Maximum Request Satisfaction in WDM Rings: Algorithms and - - PowerPoint PPT Presentation

Maximum Request Satisfaction in WDM Rings:

Algorithms and Experiments

Evangelos Bampas, Aris Pagourtzis, Katerina Potika

{ebamp,pagour,epotik}@cs.ntua.gr

National Technical University of Athens

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All-optical WDM networks

Make use of optical fibers to provide much more bandwidth than copper wire.

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All-optical WDM networks

Make use of optical fibers to provide much more bandwidth than copper wire. In the simplest case, accommodate one connection per fiber.

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All-optical WDM networks

Make use of optical fibers to provide much more bandwidth than copper wire. In the simplest case, accommodate one connection per fiber. Wavelength Division Multiplexing (WDM): Accommodate multiple connections per fiber, by exploiting different wavelengths in the same fiber.

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Wavelength restrictions

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Wavelength restrictions

Any connection uses the same wavelength on all the physical links it is routed over (wavelength continuity constraint).

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Wavelength restrictions

Any connection uses the same wavelength on all the physical links it is routed over (wavelength continuity constraint).

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Wavelength restrictions

Any connection uses the same wavelength on all the physical links it is routed over (wavelength continuity constraint). There is no pair of connections that use the same wavelength on the same physical link.

PCI 2007 3/26

Wavelength restrictions

Any connection uses the same wavelength on all the physical links it is routed over (wavelength continuity constraint). There is no pair of connections that use the same wavelength on the same physical link.

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Problem formulation (MaxPC)

Maximum Path Coloring (MaxPC) Input: a graph G, a set of paths P, and a number of available colors w. Feasible solution: a set of paths P′ ⊆ P that can be colored with w colors so that no

• verlapping paths are assigned the same color.

Goal: maximize |P′|.

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Problem formulation (MaxRPC)

Maximum Routing and Path Coloring (MaxRPC) Input: a graph G, a set of pairs of nodes (requests) R, and a number of available colors w. Feasible solution: a set of requests R′ ⊆ R that can be routed and colored conforming to the wavelength restrictions. Goal: maximize |R′|.

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Talk outline

Related results. Algorithms for MaxPC in undirected rings. Experimental results.

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Related results

MaxPC in chains is solvable in polynomial time [Carlisle, Lloyd, 1995]. MaxPC and MaxRPC are NP-hard in rings.

• 1 − 1

e

• approximation [Wan, Liu, 1998].

2 3-approximation [Nomikos, Pagourtzis,

Zachos, 2003a & 2003b].

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This work: comparison of algo- rithms for MaxPC in rings

Chain [CL98] SF (Shortest-First) BestSol: implicit in [NPZ03a] BestSol-all CombSol [NPZ03a] CombSol-all Iter [WL98] Similar study for MaxRPC.

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MaxPC-Chain

• 1. Pick a separation edge e with minimum load.
• 2. Partition the path set P into sets Pe and Pc.
• 3. Color Pc using the Carlisle-Lloyd algorithm

for MaxPC in chains.

e Pe Pc

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MaxPC-Chain (cont’d)

Achieves an approximation guarantee of 1

2.

Its time complexity is O(n + m). Used as a first step of the CombSol and BestSol algorithms.

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MaxPC-SF

• 1. Sort the paths in P by non-decreasing length.
• 2. For each path p, either assign to it the first

color that is available on all edges of p, or skip p (if no color is available). Achieves an approximation guarantee of 1

3.

Its time complexity is O(nmw).

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MaxPC-BestSol

• 1. Chain step (call MaxPC-Chain).
• 2. Matching step (find a maximum matching on

the path compatibility graph).

• 3. Output the best among the previous solutions.

p1 p2 p3 p4 p5 p6 e

p1 p2 p3 p4 p5 p6

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MaxPC-BestSol (cont’d)

Achieves an approximation guarantee of 2

3.

Its time complexity is dominated by the bipartite matching computation which can be done in O(m2). MaxPC-BestSol-all: considers all possible separation edges. Time complexity O(nm2).

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MaxPC-CombSol

• 1. Chain step (call MaxPC-Chain).
• 2. Find a maximum matching M on the path

compatibility graph.

• 3. Uncolor lonely paths.
• 4. while ∃(p, q) ∈ M and ∃ free color c do

5. Color p and q with c. 6. Remove (p, q) from M. 7. Uncolor lonely paths.

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MaxPC-CombSol (cont’d)

Achieves an approximation guarantee of 2

3.

Its time complexity is O(nmw + m2). MaxPC-CombSol-all: considers all possible separation edges. Time complexity O(n2mw + nm2).

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MaxPC-Iter

• 1. for each color c do

2. for each p ∈ P do 3. Find the maximum Sp ⊆ P that can be colored with the same color as p (activity selection). 4. Pick the best such p. 5. Color Sp and p with c. 6. Remove Sp and p from P.

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MaxPC-Iter (cont’d)

Achieves an approximation guarantee of 1 −

• 1 − 1

w

w > 1 − 1

e.

Its time complexity is O(wm2 log m).

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Experimental setup

C++, LEDATM class library. 2 sets of 60 randomly generated instances for each combination of (n, m, w). Uniform distribution of endpoints. Normal distribution of endpoints with σ ≈ 2n

15.

Measured average execution time and average number of satisfied paths/requests, with a 95% confidence interval.

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An upper bound on OPT

nw is a trivial upper bound on OPT.

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An upper bound on OPT

nw is a trivial upper bound on OPT. It is also an upper bound on the total length

• f paths in any feasible solution.

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An upper bound on OPT

nw is a trivial upper bound on OPT. It is also an upper bound on the total length

• f paths in any feasible solution.

This gives the following upper bound:

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An upper bound on OPT

nw is a trivial upper bound on OPT. It is also an upper bound on the total length

• f paths in any feasible solution.

This gives the following upper bound: Let the paths be indexed in order of non-decreasing length. UB = minimum k s.t.

k+1

• i=1

length(pi) > nw

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MaxPC, n = 100, m = 200 − 600, w = 40, uniform

1e-05 1e-04 0.001 0.01 0.1 1 10 200 250 300 350 400 450 500 550 600 time (s) m SF Chain BestSol BestSol-all CombSol CombSol-all Iter

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MaxPC, n = 100, m = 200 − 600, w = 40, uniform

80 100 120 140 160 180 200 220 200 250 300 350 400 450 500 550 600 # satisfied paths m SF Chain UB

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MaxPC, n = 100, m = 200 − 600, w = 40, uniform

90 100 110 120 130 140 150 160 170 180 190 200 200 250 300 350 400 450 500 550 600 # satisfied paths m CombSol BestSol-all

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MaxPC, n = 100, m = 200 − 600, w = 40, uniform

100 110 120 130 140 150 160 170 180 190 200 200 250 300 350 400 450 500 550 600 # satisfied paths m CombSol CombSol-all Iter

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MaxPC, n = 100, m = 500, w = 20 − 100, normal

100 120 140 160 180 200 220 240 260 280 20 30 40 50 60 70 80 90 100 # satisfied paths m SF CombSol

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Discussion

Iter and CombSol-all are the best in terms of satisfaction of paths/requests. CombSol-all is very slow. Iter is noticeably faster but still slower than the competitive CombSol. The greedy SF outperforms Chain and is a decent choice whenever time is crucial. BestSol-all consumes too much time and fails to provide satisfactory solutions. BestSol is quite fast but is outperformed by SF and CombSol.

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Conclusions – further work

Algorithm Request satisfaction Execution speed SF ⋆⋆ ⋆⋆⋆⋆⋆ Chain ⋆ ⋆⋆⋆⋆⋆ BestSol ⋆ ⋆⋆⋆⋆⋆ BestSol-all ⋆⋆ ⋆ CombSol ⋆⋆⋆⋆ ⋆⋆⋆⋆⋆ CombSol-all ⋆⋆⋆⋆⋆ ⋆ Iter ⋆⋆⋆⋆⋆ ⋆⋆⋆

Improve approximation ratio of SF. Include the 3

4-approximation algorithms of

[Caragiannis, 2007] in our comparison.

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