October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 1/104 OPTIMA 2001 Routing in communication networks and advances in metaheuristics IV Congreso Chileno de Investigación Operativa Curicó, Chile, October 2001 Celso C. Ribeiro Catholic University of Rio de Janeiro, Brazil

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 2/104 Summary PVC routing Integer multicommodity flow formulation Cost function Solution method: GRASP with path-relinking Numerical results and conclusions Weight setting in OSPF routing Genetic algorithm for OSPF routing Population dynamics Parallel GA for OSPF routing Numerical results and conclusions Experiments with // in GRASP and path- relinking

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 3/104 PVC routing Integer multicommodity flow formulation Cost function Solution method: GRASP with path- relinking Numerical results and conclusions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 4/104 PVC routing: application Frame relay service offers virtual private networks: permanent (long- term) virtual circuits (PVCs) between customer endpoints on a backbone network Routing: either automatically by switch or by network designer without any knowledge of future requests Inefficiencies and occasional need for off-line rerouting of the PVCs

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 5/104 PVC routing: application Reorder PVCs and apply algorithm on switch to reroute: –taking advantage of factors not considered by switch algorithm may lead to greater network efficiency –FR switch algorithm is typically fast since it is also used to reroute in case of switch or trunk failures –this can be traded off for improved network resource utilization when routing off-line

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 6/104 PVC routing: application Other algorithms simply handle the number of hops (e.g. routing algorithm in Cisco switches) Handling delays is particularly important in international networks, where distances between backbone nodes vary considerably Cisco Catalystic 5505 switch

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 7/104 PVC routing: application Load balancing is important for providing flexibility to handle: –overbooking: typically used by network designers to account for non-coincidence of traffic –PVC rerouting: due to failures –bursting above the committed rate: not only allowed, but also sold to customers as one of the attractive features of frame relay Integer multicommodity network flow problem

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 8/104 PVC routing: example

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 9/104 PVC routing: example

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 10/104 PVC routing: example

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 11/104 PVC routing: example

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 12/104 PVC routing: example max capacity = 3

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 13/104 PVC routing: example max capacity = 3very long path!

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 14/104 PVC routing: example max capacity = 3very long path! reroute

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 15/104 PVC routing: example max capacity = 3

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 16/104 PVC routing: example max capacity = 3 feasible and optimal!

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 17/104 PVC routing Integer multicommodity flow formulation Cost function Solution method: GRASP with path- relinking Numerical results and conclusions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 18/104 Problem formulation Given undirected FR network G = (V, E), where –V denotes n backbone nodes (FR switches) –E denotes m trunks connecting backbone nodes for each trunk e = (i,j ) –b (e ): maximum bandwidth (max kbits/sec rate) –c (e ): maximum number of PVCs that can be routed on it –d (e ): propagation and hopping delay

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 19/104 Problem formulation Demands K = {1,…,p } defined by –Origin-destination pairs –r (p): effective bandwidth requirement (forward, backward, overbooking) for PVC p Objective is to minimize –delays –network load unbalance subject to –technological constraints

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 20/104 Problem formulation route for PVC (o, d ) is a sequence of adjacent trunks from node o to node d set of routing assignments is feasible if for all trunks e –total bandwidth requirements routed on e does exceed b (e) –number of PVCs routed on e not greater than c(e)

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 21/104 Problem formulation = 1, iff trunk (i,j ) is used to route PVC k.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 22/104 PVC routing Integer multicommodity flow formulation Cost function Solution method: GRASP with path- relinking Numerical results and conclusions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 23/104 Cost function Linear combination of –delay component - weighted by (1-  ) –load balancing component - weighted by  Delay component:

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 24/104 Cost function Load balancing component: measure of Fortz & Thorup (2000) to compute congestion:  =  1 (L 1 ) +  2 (L 2 ) + … +  |E| (L |E| ) where L e is the load on link e  E,  e (L e ) is piecewise linear and convex,  e (0) = 0, for all e  E.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 25/104 Piecewise linear and convex  e (L e ) link congestion measure slope = 1 slope = 3slope = 10 slope = 70 slope = 500 slope = 5000 (Lece)(Lece)

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 26/104 Some recent applications Laguna & Glover (1993): tabu search, different cost function, no constraints on PVCs routed on the same trunk (assign calls to paths) Sung & Park (1995): Lagrangean heuristic, very small graphs Amiri et al. (1999): Lagrangean heuristic, min delay Dahl et al. (1999): cutting planes (traffic assignment) Barnhart et al (2000): branch-and-price, different cost function, no constraints on PVCs routed on same trunk Shyur & Wen (2000): tabu search, min hubs

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 27/104 PVC routing Integer multicommodity flow formulation Cost function Solution method: GRASP with path- relinking Numerical results and conclusions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 28/104 Solution method: GRASP with path-relinking GRASP: Multistart metaheuristic, Feo & Resende 1989 Path-relinking: intensification, Glover (1996) Repeat for Max_Iterations: –Construct greedy randomized solution –Use local search to improve constructed solution –Apply path-relinking to further improve solution –Update pool of elite solutions –Update best solution found

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 29/104 Solution method: GRASP GRASP –Construction : RCL: n c unrouted PVCs with largest demands choose unrouted pair k biasing in favor of high bandwidth requirements, with probablity  k = r k / (  p  RCL r p ) capacity constraints relaxed and handled via the penalty function introduced by the load- balance component length of each edge (i,j) is the incremental cost of routing r k additional units of demand on it route pair k using shortest route between its endpoints

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 30/104 Solution method: GRASP GRASP –Local search: for each PVC k  K, remove r k units of flow from each edge in its current route recompute incremental weights of routing r k additional units of flow for all edges reroute PVC k using new shortest path

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 31/104 Solution method: path- relinking Introduced in the context of tabu search by Glover (1996) –Intensification strategy using set of elite solutions Consists in exploring trajectories that connect high quality solutions. initial solution guiding solution path in neighborhood of solutions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 32/104 Solution method: path- relinking Path is generated by selecting moves that introduce in the initial solution attributes of the guiding solution. At each step, all moves that incorporate attributes of the guiding solution are evaluated and the best move is taken: Initial solution guiding solution

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 33/104 Elite solutions x and y  (x,y): symmetric difference between S and T while ( |  (x,y)| > 0 ) { evaluate moves corresponding in  (x,y) make best move update  (x,y) } Solution method: path- relinking

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 34/104 Path-relinking in GRASP Introduced by Laguna & Martí (1999) Maintain an elite set of solutions found during GRASP iterations. After each GRASP iteration (construction & local search): –Select an elite solution at random: guiding solution. –Use GRASP solution as initial solution. –Perform path-relinking between these two solutions.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 35/104 Path-relinking in GRASP Successful applications: –Prize-collecting Steiner tree problem Canuto, Resende, & Ribeiro (2000) –Steiner tree problem Ribeiro, Uchoa, & Werneck (2000) (e.g., best known results for open problems in series dv640 of the SteinLib) –Three-index assignment problem Aiex, Pardalos, Resende, & Toraldo (2000)

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 36/104 Path-relinking: elite set P is set of elite solutions Each iteration of first |P | GRASP iterations adds one solution to P (if different from others). After that: solution x is promoted to P if: –x is better than best solution in P. –x is not better than best solution in P, but is better than worst and it is sufficiently different from all solutions in P.

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October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 38/104 PVC routing Integer multicommodity flow formulation Cost function Solution method: GRASP with path- relinking Numerical results and conclusions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 39/104 Experiment Heuristics: –H1: sorts demands in decreasing order and routes them using minimum hops paths –H2: sorts demands in decreasing order and routes using same cost function as GRASP –H3: adds the same local search to H2 –GPRb: GRASP with backwards path- relinking SGI Challenge 196 MHz

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 40/104 Experiment Test problems: The Cartesian product of a family of  Theorem: algorithms by a family of test problems is an unreadable table!

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 41/104 Variants of path-relinking: –G: pure GRASP –GPRb: GRASP with backward PR –GPRf: GRASP with forward PR –GPRbf: GRASP with two-way PR Other strategies: –Truncated path-relinking –Do not apply PR at every iteration (frequency) Variants of GRASP and path- relinking S T T S S T S T

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 42/104 Variants of GRASP and path- relinking time Probability Each variant: 200 runs for one instance of PVC routing problem

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 43/104 Variants of GRASP and path- relinking Same computation time: probability of finding a solution at least as good as the target value increases from G  GPRf  GPRfb  GPRb P(h,t) = probability variant h finds solution as good as target value in time no greater than t –P(GPRfb,100s)=9.25% P(GPRb,100s)=28.75% –P(G,2000s)=8.33% P(GPRf,2000s)=65.25% P(h,time)=50% Times for each variant: –GPRb:129s G:10933s GPRf:1727s GPRfb:172s

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 44/104 Comparisons Distribution: 86/60/2: 86 edges with utilization in [0,1/3), 60 in [1/3,2/3), and two in [2/3,9/10) In general: GPRB > H3 > H2 > H1 (cost, max utilization, distribution) cost max util.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 45/104 Parameter of the objective function Objective function  (solution) = Delay x (1-  ) + Load imbalance cost x  if  = 1: consider only trunk utilization rates if  = 0: consider only delays (capacities relaxed) increasing  0  1  minimization of maximum utilization rate dominates  reduction of flows in edges with higher loads  increase of flows in underloaded edges until the next breakpoint  flows concentrate around breakpoint levels  useful strategy for setting appropriate value of  to achieve some level of quality of service (max util.)

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 46/104 Parameter of the objective function

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 47/104 Concluding remarks (1/3) New formulation with flexible objective function Family of heuristics (greedy, greedy+LS, GRASP, GRASP+PR) Simple greedy heuristic improves algorithm used in traffic engineering by network planners Objective function provides effective strategy for setting the weight parameter to achieve some quality of service level

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 48/104 Concluding remarks (2/3) Path-relinking adds memory and intensification mechanisms to GRASP, systematically contributing to improve solution quality. Some implementation strategies appear to be more effective than others (e.g., backwards from better, elite solution to current locally optimal solution).

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 49/104 Concluding remarks (3/3) NETROUTER – Tool for optimally loading demands on single-path routes on a capacitated network. It uses the GPRb variant of the combination of GRASP and path-relinking, minimizing delays while balancing network load. Application - Netrouter is currently being used for the design of AT&T's next generation frame-relay and MPLS core architecture, to assess if the current and forecasted demands can be handled by the proposed trunking plan.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 50/104 Slides and publications Slides of this talk can be downloaded from: rio/~celso/talks/curico.ppt Recent survey about GRASP available at: rio.br/~celso/publicacoes Paper about PVC routing available at: rio.br/~celso/publicacoes

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 51/104 OPTIMA 2001 Routing in communication networks and advances in metaheuristics Part II IV Congreso Chileno de Investigación Operativa Curicó, Chile, October 2001 Celso C. Ribeiro Catholic University of Rio de Janeiro, Brazil

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 52/104 Summary PVC routing Integer multicommodity flow formulation Cost function Solution method: GRASP with path-relinking Numerical results and conclusions Weight setting in OSPF routing Genetic algorithm for OSPF routing Population dynamics Parallel GA for OSPF routing Numerical results and conclusions Experiments with // in GRASP and path- relinking

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 53/104 Weight setting in OSPF routing Genetic algorithm for OSPF routing Population dynamics Parallel GA for OSPF routing Numerical results and conclusions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 54/104 Weight setting in OSPF routing Internet traffic has been doubling each year Coffman & Odlyzko (2001): in the period (introduction of web browsers), traffic doubled every three months! Increasingly heavy traffic (due to video, voice, etc.) is raising the requirements of the Internet of tomorrow.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 55/104 Weight setting in OSPF routing Objective of traffic engineering: make more efficient use of existing network resources Routing of traffic can have a major impact on the efficiency of network resource utilization

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 56/104 Packets of information bodyheader Information sent over the Internet is broken into chunks, called packets or datagrams. Contains necessary routing information, such as IP destination address.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 57/104 Packet routing router When packet arrives at router, router must decide where to send it next. Packet’s final destination. Routing consists in finding a path from source to destination. D1D1 D2D2 D3D3 D4D4 R1R1 R2R2 R3R3 R4R4 Routing table

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 58/104 Autonomous systems To decrease the complexity of routing, the Internet is divided into smaller domains, called Autonomous Systems. AS 1 AS 2 AS 3 AS 4 Routing within an AS is done via Interior Gateway Protocols (IGP), while between AS’s Exterior Gateway Protocols (EGP) are used.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 59/104 OSPF (Open Shortest Path First) OSPF is the most commonly used intra-domain routing protocol (IGP). It requires routers to exchange routing information with all other routers in the AS. –Complete network topology knowledge is available to all routers, i.e. state of all routers and links in the AS.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 60/104 Weight setting in OSPF routing Each link in the AS is assigned an integer weight  [1,65535=2 16  1] –Smaller weights may be used: MAX Each router computes tree of shortest weight paths to all other routers in the AS, with itself as the root, using Dijkstra’s algorithm. Bottom: Cisco 7000 router Top: ForeRunner ASX-200 ATM switch

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 61/104 Weight setting in OSPF routing D1D1 D2D2 D3D3 D4D4 R1R1 R1R1 R2R2 R3R3 root First hop routers. Routing table Destination routers Routing table is filled with first hop routers for each possible destination. In case of multiple shortest paths, flow is evenly split. D5D5 D6D6 R1R1 R3R3 6 Cisco routers

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 62/104 Weight setting in OSPF routing OSPF weights are assigned by network operator –CISCO assigns, by default, a weight proportional to the inverse of the available link bandwidth. –If all weights are unit, the cost of a path is the number of hops in the path. Fortz & Thorup (2000): weight setting by using local search on large networks with up to 100 nodes and 503 links Ericsson, Pardalos, & Resende (2001): genetic algorithm

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 63/104 Minimization of congestion Directed capacitated network G = (N,A,c), where N are routers, A are links, and c a is the capacity of link a  A. Same measure of Fortz & Thorup (2000) to compute congestion (also used for PVC routing):  =  1 (L 1 ) +  2 (L 2 ) + … +  |A| (L |A| ) L a is the load on link a  A,  a (L a ) is piecewise linear and convex, and  a (0) = 0, for all a  A.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 64/104 Piecewise linear and convex  a (L a ) link congestion measure slope = 1 slope = 3slope = 10 slope = 70 slope = 500 slope = 5000 (L a  c a )

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 65/104 Weight setting in OSPF routing Given a directed network G = (N, A ) with link capacities c a  A and demand matrix D = (d s,t ) specifying a demand to be sent from node s to node t : –Assign weights w a  [1,65535] to each link a  A, such that the objective function  is minimized when demand is routed according to the OSPF protocol. Weights are computed off-line and do not change often

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 66/104 Weight setting in OSPF routing Genetic algorithm for OSPF routing Population dynamics Parallel GA for OSPF routing Numerical results and conclusions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 67/104 Genetic algorithms Initialize and evaluate P (0); Set t = 1 Test termination Generate P (t ) from P (t  1) Alter P (t ) Evaluate P (t )t = t + 1 done crossover mutation P (t ) is population of solutions at generation t.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 68/104 GA for OSPF: solution encoding Ericsson, Pardalos, & Resende (2001) A population consists of nPop integer weight arrays: w = (w 1, w 2,…, w |A| ), where w a  [1,MAX] All possible weight arrays correspond to feasible solutions, i.e., every weight setting is feasible –nice problem feature for application of a GA

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 69/104 GA for OSPF: fitness evaluation Route each demand pair (s,t ) using OSPF Compute loads L a s,t on each link a  A Add up loads on each link a  A, yielding total load L a on link Compute link congestion cost  a (L a ) for each link a  A Add up costs:  =  1 (L 1 ) +  2 (L 2 ) + … +  |A| (L |A| )

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 70/104 Weight setting in OSPF routing Genetic algorithm for OSPF routing Population dynamics Parallel GA for OSPF routing Numerical results and conclusions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 71/104 Initial population nPop  10 solutions with randomly generated arc weights, uniformly in the interval [1,MAX] Weight settings of two other common heuristics: –OSPF (unit): all weights set to 1 –OSPF (invCap): each arc weight is set inversely proportional to its arc capacity –OSPF (fractions): all weights set to .MAX, with  = 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, 1 all but invCap lead to the same routing decisions (all weights are equal)

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 72/104 Population partitioning Class A Class C Class B 20% most fit 10% least fit Population is sorted according to fitness (solution value)  and solutions are classified into three categories.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 73/104 Population dynamics Class A Class C Class B generation t generation t + 1 Class A Class A is promoted unchanged Class C is replaced by randomly generated solutions. Class C Class B is replaced by crossover of: one Class A parent and one Class B or C parent. Class B

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 74/104 Parent selection Parents are chosen at random: –one parent from Class A (elite) –one parent from Class B or C (non-elite) Reselection is allowed, i.e. parents can breed more than once per generation Better individuals are more likely to reproduce

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 75/104 Crossover with random keys Bean (1994): crossover combines elite parent p 1 with non-elite parent p 2 to produce child c : for all genes i = 1,2,…,|A | do if rrandom[0,1] < 0.01 then c [i ] = irandom[1,MAX] else if rrandom[0,1] < 0.7 then c [i ] = p 1 [i ] else c [i ] = p 2 [i ] end With small probability child has single gene mutation. Child is more likely to inherit gene of elite parent.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 76/104 Weight setting in OSPF routing Genetic algorithm for OSPF routing Population dynamics Parallel GA for OSPF routing Numerical results and conclusions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 77/104 Parallel GA: local search Combine GA with local search LS with cost recomputations from scratch: –For each arc e with current weight w e do: Temporarily replace arc weight by  (1+ w e )/2  Evaluate fitness If new improved solution, update weight and go to next arc Otherwise, temporarily replace arc weight by  (MAX+ w e )/2  Evaluate fitness If new improved solution, update weight Go to next arc –Until no further improvement is possible

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 78/104 Parallel GA: local search Variants: –V-1: at each processor, apply LS to the best solution whenever it is improved –V-2: at each processor, always apply LS to the best non-locally optimal solution

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 79/104 Parallel GA: cooperation P processors Whenever a processor improves its incumbent, the latter is broadcasted to: –all other processors –all closest log P processors (logical organization) At the beginning of each generation, every processor replaces its worst solutions by those sent by other processors

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 80/104 Weight setting in OSPF routing Genetic algorithm for OSPF routing Population dynamics Parallel GA for OSPF routing Numerical results and conclusions

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 81/104 Numerical results Work-in-progress, preliminary results: GA, LS –Combine GA+LS? Cooperative // GA? Scatter search? One real world network: AT&T Worldnet backbone with 90 nodes, 274 links, and 272 pairs Compared with cost and maximum utilizations of the LB lower bound and several heuristics: –OSPF(invCap) –Local search of Fortz and Thorup (2000) –Original sequential GA of Ericsson et al. (2001) –LP lower bound

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October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 86/104 Concluding remarks (1/1) Sequential GAOSPF produced as good solutions as LS for most instances, even better in some cases. GA generally finds good solutions close to the LP lower bound. //GA+LS works very well on real-world AT&T Worldnet backbone network, significantly increasing traffic and Internet capacity over CISCO’s recommended weight setting strategy. Extensions: speedup LS, improve cooperation, evaluate effects, scatter search

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 87/104 Experiments with // in GRASP and path-relinking

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 88/104 Some experiments with parallelism in GRASP and path-relinking Parallel implementations of GRASP Aiex, Resende, & Ribeiro (2000): speedups in independent multi- thread parallel GRASP implementations random variable time to target solution value fits a two-parameter exponential distribution  approximate linear speedups with straightforward implementations

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 89/104 Using standard graphical methodology ( Aiex, Resende, & Ribeiro, 2000), one observes that random variable time to target solution value fits a two-parameter exponential distribution. Therefore, one should expect approximate linear speedup in a straightforward parallel implementation.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 90/104 3-index assignment 60 independent runs of each algorithm. MPI implementation. 196Mhz MIPS R10000

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 91/104 3-index assignment Average speedup of 60 independent runs. MPI implementation.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 92/104 Some experiments with parallelism in GRASP and path-relinking Path-relinking in parallel Aiex, Pardalos, Resende, & Toraldo 2000 Stopping criteria Independent strategy Cooperative strategy Message Passing Interface (MPI) implementation SGI Challenge computer with 28 processors

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 93/104 Stopping strategy If process finds target solution –it stops and sends a message to other processes, which stop. If process completes maximum number of iterations –it sends a message to other processes, which do not stop until all processes complete maximum number of iterations.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 94/104 Independent strategy 213p4 seed(1)seed(2)seed(3)seed(4)seed(p) Stopping criteria are communicated among processes.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 95/104 Cooperative strategy Elite set 1 p 3 2 Solutions accepted into elite sets are communicated among processes. Stopping criteria are communicated among processes as before.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 96/104 Elite set communication Each process checks if there is any message to receive before each PR leg. If messages are waiting: –receive messages: one or more candidate elite solutions –apply acceptance criteria to each candidate solution –update elite set of process if necessary

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 97/104 Elite set communication In order to minimize communication: –During a GRASP+PR iteration, each process bufferizes all solutions accepted into its elite set. –At end of GRASP+PR iteration, bufferized solutions are sent to all other processes.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 98/104 3-index assignment (AP3) cost = 10 Complete tripartite graph: Each triangle made up of three distinctly colored nodes has a cost. cost = 5 AP3: Find a set of triangles such that each node appears in exactly one triangle and the sum of the costs of the triangles is minimized.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 99/104 Independent on 3-index assignment: bs26

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 100/104 Collaborative on 3-index assignment: bs26

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 101/104 Speedup on 3-index assignment: bs26

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 102/104 Concluding remarks (1/1) Path-relinking adds intensification and memory mechanisms to GRASP. Time to target solution fits a two- parameter exponential distribution, so approximate linear speedups can be expected using independent processors. Exchange of information by processors can improve performance of parallel implementation.

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 103/104 Slides and publications Slides of this talk can be downloaded from: rio/~celso/talks/curico.ppt Chapter about GRASP and PR available at: rio.br/~celso/publicacoes Paper about sequential GA for OSPF setting available at: ospf.pdf Paper about parallel GA for OSPF setting in preparation

October 10, 2001 Routing in communication networks (OPTIMA 2001) Page 104/104 Curicó (Chile), October 9, 2001, 7:30 PM