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Wavelength Assignment in Optical Network Design Team 6: Lisa Zhang (Mentor) Brendan Farrell, Yi Huang, Mark Iwen, Ting Wang, Jintong Zheng Progress Report Presenters: Mark Iwen

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Wavelength Assignment Motivated by WDM (wavelength division multiplexing) network optimization Input A network G=(V,E) A set of demands with specified src, dest and routes demand d i = (s i, t i, R i ) WDM fibers U: fiber capacity, number of wavelengths per fiber Output Assign a wavelength for each demand route Demand paths sharing same fiber have distinct wavelengths

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Example

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Model 1: Min conversion Routes given L(e): load on link e u: fiber capacity f(e) = L(e) / u Deploy f(e) fibers on link e : no extra fibers Use converters if necessary Min number of converters Fiber capacity u = 2 Demand routes: AOB, BOC, COA A B C O converter

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Model 1: Min conversion Routes given L(e): load on link e u: fiber capacity f(e) = L(e) / u Deploy f(e) fibers on link e : no extra fibers Use converters if necessary Min number of converters converter Each demand path assigned one wavelength from src to dest – no conversion Deploy extra fibers if necessary Min total fibers Model 2: Min fiber

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Model 1: Min conversion Routes given L(e): load on link e u: fiber capacity f(e) = L(e) / u Deploy f(e) fibers on link e : no extra fibers Use converters if necessary Min number of converters converter Each demand path assigned one wavelength from src to dest – no conversion Deploy extra fibers if necessary Min total fibers Model 2: Min fiber Extra fiber

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Complexity Perspective of worst-case analysis NP hard Cannot expect to find optimal solution efficiently for all instances Hard to approximate Cannot approximate within any constant [AndrewsZhang] For any algorithm, there exist instances for which the algo returns a solution more than any constant factor larger than the optimal.

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Heuristics Focus: Simple/flexible/scalable heuristics “Typical” input instances: not worst-case analysis A greedy heuristic For every demand d in an ordered demand set: Choose a locally optimal solution for d

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Why greedy? Viable approach for many hard problems Set Cover Problem (NP-hard) SAT solving (NP-hard) Planning Problems (PSPACE-hard) Vertex Coloring (NP-hard) …

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Vertex coloring: A closely related problem A classic problem from combinatorial optimization and graph theory Problem statement Graph D Color each vertex of D such that neighboring vertices have distinct colors Minimize the total number of colors needed

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Connection to vertex coloring Create a demand graph D from wavelength assignment instance G: One vertex for each demand Two demands vertices adjacent iff demand routes share common link Demand graph D is u colorable iff wavelength assignment feasible with 0 extra fibers and 0 conversion.

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What we know about vertex coloring Complexity – worst case NP-hard Hard to approximate: cannot be approximated to within a factor of n 1- [FeigeKilian][KnotPonnuswami] Heuristic solutions – common cases Greedy approaches extremely effective For vertex v in an ordering of vertices: Color v with smallest color not used by v’s neighbors Example: Brelaz’s algorithm [Turner] gives priority to“most constrained” vertex

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Try greedy wavelength assignment For every demand d in an ordered demand set: Choose a locally optimal solution for d - Is there good ordering? - Is it easy to find a good ordering? - Local optimality is easy!

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Local optimality for model 1 : min conversion 1.Starting at first link, assign wavelength available for greatest number of consecutive links. 2.Convert and continue on a different wavelength until the entire demand path is assigned wavelength(s). Strategy locally optimal

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Local optimality for model 1 : min conversion 1.Starting at first link, assign wavelength available for greatest number of consecutive links. 2.Convert and continue on a different wavelength until the entire demand path is assigned wavelength(s). Strategy locally optimal

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Local optimality for model 2: Min fiber 1.Choose wavelength w such that assigning w to demand d requires minimum number of extra fibers

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Local optimality for model 2: Min fiber 1.Choose wavelength w such that assigning w to demand d requires minimum number of extra fibers Extra fiber on first link

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Ordering in Greedy approach Global ordering: 1.Longest first : Order demands according to number of links each demand travels. 2.Heaviest : Weigh each link according to the number of demands that traverse it. Sum the weights on each link of a demand. 3.Ordering suggested by vertex coloring on demand graph 4.Random sampling: choose a random permutation.

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Ordering in Greedy approach Local perturbation: d1, d2, d3, d4, … 1.Coin toss : -Reshuffle initial demand ordering by: -Flipping a coin for each entry in order -With a success, remove the demand and move it to new ordering 2.Top-n : -Reshuffle initial demand ordering by: -Randomly choosing a first n demands -Removing the demand to new ordering

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Iterative refinement Global ordering Greedy Local perturbation Greedy

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Generating instances Characteristics of network topology: Sparse networks; average node degree < 3 Planar Small networks (~ 20 nodes) Large network (~ 50 nodes) Characteristics of traffic: Fiber Capacity ~ [20,100] Lightly loaded networks: 1 fiber per link, fibers half full Heavily loaded networks: ~ 2 fibers per links

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Topologies of real networks

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Experimental data Group 1: real networks (light load)

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Experimental data Group 1: real networks (light load)

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Probability of No Wavelength Conflict vs. Link Load -O(log u) approx.: choose a wavelength uniformly at random for each demand -Birthday Paradox!

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Experimental data Group 2: simulated networks (heavy + small)

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Experimental data Group 2: simulated networks (heavy + small)

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Experimental data: Large Networks Group 3: simulated networks (heavy + large)

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Experimental data Group 3: simulated networks (heavy + large)

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Summary – Preliminary observations Small + light (real networks) All greedy solutions close to optimal Log approx behaves poorly Small + heavy Random sampling has advantage Longest/heaviest less meaningful for shortest paths in small networks Large + heavy Longest/heaviest more meaningful

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Combined minimization New territory: Ultimate cost optimization Combined minimization of fiber and conversion Proposed approach Compute a min fiber solution (x extra fibers) From empty network, add one fiber at a time Compute a min conversion solution for fixed additional fibers.

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Combined minimization

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QUESTIONS???

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