problemas de otimização em network design e biologia: ajuste de pesos em roteamento sob ospf

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Problemas de otimização em network design e biologia

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Problemas de otimização em network design e biologia: •Ajuste de pesos em roteamento sob OSPF• Correspondência de grafos em reconhecimento de cenasCelso Carneiro Ribeiro

Seminário do Departamento de InformáticaPUC-Rio, Junho 2002

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Weight setting in OSPF routing

• 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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• 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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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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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• 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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Graph matching for scene recognition

• Scene recognition• Graph matching• Problem formulation• GRASP heuristic

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Scene recognition

• The recognition of complex scenes requires not only a detailed description of the objects involved, but also of the spatial relationships between them.

• The diversity of the forms of the same object in different instantiations of a scene, and also the similarities of different objects in the same scene make relationships between objects of prime importance in order to remove ambiguities in the recognition of objects with similar appearance.

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Scene recognition• Graph based representations often used

for scene representation in image processing: – Vertices represent the objects in the scene– Edges represent the relationships between

the objects• Relevant information for the recognition

is extracted from the scene and represented by relational attributed graphs.

• Model-based recognition: both the model and the scene are represented by graphs– graph matching

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Motivation• Medical imaging: recognition of brain

structures from magnetic resonance images

• Model: each node represents one anatomical structure, while edges represent spatial relationships.

• Inaccuracies are one of the main problem characteristics: – Unexpected objects (pathologies, for

instance) – Expected objects not found– Deformations of objects – Attributes in the image and in the model

may be imprecise

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(a) normal brain(b) pathological brain with a tumor

(c) representation of a

brain atlas (each grey

level corresponds to a

unique connected structure)

Motivation

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Scene recognition

• Similar problems: character recognition, aerial or satellite image interpretation using a map, face recognition, etc.

• Search for the best correspondence formulated as a combinatorial optimization problem defined by the relational attributed graphs representing the model and the image, together with node and edge similarities between them.

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Graph matching• Attributed graphs widely used in pattern

recognition problems: construction depends on the imperfections of the scene or of the model, and on the attributes of the object relations

• One single vertex for each region of each image

• Once the graph has been constructed, vertex and edge attributes are computed.

• Finally, vertex-similarity and edge-similarity matrices are computed from the values of the attributed graphs, relating each pair of vertices and each pair of edges (one from the model, the other from the segmented image).

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Graph matching

• G1 = (N1,E1): model

• G2 = (N2,E2): over-segmented image

• Each solution of the graph matching problem is a correspondence between the vertex and edges of the graphs representing the model and the image

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Example: two solutions

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Graph matching• Each node of G2 is associated with one

node of G1: – xij = 1, if nodes i N1and j N2 are associated– xij = 0, otherwise.

• Each solution may be represented by a bipartite graph G' = (N1 N2,E'):– each edge (i,j) E‘ = {(i,j), i N1, j N2 | xij =

1} corresponds to the association of i N1 with j N2.

– A(i) = { j N2 | (i,j) E'} is the subset of nodes of N2 associated with i N1

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Objective function

• Similarity matrices constructed from similarity values calculated from graph attributes.

• sv(i,j) [0,1] represents the vertex-similarity between vertices i N1 and j N2

• se(u,v) [0,1] represents the edge-similarity between edges u E1 and v E2

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Objective function

Maximize G’

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Constraints

1. exactly one node i N1 such that (i,j) E‘ (i.e. xij = 1), j N2

2. The subgraph induced in G2 by A(i) is connected, i N1

3. at least one node j N2 such that (i,j) E', i N1

4. xij = 1 sv(i,j) > 0, i N1, j N2

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GRASP

For i = 1,…,MaxIterationsBuild a feasible correspondence;Apply local search to improve the correspondence;Update the best solution found;

End;

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ConstructionFor k = 1,…,MaxTrials

Generate a random permutation of N2;

For each node j N2

Randomly search all nodes in N1;Associate to j a node i N1 satisfying the

constraints;Remove i from N1;

End; If a feasible solution was built, then return;

End;

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Local search

• Iterative improvement (first improvement)

• Neighborhood:– Consider every pair of nodes j1 N2, j2 N2

– Let i1 = A-1(j1) and i2 = A-1(j2)

– Exchange the associations: A(i1) A(i1) – {j1} {j2}A(i2) A(i2) – {j2} {j1}

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ExampleCut of a muscle:

(a) original 2-D image (b) model (c) over-segmented image

14 vertices26 edges

28 vertices62 edges

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Summary• Data:

– Image to be recognized– Atlas of images

• Examples:– Identification of tumors– Facial recognition

• Images represented by graphs:– Nodes: image regions– Edges: relations among regions

(boundaries)– Attributes: colors, intensities,

adjacencies, etc.

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• Similarities between pairs of vertices and pairs of edges (model-image)

• Problem: – Determine the most similar model in the

atlas with respect to the image to be recognized

– Determine the optimal correspondence between the regions of the model and those in the image

• Other applications

Summary

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Problema da filogenia• Uma filogenia é uma árvore que relaciona espécies

ou genes homólogos de espécies distintas (taxons)• Dado um conjunto de taxons caracterizados por um

conjunto de caracteres, obter uma filogenia com o menor número de passos evolucionários (mudança de um caracter)

• Exemplo:

a b c d e f g

O 0 0 0 0 0 0 0

A 1 1 0 0 1 1 1

B 1 1 1 1 1 1 1

C 1 1 1 1 0 0 0

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Problema da filogeniaa b c d e f g

O 0 0 0 0 0 0 0

A 1 1 0 0 1 1 1

B 1 1 1 1 1 1 1

C 1 1 1 1 0 0 0

Exemplo:

Solução com 10 passos:

Solução com 9 passos:

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Roteamento de circuitos virtuais privados

•Circuitos virtuais privados (PVCs) entre os pares origem e destino de cada cliente em um backbone

•Roteamento de cada cliente feito pelo projetista sem conhecimento de demandas futuras

• Ineficiência e necessidade ocasional de rotear os PVCs offline

•Problema: minimizar atrasos e congestão, satisfazendo as demandas e restrições de capacidade

•Aplicação desenvolvida para a AT&T: melhoria sensível do desempenho e do aproveitamento de redes reais

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Roteamento de PVCs

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Roteamento de PVCscapacidade máxima = 3

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Roteamento de PVCs

Caminho longo!

rerotear

capacidade máxima = 3

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Roteamento de PVCscapacidade máxima = 3

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Roteamento de PVCsViável e ótima! capacidade máxima = 3

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Projeto de redes locais de acesso

• Construir uma rede de fibras óticas para prover serviços de banda larga a consumidores comerciais e residenciais, levando em consideração o balanceamento entre os custos de construção (colocação das fibras) e os rendimentos potenciais dos clientes atendidos (perda de receita no caso do cliente não ser atendido).

• Problema de Steiner com prêmios: aplicação desenvolvida para a AT&T

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cliente(prêmio)

ruapenalidade nula

raiz

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Projeto de redes a k-caminhos

• Dados:– Grafo suporte para a construção de uma rede– Conjunto de pares origem-destino– Custo de construção de cada arco

• Problema: construir uma rede de custo mínimo, na qual todos os pares origem-destino são conectados por caminhos usando no máximo k arcos

• Para que a confiabilidade da rede seja alta, todos os caminhos devem ser formados por k ≤3 arcos

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Slides and publications

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

• 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

(with L. Buriol, M.G.C. Resende, and M. Thorup)• Paper about GRASP for graph matching also in

preparation (with I. Bloch and M.C. Boeres)

problemas de otimização em network design e biologia: ajuste de pesos em roteamento sob ospf

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