optima 2001

October 10, 2001

Routing in communication networks (OPTIMA 2001)

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OPTIMA 2001

Routing in communication networks and advances in metaheuristics

IV Congreso Chileno de Investigación Operativa

Curicó, Chile, October 2001

Celso C. RibeiroCatholic University of Rio de Janeiro, Brazil

October 10, 2001


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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

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• PVC routing• Integer multicommodity flow

formulation• Cost function• Solution method: GRASP with path-

relinking• Numerical results and conclusions

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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

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• 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

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• 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

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• 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

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PVC routing: example

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PVC routing: examplemax capacity = 3

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PVC routing: examplemax capacity = 3very long path!

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PVC routing: examplemax capacity = 3very long path!

reroute

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PVC routing: examplemax capacity = 3

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PVC routing: examplemax capacity = 3feasible and

optimal!

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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

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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

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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)

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Problem formulation

, ,( , ) ( , )

1

, , ,

1, , , , ,

,

, , ,

( ) ,

1,

min ( ) ( ,..., , ,..., )

subject

if is source for

1

to

( ) , (

, if

, ) ,

( ) , ( , ) ,k kk i j j i i j

k

k ki j j i i j

k K

k ki j

p ki j i j i j j i j i

i j E i j

j i

K

i j E i j E

x x x

r

i V k

x x b i j E

x x c i j E i j

x

j

x

i

x

K

x

,

is destination for

0, other

0,1 , ( , ) ,

wise

.ki jx i j

i K

E k K

V k

,ki jx= 1, iff trunk (i,j )

is used to route PVC k.

,ki jx

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Cost function

• Linear combination of – delay component - weighted by (1-)– load balancing component - weighted

by

• Delay component: , , ,( )k ki j k i j j ik K

d x x

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Cost function

• Load balancing component: measure of Fortz & Thorup (2000) to compute congestion:

= 1(L1) + 2(L2) + … + |E|(L|

E|)

where Le is the load on link e E,

e(Le) is piecewise linear and convex,

e(0) = 0, for all e E.

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Piecewise linear and convex e(Le) link

congestion measure

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1 1.2

cost

per

unit

of ca

paci

ty

trunk utilization rate

slope = 1slope = 3 slope = 10

slope = 70

slope = 500

slope = 5000

(Lece)

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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

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formulation• Cost function• Solution method: GRASP with

path-relinking• Numerical results and conclusions

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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

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Solution method: GRASP

• GRASP– Construction:

• RCL: nc unrouted PVCs with largest demands• choose unrouted pair k biasing in favor of high

bandwidth requirements, with probablity k = rk / (pRCL rp)

• 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 rk additional units of demand on it

• route pair k using shortest route between its endpoints

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Solution method: GRASP

• GRASP– Local search:

• for each PVC k K , remove rk units of flow from each edge in its current route

• recompute incremental weights of routing rk additional units of flow for all edges

• reroute PVC k using new shortest path

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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.

initialsolution

guidingsolution

path in neighborhood of solutions

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• 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:

Initialsolution

guiding solution

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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)}


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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.

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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)

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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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relinking• Numerical results and

conclusions

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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

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Experiment• Test problems:

The Cartesian product of a family of Theorem:algorithms by a family of test problems is

an unreadable table!

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• 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

TS

S T

S T

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 100 1000 10000 100000 1e+06

GGPRfGPRb

GPRfb

time

Pro

bab

ility

Each variant: 200 runs for one instance of PVC routing problem

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• 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

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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)

costmax util.

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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.)

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Parameter of the objective function

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0.0001 0.001 0.01 0.1 192000

94000

96000

98000

100000

102000

104000

maxim

um

utiliz

ation

dela

y

delta

delay

maximum utilization

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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

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• 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).

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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.

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Slides and publications

• Slides of this talk can be downloaded from: http://www.inf.puc-rio/~celso/talks/curico.ppt

• Recent survey about GRASP available at: http://www.inf.puc-rio.br/~celso/publicacoes• Paper about PVC routing available at: http://www.inf.puc-rio.br/~celso/publicacoes

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OPTIMA 2001

Routing in communication networks and advances in metaheuristicsPart II

IV Congreso Chileno de Investigación Operativa

Curicó, Chile, October 2001

Celso C. RibeiroCatholic University of Rio de Janeiro, Brazil

October 10, 2001


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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

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• Weight setting in OSPF routing• Genetic algorithm for OSPF routing• Population dynamics• Parallel GA for OSPF routing• Numerical results and conclusions

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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.

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• 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

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Packets of information

body header

Information sent over the Internet is broken into chunks, calledpackets or datagrams.

Contains necessary routinginformation, such as IP destination address.

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Packet routing

router

router

router

router

router

When packet arrives at router,router must decide where tosend it next.

Packet’s final destination.

Routing consists in finding apath from source to destination.

D1

D2

D3

D4

R1

R2

R3

R4Routing table

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Autonomous systems

To decrease the complexity ofrouting, the Internet is divided intosmaller domains, called AutonomousSystems.

AS1

AS2

AS3

AS4Routing within an AS is done viaInterior Gateway Protocols (IGP),while between AS’s Exterior GatewayProtocols (EGP) are used.

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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.

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• Each link in the AS is assigned an integer weight [1,65535=2161]– 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

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321

351

2

4

D1

D2

D3

D4

R1

R1

R2

R3

root

First hop routers.

Routing table

Destination routers

Routing table is filledwith first hop routersfor each possible destination.In case of multiple shortest paths, flow is evenly split.

D5

D6

R1

R36

Cisco 12400 routers

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• 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

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Minimization of congestion

• Directed capacitated network G = (N,A,c), where N are routers, A are links, and ca is the capacity of link a A.

• Same measure of Fortz & Thorup (2000) to compute congestion (also used for PVC routing):

= 1(L1) + 2(L2) + … + |A|(L|A|)

La is the load on link a A, a(La) is piecewise linear and convex, and a(0) = 0, for all a A.

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Piecewise linear and convex a(La) link

congestion measure

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1 1.2

cost

per

unit

of ca

paci

ty

trunk utilization rate

slope = 1slope = 3 slope = 10

slope = 70

slope = 500

slope = 5000

(Laca )

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• Given a directed network G = (N, A ) with link capacities ca A and demand matrix D = (ds,t ) specifying a demand to be sent from node s to node t :– Assign weights wa [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

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• Weight setting in OSPF routing• Genetic algorithm for OSPF

routing• Population dynamics• Parallel GA for OSPF routing• Numerical results and conclusions

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Genetic algorithms

Initialize and evaluate P (0);

Set t = 1Test termination

Generate P (t ) from P (t1)

Alter P (t )

Evaluate P (t )t = t + 1

done

crossover

mutationP (t ) is population ofsolutions at generation t.

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GA for OSPF: solution encoding

• Ericsson, Pardalos, & Resende (2001)• A population consists of nPop integer

weight arrays: w = (w1, w2 ,…, w|A| ),

where wa [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

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GA for OSPF: fitness evaluation

• Route each demand pair (s,t ) using OSPF

• Compute loads Las,t

on each link a A• Add up loads on each link a A,

yielding total load La on link• Compute link congestion cost a(La) for

each link a A• Add up costs:

= 1(L1) + 2(L2) + … + |A|(L|A|)

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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)

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Population partitioning

Class A

Class C

Class B

20% most fit

10% least fit

Population is sorted according tofitness (solution value) and solutions are classified into three categories.

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Population dynamics

Class A

Class C

Class B

generation t generation t + 1

Class AClass A is promoted unchanged

Class C is replaced by randomlygenerated solutions.

Class C

Class B is replaced by crossover of: one Class A parent and

one Class B or Cparent.

Class B

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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

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Crossover with random keys

Bean (1994): crossover combines elite parent p1 with non-elite parent p2 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 ] = p1[i ]

else c [i ] = p2[i ]

end

With small probability childhas single gene mutation.

Child is more likely to inheritgene of elite parent.

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Parallel GA: local search• Combine GA with local search• LS with cost recomputations from scratch:

– For each arc e with current weight we do: • Temporarily replace arc weight by (1+ we)/2• Evaluate fitness• If new improved solution, update weight and go to

next arc• Otherwise, temporarily replace arc weight by (MAX+

we)/2• Evaluate fitness• If new improved solution, update weight• Go to next arc

– Until no further improvement is possible

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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

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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

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• Weight setting in OSPF routing• Genetic algorithm for OSPF routing• Population dynamics• Parallel GA for OSPF routing• Numerical results and

conclusions

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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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2.35

2.4

2.45

2.5

2.55

2.6

2.65

2.7

2.75

2.8

2.85

0 2 4 6 8 10 12 14 16

co

st

processors

Collaborative vs. noncollaborative versions

collab_v1nocollab_v1

collab_v2nocollab_v2

F&TseqGA-500itr

LPLB

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2.35

2.4

2.45

2.5

2.55

2.6

2.65

2.7

2.75

2.8

2.85

0 2 4 6 8 10 12 14 16

co

st

processors

Number of generations: 500 and 8000

collab_v1-500itrnocollab_v1-500itr

collab_v1-8000itrnocollab_v1-8000itr

F&TseqGA-500itr

LPLB

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0

2

4

6

8

10

12

0 5000 100001500020000250003000035000400004500050000

cost

scaled internet traffic

InvCapGA //LPLB

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0

0.5

1

1.5

2

0 5000 100001500020000250003000035000400004500050000

ma

xim

um

utiliza

tio

n

scaled internet traffic

InvCapGA //LPLB

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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

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• Experiments with // in GRASP and path-relinking

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Some experiments with parallelism in GRASP and

path-relinkingParallel 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

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0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

pro

babili

ty

time to target solution value (seconds)

0

1

2

3

4

5

6

0 1 2 3 4 5 6

measu

red tim

es

exponential quantiles

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.

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 100 1000 10000 100000

pro

babili

ty

time to sub-optimal

1 processor2 processors4 processors8 processors

16 processors

3-index assignment

60 independent runsof each algorithm.

MPI implementation.

196Mhz MIPS R10000

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2

4

6

8

10

12

14

16

18

2 4 6 8 10 12 14 16

speedup

number of processors

linear speedupparallel implementation

3-index assignment

Average speedup of 60 independent runs.

MPI implementation.

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Some experiments with parallelism in GRASP and

path-relinkingPath-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

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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.

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Independent strategy

21 3 p4

seed(1) seed(2) seed(3) seed(4) seed(p)

Stopping criteria arecommunicated amongprocesses.

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Cooperative strategy

Elite set

1

Elite set

p

Elite set

3

Elite set

2

Solutions accepted into elite sets are communicated among processes.

Stopping criteriaare communicated among processes asbefore.

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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

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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.

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3-index assignment (AP3)

cost = 10

Complete tripartite graph:Each triangle made up ofthree distinctly colored nodes has a cost.

cost = 5

AP3: Find a set of trianglessuch that each node appearsin exactly one triangle and thesum of the costs of the triangles is minimized.

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16 processors

Independent on 3-index assignment: bs26

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Collaborative on 3-index assignment: bs26

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Speedup on 3-index assignment: bs26

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• 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.

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Slides and publications

• Slides of this talk can be downloaded from: http://www.inf.puc-rio/~celso/talks/curico.ppt

• Chapter about GRASP and PR 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/gaospf.pdf• Paper about parallel GA for OSPF setting in

preparation

October 10, 2001


/104Curicó (Chile), October 9, 2001, 7:30 PM

optima 2001

Documents