xxxv sbpo natal , 4-7 de novembro de 200 3
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XXXV SBPO Natal , 4-7 de novembro de 200 3. A Tabu Search Heuristic for Partition Coloring with an Application to Routing and Wavelength Assignment. Thiago NORONHA Celso C. RIBEIRO. Introduction. The partition coloring problem (PCP) - PowerPoint PPT PresentationTRANSCRIPT
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XXXV SBPONatal, 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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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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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.
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Partition coloring problem1
22
4
6
1
22
4
6
0
22
3
6
0
2
3
6
2
1 0
22
34
5
6
7
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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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• 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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• 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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Optical network
Shortest path routing: three wavelengths are needed
Routing and wavelength assignment in circuit-switched
WDM all-optical networksFrom SLE to PCP
Lightpaths:A DB EC F
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Routing and wavelength assignment in circuit-switched
WDM all-optical networksFrom SLE to PCP Optical network
Lightpaths:A DB EC F
2-shortest path routing
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Routing and wavelength assignment in circuit-switched
WDM all-optical networksFrom SLE to PCP Optical network
Lightpaths:A DB EC F
2-shortest path routing: only two wavelengths are needed!
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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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1 0
3
2
4 5 6
7
8
CD: 0UD: 4CD: 0
UD: 3CD: 0UD: 2
CD: 0UD: 2
1 0
3
2
4 5 6
7
8
CD: 0UD: 3
CD: 0UD: 2
CD: 0UD: 2
0
3
2
4 5 6
7
8CD: 1UD: 0
CD: 1UD: 0
0
2
4 5 6
7
8
CD: 1UD: 0
0
2
4 5 6
8
0
2
6
8
Algorithms for PCP: OnestepCD
• Color degree: number of colored neighborsUncolored degree: number of uncolored neighbors
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• 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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• 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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1 0
3
2
4 5 6
7
8
1 0
3
2
4 5 6
7
8
1 0
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4 5 6
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1 0
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4 5 6
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Algorithms for PCP: Local search
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• 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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Computational results
• Random instances: – eight PCP instances generated from graph
coloring instances DJSC-250.5 and DJSC-500.5Aragon, 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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Computational resultsAverage results: construction, local search, tabu search
OnestepCD
Local search
Tabu search
Instance nodes
colors colors
% red.
colors
% red.
DSJC-250.5-1
250 41.7 40.6 3 29.6 29
DSJC-250.5-2
500 40.4 38.1 6 25.8 36
DSJC-250.5-3
750 38.8 35.6 8 24.0 38
DSJC-250.5-4
1000 38.3 34.7 9 23.0 40
DSJC-500.5-1
500 71.2 69.3 3 52.6 26
DSJC-500.5-2
1000 69.5 67.3 3 46.6 33
DSJC-500.5-3
1500 68.8 65.4 5 43.9 36
DSJC-500.5-4
2000 68.7 62.5 9 42.4 38
35%
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Computational resultsTabu search: solution values and times (10 runs)Colors Time (s)
Instance bestavera
geworst
to best
total
DSJC-250.5-1
29 29.6 30 6.7 21.4
DSJC-250.5-2
25 25.8 26 11.7 62.4
DSJC-250.5-3
24 24.0 24 35.2 164.7
DSJC-250.5-4
23 23.0 23 65.3 300.8
DSJC-500.5-1
52 52.6 53 41.9 197.2
DSJC-500.5-2
46 46.6 47286.
51068.
3
DSJC-500.5-3
43 43.9 44533.
82187.
5
DSJC-500.5-4
42 42.4 43777.
73349.
6
Robust!
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• 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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Instance DSJC-250.5-4
Time-to-target-value plots
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• 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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• 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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SLE instance #1: 14 nodes, 21 links, and 182
connections
Application: SLE
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SLE instance #1: target = 13 (optimal)
Application: SLE
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Application: SLESLE instance #2:
27 nodes, 70 links, and 702 connections
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Application: SLESLE instance #2: target = 24
(optimal)
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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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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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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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Computational resultsRandom instances: varying the
number of subsets
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Computational resultsRandom instances: varying the graph
density