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 1/26 PCI 2007

All-optical WDM networks Make use of optical fibers to provide much more bandwidth than copper wire. 2/26 PCI 2007

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. 2/26 PCI 2007

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. 2/26 PCI 2007

Wavelength restrictions 3/26 PCI 2007

Wavelength restrictions Any connection uses the same wavelength on all the physical links it is routed over (wavelength continuity constraint). 3/26 PCI 2007

Wavelength restrictions Any connection uses the same wavelength on all the physical links it is routed over (wavelength continuity constraint). 3/26 PCI 2007

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. 3/26 PCI 2007

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. 3/26 PCI 2007

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 overlapping paths are assigned the same color. Goal: maximize |P ′ | . 4/26 PCI 2007

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 ′ | . 5/26 PCI 2007

Talk outline Related results. Algorithms for MaxPC in undirected rings. Experimental results. 6/26 PCI 2007

Related results MaxPC in chains is solvable in polynomial time [Carlisle, Lloyd, 1995]. MaxPC and MaxRPC are NP-hard in rings. 1 − 1 � � -approximation [Wan, Liu, 1998]. e 2 3 -approximation [Nomikos, Pagourtzis, Zachos, 2003a & 2003b]. 7/26 PCI 2007

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. 8/26 PCI 2007

MaxPC-Chain 1. Pick a separation edge e with minimum load. 2. Partition the path set P into sets P e and P c . 3. Color P c using the Carlisle-Lloyd algorithm for MaxPC in chains. P e e P c 9/26 PCI 2007

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. 10/26 PCI 2007

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 ). 11/26 PCI 2007

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. p 3 p 1 p 2 p 4 e p 1 p 4 p 6 p 2 p 5 p 5 p 3 p 6 12/26 PCI 2007

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 ( m 2 ). MaxPC-BestSol-all: considers all possible separation edges. Time complexity O ( nm 2 ). 13/26 PCI 2007

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. 14/26 PCI 2007

MaxPC-CombSol (cont’d) Achieves an approximation guarantee of 2 3 . Its time complexity is O ( nmw + m 2 ). MaxPC-CombSol-all: considers all possible separation edges. Time complexity O ( n 2 mw + nm 2 ). 15/26 PCI 2007

MaxPC-Iter 1. for each color c do 2. for each p ∈ P do 3. Find the maximum S p ⊆ P that can be colored with the same color as p (activity selection). 4. Pick the best such p . 5. Color S p and p with c . 6. Remove S p and p from P . 16/26 PCI 2007

MaxPC-Iter (cont’d) Achieves an approximation guarantee of � w > 1 − 1 1 − 1 � 1 − e . w Its time complexity is O ( wm 2 log m ). 17/26 PCI 2007

Experimental setup C++, LEDA TM 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 ≈ 2 n 15 . Measured average execution time and average number of satisfied paths/requests, with a 95% confidence interval. 18/26 PCI 2007

An upper bound on OPT nw is a trivial upper bound on OPT. 19/26 PCI 2007

An upper bound on OPT nw is a trivial upper bound on OPT. It is also an upper bound on the total length of paths in any feasible solution. 19/26 PCI 2007

An upper bound on OPT nw is a trivial upper bound on OPT. It is also an upper bound on the total length of paths in any feasible solution. This gives the following upper bound: 19/26 PCI 2007

An upper bound on OPT nw is a trivial upper bound on OPT. It is also an upper bound on the total length of paths in any feasible solution. This gives the following upper bound: Let the paths be indexed in order of non-decreasing length. k +1 � UB = minimum k s.t. length( p i ) > nw i =1 19/26 PCI 2007

MaxPC, n = 100, m = 200 − 600, w = 40, uniform 10 1 0.1 time (s) 0.01 0.001 SF Chain BestSol 1e-04 BestSol-all CombSol CombSol-all Iter 1e-05 200 250 300 350 400 450 500 550 600 m 20/26 PCI 2007

MaxPC, n = 100, m = 200 − 600, w = 40, uniform 220 SF Chain UB 200 180 # satisfied paths 160 140 120 100 80 200 250 300 350 400 450 500 550 600 m 21/26 PCI 2007

MaxPC, n = 100, m = 200 − 600, w = 40, uniform 200 CombSol BestSol-all 190 180 170 160 # satisfied paths 150 140 130 120 110 100 90 200 250 300 350 400 450 500 550 600 m 22/26 PCI 2007

MaxPC, n = 100, m = 200 − 600, w = 40, uniform 200 CombSol CombSol-all 190 Iter 180 170 # satisfied paths 160 150 140 130 120 110 100 200 250 300 350 400 450 500 550 600 m 23/26 PCI 2007

MaxPC, n = 100, m = 500, w = 20 − 100, normal 280 SF CombSol 260 240 220 # satisfied paths 200 180 160 140 120 100 20 30 40 50 60 70 80 90 100 m 24/26 PCI 2007

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. 25/26 PCI 2007

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. 26/26 PCI 2007