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## On Finding Disjoint Paths in Single and Dual Link Cost Networks

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### Finding a shortest path consisting of certain links (e.g. set I) is itself NP-Hard ... COLE will stop iteration after finding optimal result. ... – PowerPoint PPT presentation

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Title: On Finding Disjoint Paths in Single and Dual Link Cost Networks

1
On Finding Disjoint Paths in Single and Dual Link
Cost Networks
• Chunming Qiao
• LANDER, CSE Department
• SUNY at Buffalo
• Collaborators Dahai Xu, Yang Chen, Yizhi Xiong
and Xin He

2
Outline
• The Min-Min Problem
• Motivation and Definition
• Existing and Proposed Heuristics
• Application and Performance Evaluation
• Summary

3
Finding Disjoint Path Pairs
• Basic and important problem in survivable routing
• The Min-Min Problem
• Definition Finding a link (node) disjoint path
pair such that the length of the shorter path is
minimized.
• Applications
• Encrypted data on the shorter path, and
decryption key on the longer path
• Shared Path Protection (use the shorter path as
AP)
• Counterpart problems
• Min-Max
• Min-Sum

4
Computational Complexities
• Min-Sum (P) Suurballe-74
• Min-Max (NP Complete) Li-90
• Min-Min (P or NP Hard?)
• NP Complete! proved by Xu et. al. in INFOCOM04
• Reduction from a well-known NPC problem 3SAT
• We also proved that it is NP-hard to obtain a
k-approximation to the optimal solution for any k
gt 1

5
Solving The Min-Min Problem
• Active Path First (APF) Heuristic
• Finds a shortest path for use as AP, followed by
searching a disjoint BP.
• It may fail to find such a BP even though a
disjoint path pair does exist.
• K Shortest Path (KSP) Heuristic
• First K shortest paths are found and tested in
the increasing order of their costs (path
lengths) to see if a disjoint BP exists.
• Could be time-consuming

6
Inefficiency of KSP
• Any path from s to d consists of two sub-paths in
domain E1 and E2 respectively.
• Links in E1 is much shorter than those in E2.
• The number of all possible sub-paths in E1 is
very large

1st
2nd
7
Proposed Approach
• Find a shortest AP first (as in APF)
• If the AP doesnt have a disjoint BP, determine
the conflicting link set that are causing the
problem
• Try another AP without using these problematic

8
• Definition
• A minimal subset of the links on AP such that no
path using ALL these problematic links can find
a disjoint counterpart, e.g., e1 and e2 in the
following example.
• The Min-Min problem can be solved much faster by
avoiding using at least one link in the
conflicting link set for the next shortest AP.

9
Divide and Conquer
• Let P(I, O) be the problem of finding a disjoint
path pair where AP must use the links in set I
(Inclusion) but not the links in O (the Exclusion
set).
• Denote the original Min-Min problem by P(?, ?)
• Find a shortest AP If no disjoint BP, find the
• Divide P(?, ?) into sub-problems based on the
conflicting link set, e.g., P(?, e1) and
P(e1, ?) in the previous example.
• The same procedure may be applied recursively on
these sub-problems, e.g., P(e1, ?) can be
further divided into P(e1,e2) and P(e1,e2,
?).
• The definition of conflicting link set means that
we do not need to try to solve P(e1,e2, ?).

10
The Proposed Conflicting Link Exclusion (COLE)
Heuristic
• An algorithm to find the conflicting link set (to
be discussed)
• Usually has fewer links than the half of the
• Fewer sub-problems than KSP
• Divide and Conquer based on the conflicting
KSP)
• Then pick a best solution (with a shorter AP)
among those for the sub-problems.
• Find a optimal or near-optimal result for each
sub-problem
• Each sub-problem may be solved recursively using
Divide-and-Conquer

11
Solving the Sub-problem
• Finding a shortest path consisting of certain
links (e.g. set I) is itself NP-Hard
• Approximation method to speed up the computation.
Xu et al. OFC04

12
• Finding a link-disjoint path pair between nodes s
and d in graph G(V, E) Finding two unit-flows
in a flow network where each link's capacity is
set to 1 unit
• Assume that the network is symmetrical
• For the chosen AP, construct a new graph G0
• G0 uses the same V and E of G
• The capacity of the links in AP is set to 1
• The capacity of the reverse links in AP is set to
0.
• The capacity of all other links with non-zero
capacity in G (except those in AP) is set to
AP1 (or a larger value).

13
• Let F0(S, D) be a min-cut of G0, Ss, 3, 7
D1, 2, 4, 5, 6, d
• The set of negative links (from D to S) on AP of
F0 is a Conflicting Link Set e1, e2

14
Reason for Not Using an Ordinary Min-Cut
• Divide and conquer based on Ordinary Min-Cut
might not help reducing the computational
complexity.
• AP0 is the shortest path (and no link-disjoint BP
exists)
• APopt is the shortest path with a link-disjoint
BP
• The min-cut The partition S s, positive
• Divide the original Min-Min problem into P(?,
a) and P(a, b) (no solution in P(a, b, ?)
)
• Solving P(?, a) leads to a non-optimal
solution, and trying to solve P(a, b) will
again yield AP0

15
Shared Path Protection
• Two BPs can share backup bandwidth on a common
link as long as their APs are disjoint (with a
single failure)

16
Performance Evaluation
• Solution to Min-Min problem (Single Link Cost
Networks)
• COLE will stop iteration after finding optimal
result.
• KSP can find the optimal result with a large
enough K but has a longer running time than COLE
• In both algorithms, the time for each invocation
of the Dijkstra Algorithm to find the (next)
shortest path dominates
• Application to shared path protection (Dual Link
Cost Networks)
• COLE is compared with the optimal shared Min-Sum
and optimal shared Min-Max solutions (based on
ILP)
time.

17
Number of Dijkstra Invocations (Min-Min)
• Net 1 (46 Nodes, 76 edges), Net 2 (79 Nodes, 108
edges), Net 3 (119Nodes, 190 edges)
• KSP calls the sub-routine significantly more
times than COLE, especially for large networks

18
Performance in Shared Path Protection
• Bandwidth Overhead Percentage increase in the
total bandwidth (active backup) required over
the standard active bandwidth

Min-Max
Min-Min
Min-Sum
19
Summary
• The Min-Min problem is formulated and applied to
shared path protection
• The concept of Conflict Link Set is defined,
which helps to solve the Min-Min problem fast
• A novel heuristic algorithm COLE capable of
solving the Min-Min problem faster than KSP is
proposed
• COLE is also found to be competitive in providing
shared path protection.