1. July 14, 2001Routing in communication networks (MIC'2001)Page 1/70 Routing in communication networks Celso C. Ribeiro Computer Science Department Catholic University of Rio de Janeiro joint work with M.G.C. Resende

2. July 14, 2001Routing in communication networks (MIC'2001)Page 2/70 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, extensions, and conclusions

3. July 14, 2001Routing in communication networks (MIC'2001)Page 3/70 PVC routing Integer multicommodity flow formulation Cost function Solution method: GRASP with path- relinking Numerical results and conclusions

4. July 14, 2001Routing in communication networks (MIC'2001)Page 4/70 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

5. July 14, 2001Routing in communication networks (MIC'2001)Page 5/70 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

6. July 14, 2001Routing in communication networks (MIC'2001)Page 6/70 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

7. July 14, 2001Routing in communication networks (MIC'2001)Page 7/70 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

8. July 14, 2001Routing in communication networks (MIC'2001)Page 8/70 PVC routing: example

9. July 14, 2001Routing in communication networks (MIC'2001)Page 9/70 PVC routing: example

10. July 14, 2001Routing in communication networks (MIC'2001)Page 10/70 PVC routing: example

11. July 14, 2001Routing in communication networks (MIC'2001)Page 11/70 PVC routing: example

12. July 14, 2001Routing in communication networks (MIC'2001)Page 12/70 PVC routing: example max capacity = 3

13. July 14, 2001Routing in communication networks (MIC'2001)Page 13/70 PVC routing: example max capacity = 3very long path!

14. July 14, 2001Routing in communication networks (MIC'2001)Page 14/70 PVC routing: example max capacity = 3very long path! reroute

15. July 14, 2001Routing in communication networks (MIC'2001)Page 15/70 PVC routing: example max capacity = 3

16. July 14, 2001Routing in communication networks (MIC'2001)Page 16/70 PVC routing: example max capacity = 3 feasible and optimal!

17. July 14, 2001Routing in communication networks (MIC'2001)Page 17/70 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

18. July 14, 2001Routing in communication networks (MIC'2001)Page 18/70 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

19. July 14, 2001Routing in communication networks (MIC'2001)Page 19/70 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)

20. July 14, 2001Routing in communication networks (MIC'2001)Page 20/70 Problem formulation = 1, iff trunk (i,j ) is used to route PVC k.

21. July 14, 2001Routing in communication networks (MIC'2001)Page 21/70 Cost function Linear combination of –delay component - weighted by (1-  ) –load balancing component - weighted by  Delay component:

22. July 14, 2001Routing in communication networks (MIC'2001)Page 22/70 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.

23. July 14, 2001Routing in communication networks (MIC'2001)Page 23/70 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)

24. July 14, 2001Routing in communication networks (MIC'2001)Page 24/70 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

25. July 14, 2001Routing in communication networks (MIC'2001)Page 25/70 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

26. July 14, 2001Routing in communication networks (MIC'2001)Page 26/70 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

27. July 14, 2001Routing in communication networks (MIC'2001)Page 27/70 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

28. July 14, 2001Routing in communication networks (MIC'2001)Page 28/70 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

29. July 14, 2001Routing in communication networks (MIC'2001)Page 29/70 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

30. July 14, 2001Routing in communication networks (MIC'2001)Page 30/70 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

31. July 14, 2001Routing in communication networks (MIC'2001)Page 31/70 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.

32. July 14, 2001Routing in communication networks (MIC'2001)Page 32/70 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)

33. July 14, 2001Routing in communication networks (MIC'2001)Page 33/70 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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35. July 14, 2001Routing in communication networks (MIC'2001)Page 35/70 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

36. July 14, 2001Routing in communication networks (MIC'2001)Page 36/70 Experiment Test problems: The Cartesian product of a family of  Theorem: algorithms by a family of test problems is an unreadable table!

37. July 14, 2001Routing in communication networks (MIC'2001)Page 37/70 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

38. July 14, 2001Routing in communication networks (MIC'2001)Page 38/70 Variants of GRASP and path- relinking Iterations Probability Each variant: 200 runs for one instance of PVC routing problem

39. July 14, 2001Routing in communication networks (MIC'2001)Page 39/70 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

40. July 14, 2001Routing in communication networks (MIC'2001)Page 40/70 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.

41. July 14, 2001Routing in communication networks (MIC'2001)Page 41/70 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.)

42. July 14, 2001Routing in communication networks (MIC'2001)Page 42/70 Parameter of the objective function

43. July 14, 2001Routing in communication networks (MIC'2001)Page 43/70 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

44. July 14, 2001Routing in communication networks (MIC'2001)Page 44/70 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).

45. July 14, 2001Routing in communication networks (MIC'2001)Page 45/70 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.

46. July 14, 2001Routing in communication networks (MIC'2001)Page 46/70 Weight setting in OSPF routing Genetic algorithm for OSPF routing Population dynamics Parallel GA for OSPF routing Numerical results, extensions, and conclusions

47. July 14, 2001Routing in communication networks (MIC'2001)Page 47/70 Weight setting in OSPF routing Internet traffic has been doubling each year Coffman & Odlyzko (2001): in the 1995-96 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. Objective of traffic engineering: make more efficient use of network resources Traffic routing can have a major impact on the efficiency of network resource utilization

48. July 14, 2001Routing in communication networks (MIC'2001)Page 48/70 Traffic routing When packet arrives at router, router must decide where to send it next. Routing consists in finding a path from source to destination. To decrease the complexity of routing, the Internet is divided into smaller domains, called Autonomous Systems (AS). Routing within an AS is done via Interior Gateway Protocols (IGP), while between AS’s Exterior Gateway Protocols (EGP) are used.

49. July 14, 2001Routing in communication networks (MIC'2001)Page 49/70 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.

50. July 14, 2001Routing in communication networks (MIC'2001)Page 50/70 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

51. July 14, 2001Routing in communication networks (MIC'2001)Page 51/70 Weight setting in OSPF routing 321 351 2 4 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 12400 routers

52. July 14, 2001Routing in communication networks (MIC'2001)Page 52/70 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

53. July 14, 2001Routing in communication networks (MIC'2001)Page 53/70 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.

54. July 14, 2001Routing in communication networks (MIC'2001)Page 54/70 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

55. July 14, 2001Routing in communication networks (MIC'2001)Page 55/70 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.

56. July 14, 2001Routing in communication networks (MIC'2001)Page 56/70 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

57. July 14, 2001Routing in communication networks (MIC'2001)Page 57/70 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| )

58. July 14, 2001Routing in communication networks (MIC'2001)Page 58/70 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) Population is sorted according to fitness  and classified into three categories

59. July 14, 2001Routing in communication networks (MIC'2001)Page 59/70 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 20% most fit 10% least fit

60. July 14, 2001Routing in communication networks (MIC'2001)Page 60/70 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.

61. July 14, 2001Routing in communication networks (MIC'2001)Page 61/70 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

62. July 14, 2001Routing in communication networks (MIC'2001)Page 62/70 Parallel GA: local search Variants: –V1: at each processor, apply LS to the best solution whenever it is improved –V2: at each processor, always apply LS to the best non-locally optimal solution

63. July 14, 2001Routing in communication networks (MIC'2001)Page 63/70 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

64. July 14, 2001Routing in communication networks (MIC'2001)Page 64/70 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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69. July 14, 2001Routing in communication networks (MIC'2001)Page 69/70 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

70. July 14, 2001Routing in communication networks (MIC'2001)Page 70/70 Slides and publications Slides of this talk can be downloaded from: http://www.inf.puc-rio/~celso/talks Paper about PVC routing available at: http://www.inf.puc- rio.br/~celso/publicacoes Paper about sequential GA for OSPF setting available at: http://www.research.att.com/~mgcr/doc/ga ospf.pdf Paper about parallel GA for OSPF weight setting in preparation