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Novembro 2003 Tabu search heuristic for partition coloring1/29 XXXV SBPO XXXV SBPO Natal, 4-7 de novembro de 2003 A Tabu Search Heuristic for Partition Coloring with an Application to Routing and Wavelength Assignment Thiago NORONHA Celso C. RIBEIRO

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Novembro 2003 Tabu search heuristic for partition coloring2/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring3/29 XXXV SBPO Partition coloring problem (PCP) Graph G = (V,E) with vertex set partitioned into k disjoint subsets: V = V 1 V 2 ... V p PCP consists in coloring exactly one node in each subset V i, such that every two adjacent colored nodes have different colors. Objective: minimize the number of colors used.

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Novembro 2003 Tabu search heuristic for partition coloring4/29 XXXV SBPO Partition coloring problem 1 22 4 6 1 22 4 6 0 22 3 6 0 2 3 6 2 10 22 3 4 5 6 7

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Novembro 2003 Tabu search heuristic for partition coloring5/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring6/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring7/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring8/29 XXXV SBPO Optical network Shortest path routing: three wavelengths are needed Routing and wavelength assignment in circuit-switched WDM all-optical networks From SLE to PCP Lightpaths: A D B E C F

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Novembro 2003 Tabu search heuristic for partition coloring9/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring10/29 XXXV SBPO 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: only two wavelengths are needed!

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Novembro 2003 Tabu search heuristic for partition coloring11/29 XXXV SBPO 1. Remove all edges whose vertices are in same group. 2. Find the vertex with minimal color- degree for each uncolored group. 3. Among these vertices, find that with the largest color-degree. 4. Assign to this vertex the smallest available color and remove all other vertices in the same group. 5. Repeat the above steps until all groups are colored. Algorithms for PCP: OnestepCD (greedy)

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Novembro 2003 Tabu search heuristic for partition coloring12/29 XXXV SBPO CD: 0 UD: 4 CD: 0 UD: 3 CD: 0 UD: 2 CD: 0 UD: 2 CD: 0 UD: 3 CD: 0 UD: 2 CD: 0 UD: 2 CD: 1 UD: 0 CD: 1 UD: 0 CD: 1 UD: 0 Algorithms for PCP: OnestepCD Color degree: number of colored neighbors Uncolored degree: number of uncolored neighbors

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Novembro 2003 Tabu search heuristic for partition coloring13/29 XXXV SBPO 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)

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Novembro 2003 Tabu search heuristic for partition coloring14/29 XXXV SBPO 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)

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Novembro 2003 Tabu search heuristic for partition coloring15/29 XXXV SBPO Algorithms for PCP: Local search

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Novembro 2003 Tabu search heuristic for partition coloring16/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring17/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring18/29 XXXV SBPO Computational results Average results: construction, local search, tabu search Onestep CD Local search Tabu search Instancenode s colors % red. color s % red. DSJC- 250.5-1 25041.740.6329.629 DSJC- 250.5-2 50040.438.1625.836 DSJC- 250.5-3 75038.835.6824.038 DSJC- 250.5-4 100038.334.7923.040 DSJC- 500.5-1 50071.269.3352.626 DSJC- 500.5-2 100069.567.3346.633 DSJC- 500.5-3 150068.865.4543.936 DSJC- 500.5-4 200068.762.5942.438 35%

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Novembro 2003 Tabu search heuristic for partition coloring19/29 XXXV SBPO Computational results Tabu search: solution values and times (10 runs) ColorsTime (s) Instancebest averag e worst to best total DSJC- 250.5-1 2929.6306.721.4 DSJC- 250.5-2 2525.82611.762.4 DSJC- 250.5-3 2424.02435.2164.7 DSJC- 250.5-4 2323.02365.3300.8 DSJC- 500.5-1 5252.65341.9197.2 DSJC- 500.5-2 4646.647286.51068.3 DSJC- 500.5-3 4343.944533.82187.5 DSJC- 500.5-4 4242.443777.7 3349. 6 Robust!

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Novembro 2003 Tabu search heuristic for partition coloring20/29 XXXV SBPO 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) Time-to-target-value plots

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Novembro 2003 Tabu search heuristic for partition coloring21/29 XXXV SBPO Instance DSJC-250.5-4 Time-to-target-value plots

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Novembro 2003 Tabu search heuristic for partition coloring22/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring23/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring24/29 XXXV SBPO SLE instance #1: 14 nodes, 21 links, and 182 connections Application: SLE

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Novembro 2003 Tabu search heuristic for partition coloring25/29 XXXV SBPO SLE instance #1: target = 13 (optimal) Application: SLE

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Novembro 2003 Tabu search heuristic for partition coloring26/29 XXXV SBPO Application: SLE SLE instance #2: 27 nodes, 70 links, and 702 connections

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Novembro 2003 Tabu search heuristic for partition coloring27/29 XXXV SBPO Application: SLE SLE instance #2: target = 24 (optimal)

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Novembro 2003 Tabu search heuristic for partition coloring28/29 XXXV SBPO Conclusions Local search and 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.

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Novembro 2003 Tabu search heuristic for partition coloring29/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring30/29 XXXV SBPO 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

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Novembro 2003 Tabu search heuristic for partition coloring31/29 XXXV SBPO Computational results Random instances: varying the number of subsets

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Novembro 2003 Tabu search heuristic for partition coloring32/29 XXXV SBPO Computational results Random instances: varying the graph density

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