified mathematical
framework
Aden Grue · Wayne D. Grover
Received: 28 February 2007 / Accepted: 2 May 2007 / Published online: 17 August 2007
© Springer Science+Business Media, LLC 2007
Abstract As high-speed networks grow in capacity,
network protection becomes increasingly important.
Recently, following interest in p-cycle protection, the related
concept of p-trees has also been studied. In one line of
work, a so-called “hierarchical tree” approach is studied and
compared to p-cycles on some points. Some of the qualitative conclusions drawn, however, apply only to p-cycle
designs consisting of a single Hamiltonian p-cycle. There
are other confounding factors in the comparison between
the two, such as the fact that, while the tree-based approach
is not 100% restorable, p-cycles are. The tree and p-cycle
networks are also designed by highly dissimilar methods.
In addition, the claims regarding hierarchical trees seem to
contradict earlier work, which found pre-planned trees to
be significantly less capacity-efficient than p-cycles. These
contradictory findings need to be resolved; a correct understanding of how these two architectures rank in terms of
capacity efficiency is a basic issue of network science in
this field. We therefore revisit the question in a definitive and
novel way in which a unified optimal design framework compares minimum capacity, 100% restorable p-tree and p-cycle
network designs. Results confirm the significantly higher
capacity efficiency of p-cycles. Supporting discussion provides intuitive appreciation of why this is so, and the unified
design framework contributes a further theoretical appreciation of how pre-planned trees and pre-connected cycles are
related. In a novel further experiment we use the common
A. Grue (B) · W. D. Grover
TRLabs, 7th Floor, 9107-116St, Edmonton, AB, Canada T6G 2V4
e-mail: agrue@trlabs.ca
A. Grue · W. D. Grover
Dept. of Electrical and Computer Engineering,
University of Alberta, Edmonton, Alberta, Canada
W. D. Grover
e-mail: grover@trlabs.ca
optimal design model to study p-cycle/ p-tree hybrid designs.
This experiment answers the question “To what extent can a
selection of trees compliment a cycle-based design, or viceversa?” The results demonstrate the intrinsic merit of cycles
over trees for pre-planned protection.
Keywords p-Cycle · p-Tree · p-Segment · p-Path ·
Hierarchical tree · Restorable network · Integer linear
programming
Introduction
As the capacity of communication networks increases, protection of the network resources becomes increasingly important. One promising strategy for protecting networks via
rerouting of connections around single failed network spans
is the p-cycle approach. First proposed in [1], p-cycles are
formed out of spare capacity that is pre-cross-connected into
cycles and used to re-route working flow in the event of span
failures. The most significant property of p-cycles is that they
yield the high capacity efficiency of mesh networks, while
retaining ring-like protection switching speeds and the planning simplicity of fully pre-planned, predictable protection
reactions. A comprehensive background on p-cycles can be
found in [1] or Chapter 10 of [2]. Figure 1a shows a single
p-cycle (which may be one of several in a complete network
design), and Figs. 1b and c give examples of the p-cycle’s
restoration action for an on-cycle span and a straddling span
failure, respectively.
For later reference we note that for every span on the cycle,
the cycle offers a single surviving re-route path to the failure,
and two such re-routing paths for each span that “straddles”
the cycle (as in Fig. 1c).
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Photon Netw Commun (2007) 14:123–133
Fig. 1 (a) A unit-capacity
p-cycle, (b) its single-channel
protecting reaction to an
on-cycle span failure, and (c), its
dual-channel protecting reaction
to a straddling span failure
In this work we will be reviewing and defining a corresponding approach to pre-planned protection using trees
instead of cycles, and then comparing their spare capacity
requirements for 100% restorability to that of cycle-based
designs. In this regard, it is worth recounting that p-cycles
emerged originally out of a study of how many different types
of pre-connected spare capacity “patterns” could be shared
efficiently over multiple distinct failure scenarios to form
highly failure-ready survivable networks [3]. Failure-ready
in this sense meant mainly that—other than the cross-connections at the end-nodes of failures, used to switch affected
working traffic into the protection paths—the protection paths
themselves would be either maximally or fully pre-crossconnected before failure, in a state as needed by the failure.
It turns out this is possible by design to fully achieve this
condition using only p-cycles. But in the original studies that
lead to that finding, cycles, linear segments, trees, and even
completely arbitrary patterns, were all considered admissible
to the design problem under a genetic algorithm that aimed
to evolve a set of pre-configured patterns with minimal spare
capacity for 100% restorability. It was surprising at first that
the most efficient solutions from the GA were based almost
entirely on cycles. The same finding was sustained in comparison of an all-cycles based design obtained by integer liner
programming (ILP) in comparison to an all-tree based survivable network, developed using adaptations of spanning
tree algorithms. The surprising efficiency of p-cycles was
rendered intuitively understandable by the central role of the
availability of two protection paths for straddling span failure
scenarios [1].
It is relevant that, as far back as [3], it was appreciated
that what is significant about cycles as protection structures,
compared to any acyclic structure such as a path segment or
a tree, is that an acyclic building block element can protect
at most one working channel (per unit of its own capacity).
The instant that the pre-connected path segment is extended
to close on itself, forming a cycle, this ratio jumps to two.
This is an inherent, discontinuous jump in efficiency that
exists between cyclic structures and acyclic structures, such
as trees. We feel that any new findings that show tree-based
protection to be equal to or greater than p-cycles in capacity efficiency should be investigated to determine the cause
of the apparent contradiction between such findings and the
original work performed in [3].
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Given the impact that p-cycles have had in recent years,
there has been renewed interest in the overall idea of preconnected protection ( p-) structures in general. Next to
cycles, tree-oriented techniques are probably the most obvious and interesting counterpart to consider. Tree-based algorithms for non-survivable networking and for applications
like broadcasting and computer networking have been well
developed in past decades. The heritage of past tree-based
graph algorithms and the recent interest in protection, and
the emergence of p-cycles all have lead to a new interest
in considering tree-based protection. The increased interest in tree-based approaches makes the comparison against
p-cycles of heightened interest and relevance.
This leads to our focus here, which is on the capacity efficiency of p-cycle designs versus tree-based designs. Other,
more qualitative characteristics such as perceived flexibility,
speed, scalability, and so on will be commented on but are
not the main emphasis. To be meaningful, two ground rules
must apply to the comparison: (1) both types of design must
be 100% restorable to any single span failure and (2) for fair
comparison, both types of network should be based on optimal methods for the corresponding design problem. If (1)
did not pertain, the comparison of protection capacity is not
“apples-to-apples.” And (2) is required so that any heuristic, empirical, or algorithmic bias or approximation cannot
be attributed to one or the other set of designs as confounding our ability to gain reliable findings and insights in the
comparison.
Literature
The original work done in [3] on pre-connected patterns considered pre-connected tree structures for the purpose of span
protection, and was discussed in passing above. While useful to discover the high merit of p-cycles at the time, the
genetic algorithm method was suboptimal, and did not provide a rigorous comparison of optimal pure tree-based solutions against corresponding pure optimal p-cycle designs.
Other areas in which one finds consideration of tree-based
protection include “shared backup trees” [4–6] and redundant
or “red/blue” trees [7–9]. Shared backup trees are rooted tree
structures that protect unidirectional working paths that terminate at their roots, while redundant trees are designed such
Photon Netw Commun (2007) 14:123–133
125
Fig. 2 (a) a chain of three
nodes, (b) the impossibility of
finding a spanning h-tree for
100% restorability in such a
network, and (c) an unrestorable
failure due to the improper
choice of h-tree and root node
the failure of any single span in the network leaves the end
nodes of every working path connected to each other through
at least one of the trees. None of these works, however, offers
comparison of trees to p-cycles for span restorable mesh network design, which is the present aim.
This brings us to the aforementioned line of work on “hierarchical trees” [10–12] (there is also a related Ph.D. thesis
[13] under revisions at the time of this draft), which motivated
the present effort. In this series of publications a number of
surprising assertions are made as to how tree and cycle-based
protection schemes compare. Particularly, found in [11] are
statements that give a mistaken impression that p-cycle networking is always based on the use of a single Hamiltonian
cycle, statements that non-shortest path restoration re-routing
wastes protection capacity, and statements in regard to spare
capacity efficiency that “the performance of our tree algorithm… does come very close to cycle-based schemes”. The
last claim particularly is generally contrary to what was found
in [3], and therefore merits further investigation. Furthermore, this also brings to light the fact that, to date, there
has not been any formal, systematic comparison study of
p-cycles and trees in terms of spare capacity efficiency.
Part of the current confusion arises from different notions
of what tree-based pre-configured survivability actually
means. We are of the opinion that it should be axiomatic
that any pre-configured scheme should be able to support
100% restorability against any single span failure by design,
as was assumed in [3]. It would then be the spare capacity of
such designs that one would consider in comparison to other
schemes. In addition, using tree-based structures implies that
any failure scenario derives its survivability from immediately available fully pre-configured protection paths found
within these structures.
But neither of these are properties of the hierarchical tree
approach of [10–12]. First of all, it cannot assure 100% restorability by design on an arbitrary bi-connected network. There
are two possible causes for this. First, spans more than one
hop from each end of a chain of degree two nodes remain
un-restorable because no single hop off-tree routing exists
which would restore the failure, and secondly, the methods for root node choice and spanning-tree formation from
[10–12] have no ability to assure by design, that there will
not be on-tree spans which also turn out to be un-restorable
because the only off-tree links available from the “downstream” node of the failed link return to the tree on the same
side of the failed link. These two cases are illustrated in
Fig. 2.1
Figure 2a shows an instance in which the network contains a chain of three degree-2 nodes. Figure 2b illustrates
an attempt to find a spanning h-tree for this network. Obviously, the illustrated tree is not a spanning tree at all, because
it does not reach node X. But if we were to extend the tree
from span A to node X, span A would become unrestorable,
as there would be no off-tree hop available from either of its
nodes to reach the other side of the tree. A similar argument
applies to extending the tree from span B. Therefore no h-tree
can reach node X in the middle of the degree-2 node chain,
and a spanning tree to provide 100% restorability cannot be
found. Figure 2c illustrates the second situation, in which the
downstream node of the h-tree in the illustrated failure situation cannot find a single hop off-tree routing to the opposite
section of the tree (i.e. nodes X or Y) as needed to effect
restoration. The only adjacent nodes are on the same side of
the span failure as the downstream node itself. Nothing in the
algorithms given for h-tree in any of the papers on the topic
[10–12] includes a way to construct the spanning h-tree so
that instances of Fig. 2c are avoided by design. In both situations, the hierarchical tree scheme will be unable to restore
against the given failures.
In addition, the above-mentioned need to resort to offtree routing methods for the failure of on-tree spans introduces two separate classes of restoration in the h-tree scheme:
one for off-tree spans and one for on-tree spans. This helps
explain the apparent contradiction between the conclusions
1
Note that in both situations, the problem is not that there is no off-tree
route at all over which the illustrated failures could be restored. In all
cases the graph remains a bi-connected graph containing one failure, so
there is always at least one surviving route through the graph between
the failure end-nodes, which can be found be a generalized re-routing
procedure if a more generalized approach than the hierarchical tree was
used. Rather, the un-restorability arises from the strict adherence under
the “h-tree” scheme that generalized off-tree routing is not used. Under
the h-tree scheme, restoration for on-tree failures occurs always over a
path that begins with a single off-tree span that is adjacent to the end
of the failure span that is farthest away from (downstream of) the root
node of the h-tree. This rigid on-tree failure restoration approach results
in the two classes of problems listed.
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drawn in [11] and previous knowledge about the spare capacity efficiency of tree-based protection; the h-tree scheme is
in fact not fully tree-based for a significant proportion of
network failures. In fact, it may be viewed as just a special case of the more general span-restorable mesh scheme,
except that the restoration routes for some spans are constrained to lie on the pre-planned tree. In contrast, in p-cycle
network designs, every span is similarly protected by fully
pre-formed paths through a p-cycle. In efficient p-cycle network design, multiple cycles are used and each is an inherently local structure, the size of which individually has no
connection to the overall network size. Thus, it is problematic to compare h-tree protection to p-cycles. A meaningful
comparison of trees to p-cycles would involve a concept of
tree-based protection which provides the same key properties as the p-cycle designs, i.e., (1) capability to be designed
for 100% restorability (on any biconnected graph), and (2)
supporting fully structure-based restoration for all spans (not
a mixture of partly tree-based restoration supplemented with
an off-tree routing algorithm). Figure 3 illustrates the resulting concept of complementary tree-based protection design,
which is comparable to p-cycles in these two main regards.
It is this conception of what tree-based survivability design
embodies that will be studied here in comparison to p-cycles.
We call this approach “ p-trees”.
In other literature on tree-based survivability design, a
paper by Tang and Ruan [14] recently picked up on the single spanning tree approach from [10–12] and modified the
on-tree protection scheme such that 100% restorability could
at least be attained in all cases. The authors first prove that
there will always exist a restoration route for an on-tree span
Photon Netw Commun (2007) 14:123–133
that contains at most one off-tree span. The reason that the
original h-tree scheme fails in some cases, however, is that it
requires that this off-tree span be adjacent to the downstream
node of the failed span. When this is not the case, the restoration route is not found. Tang and Ruan simply relax the
scheme slightly such that the restoration route can be found
even when this single off-tree span is not adjacent to the failure. In [14] the authors also provide the first example, in the
literature that we know of, of an ILP model that could be used
to solve the hierarchical tree protection problem. However,
they do not make it their goal to compare trees to p-cycle
designs. And the fact remains that the h-tree concept, even
when treated by ILP methods, remains not fully tree-based;
it is a hybridization of tree-based protection with off-tree
dynamic routing concepts.
Experimental method
In this section we explain how the capacity-comparison of
p-trees and p-cycles can be approached through the use
of a single unified ILP design model in which cycles and
trees are just different classes of pre-configured patterns that
one may elect to use to achieve the survivable design. This
not only gives a single overarching theoretical framework
which unifies p-trees and p-cycles under the higher classification of pre-configured pattern protection schemes, but it
also provides an unassailably fair “apples and apples” basis
for quantitative comparison of p-tree and p-cycle network
designs.
Design assembly under a common framework
Fig. 3 A set of p-trees that provides full restoration using only treebased protection
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The central realization is that, viewed in a certain optimization framework, the p-cycle design problem is not different than the p-tree problem. p-Trees and p-cycles are both
span protecting technologies based on the idea of pre-defined
structures formed in the protection layer. This allows their
optimal design problems to be expressed as a common ILP
problem that is simply provided with different families of
candidate protection structures for its solution. An important implication is that, if the set of candidate structures provided is the set of all distinct trees on the network graph,
then all possible schemes for tree-based network design will
be inherently considered, as the ILP tree solutions will be
known-optimal solutions, the best possible that can exist by
any algorithm. This is also practically valuable because it
means that to compare p-trees and p-cycles (at the fundamental level intended here) we are not obliged to find and
implement every possible known algorithm for the tree-based
approach. The best performance of any possible tree algorithm will be reflected in the results. The same argument
Photon Netw Commun (2007) 14:123–133
127
Fig. 4 A p-tree (a), its
reactions to two protectable
span failures (b) and (c), and an
on-tree failure (d) that cannot be
protected by the tree itself, and
must be protected by another
complementary tree
applies to the p-cycle case if it is populated with the set of
all distinct (simple) cycles of the graph.
Thus, p-trees are to be formulated under the same general
paradigm of pre-configured structures of spare capacity used
by p-cycles. To do this we have to characterize trees as protection structures in a manner analogous to how we characterize candidate cycles in the p-cycle design problem. This
involves determining the parameters that describe the layout of each tree on the graph and that encode the amount
of protection any candidate tree provides to any prospective
failure. Let us illustrate by considering a specific candidate
tree in Fig. 4. In Fig. 4b the tree provides protection for one
working channel on the failed span shown (per unit capacity
of the tree itself). A tree is, however, fundamentally unable to
provide protection to spans that are on the tree itself, because
no surviving path through the structure itself remains when
a tree is cut; protection of these spans must be performed by
other, complementary trees in the design. In other words, the
only types of failures that trees can protect against are analogous to the “straddling” span failures in p-cycles, i.e., spans
that are not part of the tree but have both end-nodes on the
same tree. Even then, however, trees can only provide one
protection path for a failed “straddling” span, as opposed to
the two provided a p-cycle.
Also note that when a tree is used to restore a failed span,
some parts of the tree will remain unused (dashed lines in
Fig. 4b, c) and have to be “pruned off” in real-time, allowing
formation on the desired single path through the tree. This
implies that p-trees inherently not as amenable to strict prefailure pre-cross-connection of protection paths in the same
sense as p-cycles are.
Sets:
S is the set of spans in the network, and is usually indexed
by i for a failure span, and j for surviving spans.
K is the set of candidate structures in the graph eligible for
formation of protection structures, and is usually indexed by
k. In the traditional p-cycle model this is a set of candidate
cycles (usually all cycles in the graph if possible).
The common ILP model
The set of constraint inequalities (1) ensures that there will
be enough structures to protect all of the working capacity
on each span in the network. The set of constraint equations
(2) ensures that there will then be enough spare capacity on
each span to support the allocation of the structures set by n k .
The ILP model that has traditionally been used to solve the
p-cycle design problem can be adapted to serve also for
p-tree network design as follows.
Input parameters:
wi is the number of working channels (or capacity units)
on span i that require protection.
xik ∈ {0, 1, 2} encodes the number of protection relationships provided to span i by a unit-sized copy of structure k.
δ kj ∈ {0, 1} encodes the spans on protection structure k
itself.
C j is the cost of a unit of capacity (i.e., a single channel)
placed on span j.
Decision variables:
s j ≥ 0 is the integer number of spare channels assigned
to span j in the design.
n k ≥ 0 is the integer number of unit-capacity copies of
structure k in the design.
Objective function:
Cj · sj
Minimize
j∈S
Constraints:
wi ≤
xik · n k ∀i ∈ S
(1)
δ kj · n k ∀ j ∈ S
(2)
k∈P
sj =
k∈P
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Photon Netw Commun (2007) 14:123–133
Table 1 Meanings of the model
parameters specific to the tree
and cycle contexts
For a candidate tree
For a candidate cycle
k
Index of the candidate tree
Index of the candidate cycle
xik
1 if span i has its end nodes on the tree but
is not on the tree itself, 0 otherwise
2 if span i straddles cycle k, 1 if span i is
on cycle k, 0 otherwise
δ kj
1 if span i is on tree k, 0 otherwise
1 if span i is on cycle k, 0 otherwise
nk
The number of unit copies of this p-tree in
the solution
The number of unit copies of this p-cycle
in the solution
The objective function minimizes the total amount of spare
capacity in the network.
As stated above, the model employs generalized descriptions of the candidate “protection structures” through parameters k, xik , δ kj , and n k . For our purposes, these structures
will be either trees or cycles. Table 1 elaborates on the
meaning of each of these parameters in the tree and cycle
contexts.
Note that the basic ILP model is capable of even greater
generality than we will use it for here. Nothing in the ILP
is itself specific to cycles or trees. If the parameters xik and
δ kj are computed accordingly, they can represent any preconnected protection structures. For our purposes, though,
when the set K is populated with candidate tree structures,
the model will provide a lowest cost set of p-trees to protect the network from all single span failures. Similarly, K is
populated with candidate cycles for the pure p-cycle design
case. But note also that the model has the ability to consider
both p-cycles and p-trees at the same time to allow study
of a hybrid of the p-cycle and p-tree architectures. This is
possible by populating K with both candidate cycles and candidate trees simultaneously. The solver is indifferent to the
distinctions we make between “protection architectures” and
will simply use the best span-protecting structures out of the
set it is given to protect the network.
Figure 5 thus outlines how the unified ILP model is to be
used in this study. The hybrid experiment will be especially
interesting because it will answer the question: “What if the
unbiased optimizer were allowed to decide for itself, on a
structure by structure basis, what specific trees and what specific cycles it wanted to use together to achieve an even more
comprehensively optimized overall design?” Will a globally
optimum design that is free to use any possible choices of
tree or cycle consist of a significant hybridization of trees
and cycles? Or will the solver chose all trees, or all cycles,
based on superior ability to reduce the objective function?
And will the spare capacity decrease significantly below that
achievable by the best pure architecture alone?
Practical considerations
In reality, the size of the sets of all candidate trees may be so
large that we cannot represent the entire set in all our experi-
123
Candidate
trees
Candidate
cycles
All trees +
all cycles
Common
ILP design
model
Optimal p-tree
designs
Optimal p-cycle
designs
Optimal treecycle ìh ybrid”
designs
Fig. 5 Block diagram of the single-model/multiple architecture design
concept
ments. The candidate cycle sets are always smaller however,
and amenable to complete representation in practically-sized
problems. Although theoretically the numbers of cycles and
trees both increase exponentially with the size of a network,
for practical network sizes (up to perhaps 50 nodes at the very
most), the number of distinct simple cycles to serve as p-cycle
candidates is quite manageable. The set of all trees, however,
increases in size much faster, becoming impractical in some
of our test cases for present-day ILP software and computer
hardware. As a result, we are restricted to either solving optimal designs for small networks, or making compromises for
network designs using larger topologies by applying reasonable criteria by which to exclude trees from the candidate set
to limit its size to a manageable level. The results that follow
are based on both strategies. For the “small” networks results
are truly optimal for tree, cycles, and hybrids. Where noted,
results on larger topologies are based on a restricted set of trees
used in the candidate set. Specifically, this involved limiting
both the size of the trees (i.e. the number of spans they contain), and their maximum nodal degree (i.e., the largest number of spans that branch out from any node in the tree). Details
are given for each test case.
Photon Netw Commun (2007) 14:123–133
129
Test cases
Results and discussion
Our “small” network designs consist of p-cycle, p-tree and
hybrid designs for 10 different test networks that together
comprise a network family. A network family is generated
by starting with a “master network” with a network graph of
degree 4, and removing one randomly chosen span at a time,
while retaining bi-connectivity (necessary to ensure the possibility of 100% single span failure restorability), until no
more spans can be removed in this way. Each of the intermediate networks created this way has a different average
nodal degree but serves the same node and demand set, facilitating comparison based on the effect of varying network
connectivity while keeping other factors constant. The master network used to create this family contained 10 nodes
and 20 spans. The family itself consists of 10 networks with
11–20 spans (nodal degree of 2.2–4). All results on networks
from this family are based on the mathematically complete
sets of candidate cycles and trees.
A further set of 35 larger test networks was drawn from
two more network families, one with 15 nodes and one with
20 nodes. The 15-node family contains 15 test networks with
16–30 spans (nodal degree of 2.13–4) and the 20-node family contains 20 networks with 21–40 spans (nodal degree of
2.1–4). For these test cases, the set of candidate structures
consisted of all trees on the graph containing seven or fewer
spans, with a maximum nodal degree of 3. This means that,
even though the ILP problem instances where trees were
considered were allowed to run to completion, they are not
strictly optimal designs. However, these results are still useful
in the sense that they provide an indication of the comparative usefulness of p-cycles and p-trees in design problems
of realistic size.
All of the networks in a single network family share the
same demand pattern between their nodes, because although
the number of spans varies between the networks, the number
and placement of the nodes does not change. Three demand
patterns were used, one for each family. The demand patterns were created by allocating a random integer demand
value, chosen uniformly on the interval from 1 to 10 inclusive, to each node pair. The degree 4 master networks used
to generate each network family are shown in Fig. 6.
Spare capacity costs
Figure 7 gives the spare capacity cost of the three different
types of designs (pure tree, pure p-cycle, and hybrid) over
all test cases. Logically, because the hybrid designs are able
to make use of all the cycles and trees available to the pure
p-cycle and p-tree design problems, the spare capacity costs
of a hybrid design should never be greater than either of the
pure architecture designs for the same test case. In fact, it
turns out that the cost difference between the hybrid designs
and the pure p-cycle designs is so small that the curves for
hybrids and pure p-cycle design never graphically distinguish themselves on the chart. (This is why Fig. 7 appears to
have only six lines, although nine are expected: three design
tests for each of the three test network families). We will
return to discuss this finding further.
The main finding portrayed in Fig. 7 is that the spare capacity requirements of the p-tree designs are usually at least
double that of the p-cycle designs. In case it is thought that
this could be a result of limiting the candidate tree sizes in
the 15- and 20-node families, we can consult the 10 node
designs, where the strictly complete set of all possible trees
is represented in the problem. Even in those cases, the best of
the designs (which occurs for the 18-span network) still uses
164% as much spare capacity as the corresponding p-cycle
design.
The curves for the p-tree designs in the 15- and 20-node
network families in Fig. 7 end (viewing in a right to left sense)
before those for the p-cycle curves on the same networks.
The reason gives an additional insight about the nature of the
optimal tree network design problem. In the sparser topology members of these families, it turns out to be infeasible
to achieve 100% restorability using trees of only seven spans
or fewer. One could try to repeat these results with a higher
allowance for tree size, but the number of candidate trees
grows so rapidly that this becomes technically very difficult. Furthermore, it is not in any case required to see the
trend and the relative ranking of p-tree and p-cycle spare
capacity requirements that appears in the results that are
available.
Fig. 6 10-node (a), 15-node
(b), and 20-node (c) master
networks
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Photon Netw Commun (2007) 14:123–133
Fig. 7 Comparison of p-cycle,
p-tree, and hybrid designs
(hybrid costs are
indistinguishable from p-cycle
costs at the resolution shown)
10 node, p-Tree
15 node, p-Tree
20 node, p-Tree
10 node, p-Cycle (and Hybrid)
15 node, p-Cycle (and Hybrid)
20 node, p-Cycle (and Hybrid)
900000
Spare Capacity Cost
800000
700000
600000
500000
400000
300000
200000
100000
0
2
2.5
3
3.5
4
Network Nodal Degree
Returning to the hybrid design tests, what of the finding that the hybrid designs with complete tree and cycle
sets wound up having spare capacity costs which are almost
indistinguishable from the p-cycle designs? First of all, it
confirms the near optimality of pure p-cycle design in this
broader space of alternatives. It also suggests that, when
absolutely free to choose any mixture of structural building
blocks, the unbiased optimal solver chooses to use cycles,
not trees, and not even substantial hybrids of trees with pcycles. But were any trees employed at all in the hybrids?
In fact some small spare capacity savings (not graphically
distinguishable in Fig. 7) did arise in some cases with the
hybrid designs. Figure 8 shows a blown up view of the actual
differences in spare capacity costs attributable to adding the
set of trees to the design problem. The largest differences in
spare capacity between a pure p-cycle design and its corresponding design hybridized with p-trees is 5.8%, in the test
network with 10 nodes and 14 spans in its topology. The average improvement for the 10-node family is 1.9%, and rarely
exceeds 1% in the other families. In many cases absolutely
no improvement is found at all. The fact that improvements
exist at all, however, might be evidence of there being at
least some merit to adding trees in some cases to p-cycle
designs. The nature of these hybrid designs is discussed
further shortly.
Having noted how much more numerous the set of all
trees is than the set of all cycles, we decided to also quantify
this aspect of the p-cycle/ p-tree design comparison. Figure 9
presents the size of both these sets in each of the test cases.
Note that the 10-node designs use approximately the same
number of tree candidates as the 15-node designs, because
the 10-node designs use all trees while the tree set for the
15-node designs is limited. The plot shows that the set of
trees is usually larger than the set of cycles by up to two
orders of magnitude. But even with access to this drastically increased set of protection choices, the best possible
improvements we can obtain are on the order of only a few
percent.
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Analysis of the hybrid designs
Although there was never a practically significant overall
reduction in spare capacity by admitting trees to the design
problem, we were curious to understand what situations lead
to an improvement at all via hybridization with tree structures. We therefore checked each case in detail. It turns out
that in most cases, where a unit capacity tree or trees were
selected into the hybrid design, they were non-branching segments, not trees at all in the usual sense of the word. Segments
are, however, technically structures that are included in the
set of “all trees.” In the context of the hybrid design problems,
the ILP solver can be seen as an impartial arbiter, deciding
between p-cycles and p-trees for inclusion in the network
design solely according to which structures contribute best
towards minimizing the total spare capacity. Therefore, these
p-tree segments can be considered to be, in a sense, the best
representatives of the more general set of trees.
Surprisingly, even in the cases where trees only provide a
fraction of a percent improvement, they can sometimes make
up a significant fraction of the total number of unit-capacity
structures present. For example, the 20- and 36-span hybrid
design is only 0.29% less costly than the corresponding pure
p-cycle design, but contains a total of 73 capacitated “trees”
(all segments) and 127 capacitated cycles. These 73 trees
are all unit copies of the single segment pattern illustrated in
Fig. 10. As mentioned, this structure is from the tree set, but
it does not exhibit any branching. In fact, out of all of the
“ p-trees” that were found to be used in the hybrid designs,
86% of them (357 out of 415) are actually purely linear segments. In fact these structures are already know under another
name as p-segments or “ p-paths” [15]. The segments that
provide this slight enhancement to predominantly p-cycle
designs can be thought of as cycles for which including the
last remaining working channels to close the cycle cannot be
justified. In other words a p-segment is like a p-cycle but
one for which, upon checking, it is found that one or more
spare channels of the p-cycle can be removed, because the
Photon Netw Commun (2007) 14:123–133
7
10 node family
15 node family
20 node family
6
Hybrid Cost Reduction (%)
Fig. 8 Cost differences
between p-cycle designs and
their corresponding hybrid
p-cycle/ p-tree designs
131
5
4
3
2
1
0
2
3
Network Nodal Degree
3.5
4
1000000
Number of Candidate Structures
Fig. 9 Number of candidate
trees and cycles in the test
networks
2.5
10 node Trees
15 node Trees
20 node Trees
100000
10 node Cycles
15 node Cycles
20 node Cycles
10000
1000
100
10
1
2
Fig. 10 (a) A “ p-segment” from the 20-node, 36-span test network
hybrid design, and (b) a true degree-3 tree from the 15-node, 27-span
network
extra protection relationships provided by closing the cycle
are not needed. This effect can also be thought of as a way of
fine-tuning the amount of spare capacity in a p-cycle design
to only a very specific demand pattern. As soon as any ongoing growth in the demand occurs, p-segments will almost
instantly be preferred to have been fully closed p-cycles from
the start to accommodate the new working capacity.
In the remaining cases (about 15% by comparing quantities of unit capacity structures and 29% by counting only
2.5
3
Network Nodal Degree
3.5
4
distinct structures), the trees employed in the hybrids were
small degree-3 structures, such as the one pictured in Fig. 10b.
The details of the topology in Fig. 10a show how a 7-hop
segment can strictly require less spare capacity than an 8-hop
cycle in these specific circumstances. The failure of the protected (dashed) span in the figure leaves the two end-nodes
separated by quite a distance in the remaining topology. Evidently, any structure that protects this span must be quite
long; at least as long as the shortest possible protection path
that remains between them, which consists of 7 hops. Therefore the p-segment that protects them is the shortest possible
structure that is able to do so. This is characteristic of most
of the p-segments found in the designs. Of course, if the
segment were closed by adding the one remaining span, the
resulting 8-hop p-cycle could also protect that span, but in
this particular situation the extra protection relationships that
this would provide to other spans are not required. So under
strictly optimal design, the cycle is not “closed” and manifests itself as a p-segment, not a p-cycle.
Comparison to ILP-computed hierarchical trees
As mentioned previously, prior work in [14] used an ILP model
to compute optimal designs using a slight modification of the
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132
Table 2 Comparison of results
from [14] with p-cycle, p-tree,
and hybrid results produced by
our model
Photon Netw Commun (2007) 14:123–133
Network
Spare capacity (working capacity is identical in all cases)
ILP h-trees [14]
p-cycle design
p-tree design
Hybrid design
NET1
96
56
102
56
NET2
424
279
559
270
NET3
178
106
180
105
hierarchical tree scheme [10–12]. They give spare capacity
costs for such designs on three test networks. The network
topologies and demand patterns are given in [14]. For completeness, we present here in Table 2 spare capacity results for
p-cycle, p-tree, and hybrid designs, obtained using our ILP
model, for these same networks. Working capacity values are
the same as those given in [14], as the same shortest path working capacity routing method was used. Note that the p-tree and
hybrid designs for NET1 and NET3 were given all network
trees as candidates, while the designs for NET2 were limited
to using trees with maximum size of 7 and maximum degree
3, because of the size of the network.
negligible spare capacity cost improvements (less than 1% on
average). In most cases pure p-cycle designs resulted, even
though orders of magnitude more tree candidates were present. In a few cases, degenerate degree-2 “trees” (segments in
actuality) were recruited into the p-cycle designs for small
reductions in spare channel counts, but this is an already
known effect, which we explain with the conceptualization
of p-segments as an incremental, special case fine-tuning of
p-cycle designs. Finally, in some rare and detailed circumstances, small but true trees (degree 3 or more) were found
in the hybrid designs.
References
Concluding discussion
Through the investigations presented here, we conclude that
the use of pre-planned trees to protect networks against span
failures is not capacity efficient in general, as compared to
p-cycles. The difference is usually a factor of two or more
in total spare capacity requirements. The claims to the contrary in some recent work on “hierarchical tree” protection
are partly explainable by the fact that proposed “hierarchical tree” scheme is not 100% restorable by design, and does
not use fully tree-based protection. Furthermore, it did not
make the comparison of p-trees to p-cycles in the directly
comparable quantitative manner used here.
The work also makes some other contributions to
survivable networking science: we have also shown that a
unified mathematical framework exists for design of any
span-protecting network using trees, cycles or any other predefined and enumerated sets of protection structures. Using
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structures. In these hybrid design experiments, all cycles and
trees were presented as candidate structures. Overwhelmingly the solver adopted and employed cycles as its preferred building blocks. Even including a set of trees many
times larger than the set of candidate cycles yielded zero or
123
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Aden Grue received his B.Sc.
Degree, with honours, in Computer Engineering from the University of Alberta in April
2004. He is currently pursuing a
Ph. D. degree in the Department
of Electrical and Computer Engineering at the same institution, in
affiliation and with support from
TRLabs. In November of 2006
he also assumed a position on
TRLabs staff as a Research Engineer. His main research interests include network design,
restorable networks, and precross-connected network protection architectures, including p-cycles,
p-trees, PXTs, and FIPP p-cycles.
133
Wayne D. Grover received
the B.Eng. degree from Carleton University, Canada, the
M.Sc. degree from the University of Essex, U.K., and the Ph.D.
degree from the University of
Alberta, Canada, all in electrical engineering. He had 10 years
of experience as scientific staff
and manager at BNR (now Nortel Networks) working on fiber
optics, switching systems, digital radio, and other areas before
joining TRLabs as its founding
Technical VP in 1986. He now
functions as Chief Scientist—Network Systems, TRLabs, and as Professor, Electrical and Computer Engineering, at the University of Alberta.
He has 32 patented inventions, and amid his nearly 200 peerreviewed publications are “highly cited” papers in clock distribution,
error-correction coding for fiber optics, digital subscriber loops, and
transport network design and survivability, including recent origination of the p-cycles concept and the “protected working capacity envelope” (PWCE) concept for dynamic survivable service provisioning.
He is a recipient of the IEEE Baker Prize Paper Award and IEEE Fellow cited for work on survivable and self-organizing networks. Among
other awards are the IEEE Canada Outstanding Engineer Award, the
Alberta Science and Technology Leadership Award, and the prestigious NSERC Steacie Fellowship. He has received TRLabs Technology
Commercialization Awards for the licensing of restoration and networkdesign-related technologies. He is also author of Mesh-based Survivable
Networks, Prentice-Hall PTR, 2004 and a co-author of Next Generation
Transport Networks, Springer Science, 2005. Current research interests focus on optical network design optimization, new survivability architectures including p-cycles, and new approaches to operation
and ongoing re-optimization of dynamic transport networks. www.ece.
ualberta.ca/∼grover/
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