1. 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

2. 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

3. Example

4. 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

5. 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

6. 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

7. 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.

8. 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

9. 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)  …

10. 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

11. 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.

13. 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

14. 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!

15. 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

16. 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

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

18. 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

19. 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.

20. 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

21. Iterative refinement Global ordering Greedy Local perturbation Greedy

22. 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

23. Topologies of real networks

26. Experimental data Group 1: real networks (light load)

27. Experimental data Group 1: real networks (light load)

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

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

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

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

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

34. 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

35. 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.

36. Combined minimization

37. QUESTIONS???