Md. Tanveer Anwar University of Arkansas PRSA For WDM Md. Tanveer Anwar University of Arkansas

PRSA = GA + SA (parallel) Genetic Algorithms Heuristic optimization technique Approximates global solution Inherently parallel Simulated Annealing Global Optimum solution Not Inherently parallel Parallel Recombinative Simulated Annealing PRSA = GA + SA (parallel using MPI) GA is a heuristic optimization technique for approximating the global solution to a complex problem space. GA is good at coarsely exploring the search space but is poor at precisely finding the local minima in the region that it finally selects. Therefore, the combination of a simple local search algorithm with GA is common. In theory, SA with a logarithmic cooling schedule converges with probability one to the global optimum solution, but requires an infinite number of iterations.

Wavelength-Division Multiplexing Coarse WDM - Channel spacing of 20 nanometers (nm) - Avoid temperature control problems - Less expensive Dense WDM - Channel spacing < 1 nm - 160 channels possible in 2001 Ultra Dense WDM - 1,022 channels (Bell Labs) WDM is the technology that enables multiple optical signals over a single fiber by carrying signals on separate wavelengths.

Physical Topology for WDM Network We are provided with a physical topology composed 25 Nodes. Each node has some traffic to every other node in the topology. At each node there are WDM cross-connects. There are two general categories of WDM cross-connects. Wavelength Selective Cross-Connects (WSXC) Wavelength Interchanging Cross-Connects (WIXC)

Wavelength Selective Cross-connect (WSXC) D1 S2 D2 Wavelength Interchanging Cross-connect (WIXC) WDM Wavelength Selective Cross-connect (WSXC) The wavelength on the input port is switched to an output port and the wavelength remains unchanged Wavelength Interchanging Cross-connect (WIXC) The cross-connect switches wavelengths from an input port to an output port and may change the input wavelength to a different output wavelength S1 D1 S2 D2

Example of Wavelength Assignment Constraint Routing and Wavelenght Assignment

Simple PRSA Problem GRAPH TRAFFIC 6 A 2 1 B C 5 4 3 D S-D Pairs Cost Capacity AB 2 5 AC 1 AD BC 6 BD 3 CD 4 INDEX S-D Pair l AB 1 AC 2 AD 3 BA 4 BC 5 BD 6 CA 7 CB 8 CD 9 DA 10 DB 11 DC A 2 1 B C 5 4 3 D

Simple PRSA Problem Combinatorial Problem 6 TRAFFIC K – Shortest Paths INDEX S-D Pair l AB 1 AC …. 11 DC A K0 : Shortest K1: 2nd Shortest K2: 3rd Shortest 2 1 B C 5 CHROMOSOME (12 Genomes) 1 2 3 4 5 6 7 8 9 10 11 K2 K0 K1 4 3 D 1 2 3 4 5 6 7 8 9 10 11 C0 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 Total Cost 4 Nodes : 312 25 Nodes : 3600 Combinatorial Problem

Optimal Solution (ILP) Ample/Cplex Advantages Optimal Solution Fast Disadvantages The problem must be bounded Requires High Memory (RAM)

PRSA Algorithm Preview Crossover Operation Mutation Operation Parent A 1 0 1 1 1 Parent B 1 1 0 1 0 Child A 1 0 0 1 0 Child B 1 1 1 1 1 Parent A 1 0 1 1 1 Parent B 1 1 0 1 0 Child A 1 0 0 1 0 Child B 1 1 0 1 1 Competition Between Parents and Children Parent A Child A Parent B Child B Metropolis Criteria If Child wins, accept it. If Parent wins, Accept the child with a probability: EXP((fparent – fchild)/T)

PRSA Algorithm Initialize the Temperature (SA) Initialize population with n chromosomes (GA) Repeat for max generations Do n/2 times Select 2 parent chromosomes at random (GA) Generate 2 children using crossover and mutation (GA) Hold competitions using the Metropolis criterion between children and parents (SA) Overwrite parents with trial winner Lower the Temperature (GA) Send/Receive migrants to/from other processors Simulate Annealing Initialize the Temperature Initialize the Current Solution Evaluate the Solution While Temperature > 0 Modify the current solution (Mutation) Evaluate the new solution Use Metropolis Selection Criteria to determine whether we should accept/reject the new solution Lower the Temperature Genetic Algorithm Initialize population with n chromosomes Evaluate fitness of each chromosome Save best chromosome Repeat for max generations Select chromosomes for next generation Pick two chromosomes with probability pC and exchange genetic material with a crossover operator Mutate genes with probability pM Use elitist strategy

Results p = 50, m = 2 m = 2, c = 0.99 p = 50, c = 0.99

Conclusion Thank You !! Another Method to solve Combinatorial Problems Like S.A, a smaller cooling coefficient that causes a faster decrease in temperature increases convergence rate at the expense of the final solution A large population size is preferable but not too large Keep the # of migrants to a minimum. Thank You !!