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Mar. 10th. 2004

On Finding Disjoint Pathsin Single and Dual Link Cost Networks

Chunming Qiao*Chunming Qiao*

LANDER, CSE DepartmentLANDER, CSE DepartmentSUNY at BuffaloSUNY at Buffalo

*Collaborators: Dahai Xu, Yang Chen, Yizhi Xiong and Xin He*Collaborators: Dahai Xu, Yang Chen, Yizhi Xiong and Xin He

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Outline

The Min-Min Problem Motivation and Definition Existing and Proposed Heuristics Application and Performance Evaluation

Summary

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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.

ApplicationsEncrypted data on the shorter path, and decryption

key on the longer pathShared Path Protection (use the shorter path as AP)

Counterpart problems Min-Max Min-Sum

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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 INFOCOM’04] 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 > 1

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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

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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

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Proposed Approach

Find a shortest AP first (as in APF) If the AP doesn’t have a disjoint BP, determine

the “conflicting link set” that are causing the problem

Try another AP without using these problematic links

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Conflicting Link Set

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.

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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 Conflicting Link Set 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}, ).

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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 links on AP Fewer sub-problems than KSP

“Divide and Conquer” based on the conflicting link set (rather than all the links on AP as in 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

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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. OFC’04]

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Finding Conflicting Link Set 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 |AP|+1 (or a larger value).

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Finding Conflicting Link Set (II)

Let Φ0=(S, D) be a min-cut of G0, S={s, 3, 7} D={1, 2, 4, 5, 6, d}

The set of negative links (from D to S) on AP of Φ0 is a Conflicting Link Set {e1, e2}

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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 links are a and b 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

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Shared Path Protection

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

AP2(w2)

AP1(w1)

BP1

BP2

S2 D2

S1 D1

Li nk L(max{w1, w2})

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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) Tradeoffs between bandwidth overhead and recovery

time.

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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

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Performance in Shared Path Protection

Min-Sum

Min-Min Min-Max

Bandwidth Overhead: Percentage increase in the total bandwidth (active + backup) required over the standard active bandwidth

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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.