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Introduction

Brazil. August 2003. Tabu search heuristic to partition coloring. 4 /36 ... B zios (Brazil), May 25 to 28, 2004. IV Workshop on Efficient and Experimental Algorithms ...

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Introduction

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    MIC2003 Kyoto, August 25-28, 2003 Bora Bora, Tahiti MIC2003 Kyoto, August 25-28, 2003 A Tabu Search Heuristic for Partition Coloring with an Application to Routing and Wavelength Assignment MIC2003 Kyoto, August 25-28, 2003 A Tabu Search Heuristic for Partition Coloring with an Application to Routing and Wavelength Assignment Thiago NORONHA Celso C. RIBEIRO Catholic University of Rio de Janeiro Brazil

    4. Introduction

    The partition coloring problem (PCP) Routing and wavelength assignment in all-optical networks (RWA) Algorithms for PCP: construction, LS, tabu search Computational results Application: static lightpath establishment Conclusions

    5. Partition coloring problem (PCP)

    Graph G = (V,E) with vertex set partitioned into k disjoint subsets: V = V1 ? V2 ? ... ? Vp PCP consists in coloring exactly one node in each subset Vi , such that every two adjacent colored nodes have different colors. Objective: minimize the number of colors used.

    6. Partition coloring problem

    7. Routing and wavelength assignment in circuit-switched WDM all-optical networks

    Different signals can be simultaneously transmitted in a fiber, using different wavelengths: Wavelength Division Multiplexing Connections (between origin-destination pairs) are established by lightpaths. To establish a lightpath consists in determining: a route a wavelength

    8. Each signal can be switched optically at intermediate nodes in the network. No wavelength conversion is possible. Lightpaths sharing a common link are not allowed to use the same wavelength. Traffic assumptions: Yoo & Banerjee (1997) static lightpath establishment dynamic lightpath establishment (O-D pairs are not known beforehand)

    Routing and wavelength assignment in circuit-switched WDM all-optical networks

    9. Static lightpath establishment (SLE) without wavelength conversion: Minimize the total number of used wavelengths Other objective functions may also consider the load in the most loaded link, the total number of optical switches (total length), etc.

    Routing and wavelength assignment in circuit-switched WDM all-optical networks

    Optical network

    10. Routing and wavelength assignment in circuit-switched WDM all-optical networks

    From SLE to PCP Lightpaths: A ? D B ? E C ? F

    11. Routing and wavelength assignment in circuit-switched WDM all-optical networks

    From SLE to PCP Optical network Lightpaths: A ? D B ? E C ? F 2-shortest path routing

    12. Routing and wavelength assignment in circuit-switched WDM all-optical networks

    From SLE to PCP Optical network Lightpaths: A ? D B ? E C ? F

    13. Algorithms for PCP: Greedy heuristics

    Onestep Largest First Onestep Smallest Last Onestep Color Degree (onestepCD) best in literature: Li & Simha (2000) Twostep Largest First Twostep Smallest Last Twostep Color Degree

    14. Remove all edges whose vertices are in same group. Find the vertex with minimal color-degree for each uncolored group. Among these vertices, find that with the largest color-degree. Assign to this vertex the smallest available color and remove all other vertices in the same group. Repeat the above steps until all groups are colored.

    Algorithms for PCP: OnestepCD

    15. Algorithms for PCP: OnestepCD

    Color degree: number of colored neighbors Uncolored degree: number of uncolored neighbors

    16. First, LS-PCP converts a feasible solution with C colors into an infeasible solution with C-1 colors; next, it attempts to restore solution feasibility. The local search procedure investigates the subsets whose colored node is involved in a coloring conflict. LS-PCP searches within each subset for a node that can be colored or recolored so as to reduce the overall number of coloring conflicts.

    Algorithms for PCP: Local search (1/2)

    17. In case such a node exists, the algorithm moves to a new solution. Otherwise, another subset is randomly chosen and investigated. If a feasible solution with C-1 colors is found, the feasibility of this coloring is destroyed and another coloring using C-2 colors is sought. LS-PCP stops when the number of coloring conflicts cannot be reduced and the solution is still infeasible.

    Algorithms for PCP: Local search (2/2)

    Algorithms for PCP: Local search

    19. Simple short-term memory strategy: TS-PCP Initial solutions: OnestepCD Local search strategy: LS-PCP move: pair (node,color) Tabu tenure: randomly in U[C/4,3C/4] Aspiration criterion: improve best Stopping criterion: C.P.10 iterations without finding a feasible solution, where C = number of colors and P = number of subsets in the partition

    Algorithms for PCP: Tabu search

    20. Computational results

    Random instances: eight PCP instances generated from graph coloring instances DJSC-250.5 and DJSC-500.5 Aragon, Johnson, McGeoch & C. Schevon (1991) nodes in original instance are replicated (2x, 3x, 4x) edges are additioned with density 0.5 one subset for each original node Computational experiments: Pentium IV 2.0 GHz

    21. Computational results

    Average results: construction, local search, tabu search

    22. Computational results

    Tabu search: solution values and times (10 runs) Robust!

    23. Computational results

    Random instances: varying the number of subsets

    24. Computational results

    Random instances: varying the graph density

    25. Select an instance and a target value: Perform 200 runs using different seeds. Stop when a solution value at least as good as the target is found. For each run, measure the time-to-target-value. Plot the probabilities of finding a solution at least as good as the target value within some computation time. Plots can illustrate algorithm robustness and are very useful for comparisons based on the probability distribution of the time-to-target-value Aiex, Resende & Ribeiro (2002) Resende & Ribeiro (2003, this meeting)

    Time-to-target-value plots

    Instance DSJC-250.5-4

    26. Time-to-target-value plots

    27. Possible routing algorithms: k-shortest paths Path stripping: solves LP relaxation and builds progressively longer shortest routes using edges in the fractional solution. Banerjee & Mukherjee (1995) Greedy-EDP-RWA: multistart construction using random permutations (greedy max edge-disjoint paths routing), too many restarts are needed. Manohar, Manjunath & Shevgaonkar (2002)

    Static Lightpath Establishment

    28. Comparison: n-Greedy-EDP-RWA vs. ... ... two routing iterations of Greedy-EDP-RWA followed by partition coloring using TS-PCP Both algorithms stop when a target solution value is found: Target is the optimal value of the LP relaxation of the IP formulation without optical continuity constraints.

    Application: SLE

    SLE instance #1: 14 nodes, 21 links, and 182 connections

    29. Application: SLE

    SLE instance #1: target = 13 (optimal)

    30. Application: SLE

    31. Application: SLE

    SLE instance #2: 27 nodes, 70 links, and 702 connections

    32. Application: SLE

    SLE instance #2: target = 24 (optimal)

    33. Conclusions

    We proposed a local search procedure and a tabu search heuristic for partition coloring. TS-PCP is able to significantly improve the solutions obtained by OnestepCD. TS-PCP together with a routing algorithm can be successfully used to solve SLE in RWA. Future work will consider other routing algorithms to be used with TS-PCP to solve SLE in practical applications.

    34. Slides and publications

    Slides of this talk can be downloaded from: http://www.inf.puc-rio/~celso/talks Paper will be soon available at: http://www.inf.puc-rio.br/~celso/publicacoes

    35. Announcements

    IV Workshop on Efficient and Experimental Algorithms Bzios (Brazil), May 25 to 28, 2004 IV Workshop on Efficient and Experimental Algorithms Bzios (Brazil), May 25 to 28, 2004

    XIX International Symposium on Mathematical Programming Rio de Janeiro (Brazil), July 2006
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