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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
    links

8
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.

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
    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, ?).

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
    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

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 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
    AP1 (or a larger value).

13
Finding Conflicting Link Set (II)
  • 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
    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

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)
  • Tradeoffs between bandwidth overhead and recovery
    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.
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