Min-Cut/Max-Flow Algorithm
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Min-Cut/Max-Flow Algorithm Prof. Dr. Paulo A. V. de Miranda [email protected] Instituto de Matem ´ atica e Estat´ ıstica (IME), Universidade de S ˜ ao Paulo (USP) P.A.V. Miranda – p. 1
Transcript of Min-Cut/Max-Flow Algorithm
Min-Cut/Max-Flow Algorithm
Prof. Dr. Paulo A. V. de Miranda
Instituto de Matematica e Estatıstica (IME),
Universidade de Sao Paulo (USP)
P.A.V. Miranda – p. 1
Introdução
Imagine um material fluindo através de um sistema apartir de uma fonte, onde o material é produzido, atéum destino, onde ele é consumido.
O fluxo do material em qualquer ponto do sistema édado pela taxa com que o material se move.
Redes de fluxo podem ser usadas para modelar:
líquidos fluindo ao longo de tubulações.
peças através de linhas de montagem.
corrente através de redes elétricas.
informação através de redes de comunicação.
P.A.V. Miranda – p. 2
Introdução
Problema de fluxo máximo:
Qual é a maior taxa de transmissão de material a partirda fonte até o destino sem violar as restrições decapacidade entre as várias partes da rede?
P.A.V. Miranda – p. 3
Aplicação em segmentação
Explora o Teorema do Fluxo Máximo e Corte Mínimo.
Nós fonte e destino são adicionados ao grafo daimagem e cada pixel deve ser conectado a essesnós terminais por arcos.
O peso dos arcos entre pixels deve ser maior dentroe fora do objeto do que na fronteira do objeto.
Os pesos de arco com a fonte devem ser maiores nointerior do objeto do que fora dele e o contrário emrelação ao destino.
P.A.V. Miranda – p. 4
Aplicação em segmentação
E =∑
∀(u,v)∈A| u∈S,v∈T
w(u, v)+∑
∀u∈I| u∈S
w(u, t)+∑
∀u∈I| u∈T
w(s, u)
P.A.V. Miranda – p. 5
Definição
Uma rede de fluxo G = (V,E) é um grafo direcionadono qual cada aresta (u, v) ∈ E possui uma capacidadenão negativa c(u, v) ≥ 0.
P.A.V. Miranda – p. 6
Definição
Uma rede de fluxo G = (V,E) é um grafo direcionadono qual cada aresta (u, v) ∈ E possui uma capacidadenão negativa c(u, v) ≥ 0.
O grafo possui dois vértices especiais: fonte (source) se destino (sink) t.
P.A.V. Miranda – p. 6
Definição
Uma rede de fluxo G = (V,E) é um grafo direcionadono qual cada aresta (u, v) ∈ E possui uma capacidadenão negativa c(u, v) ≥ 0.
O grafo possui dois vértices especiais: fonte (source) se destino (sink) t.
Exemplo:
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4
1
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4
s t
P.A.V. Miranda – p. 6
Fluxo no grafo
O fluxo em G é uma função f : V × V → R que satisfazas seguintes três propriedades:
Restrição de capacidade: Para todos u, v ∈ V ,exigimos que f(u, v) ≤ c(u, v).
Anti-simetria: Para todos u, v ∈ V , temos quef(u, v) = −f(v, u).
Conservação de fluxo: Para todo u ∈ V − {s, t},temos que ∑
v∈V
f(u, v) = 0
P.A.V. Miranda – p. 7
Fluxo total no grafo
O valor total de fluxo é definido como a soma do fluxoque sai da fonte s:
|f | =∑
v∈V
f(s, v)
P.A.V. Miranda – p. 8
Fluxo total no grafo
O valor total de fluxo é definido como a soma do fluxoque sai da fonte s:
|f | =∑
v∈V
f(s, v)
No problema de fluxo máximo, nos é dado uma rede defluxo G com fonte s e destino t, e queremos encontrarum fluxo de valor máximo indo de s para t.
P.A.V. Miranda – p. 8
Visão geral da solução
Método de Ford-Fulkerson:
Começamos com fluxo inicial zero (i.e., f(u, v) = 0 paratodos u, v ∈ V ).
P.A.V. Miranda – p. 9
Visão geral da solução
Método de Ford-Fulkerson:
Começamos com fluxo inicial zero (i.e., f(u, v) = 0 paratodos u, v ∈ V ).
A cada iteração, aumentamos o fluxo total encontrandoalgum caminho (a partir da fonte s até o destino t) aolongo do qual podemos empurrar mais fluxo(augmenting paths).
P.A.V. Miranda – p. 9
Visão geral da solução
Método de Ford-Fulkerson:
Começamos com fluxo inicial zero (i.e., f(u, v) = 0 paratodos u, v ∈ V ).
A cada iteração, aumentamos o fluxo total encontrandoalgum caminho (a partir da fonte s até o destino t) aolongo do qual podemos empurrar mais fluxo(augmenting paths).
Repetimos este processo até que nenhum caminho deaumento pode ser encontrado.
P.A.V. Miranda – p. 9
Redes Residuais
Dados uma rede de fluxo G e um fluxo f , a rederesidual consiste de arestas que podem admitir maisfluxo adicional.
Para todos u, v ∈ V , a quantidade de fluxo adicional quepodemos empurrar de u para v antes de exceder acapacidade c(u, v) é a capacidade residual de (u, v),dada por:
cf (u, v) = c(u, v)− f(u, v)
P.A.V. Miranda – p. 10
Exemplo: augmenting paths
capacidade residual fluxo
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s t
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P.A.V. Miranda – p. 11
Alg. básico de Min-Cut/Max-Flow
Algorithm 1 – FORD-FULKERSON ALGORITHM
INPUT: A flow network G = (V,E) with nodes s and t.
OUTPUT: The maximum flow f in G.
AUXILIARY: The residual network Gf .
1. For each edge (u, v) ∈ E[G], do
2. f [u, v]← 0 and f [v, u]← 0.
3. While there exists a path π from s to t in Gf do
4. cf (π)← min{cf (u, v) : (u, v) is in π}
5. For each edge (u, v) in π, do
6. f [u, v]← f [u, v] + cf (π) and f [v, u]← −f [u, v].
7. cf [u, v] = c[u, v]− f [u, v] and cf [v, u] = c[v, u]− f [v, u].
P.A.V. Miranda – p. 12
Análise de Complexidade
O tempo de execução depende de como os caminhos sãodeterminados.
S t
u
v
1.000.000
1.000.0001.000.000
1.000.000
1
P.A.V. Miranda – p. 13
Análise de Complexidade
O tempo de execução depende de como os caminhos sãodeterminados.
S t
u
v
1.000.000
1.000.0001.000.000
1.000.000
1
P.A.V. Miranda – p. 13
Análise de Complexidade
O tempo de execução depende de como os caminhos sãodeterminados.
S t
u
v
1.000.000
1.000.000999.999
11
999.999
1
após 1 iteração.
P.A.V. Miranda – p. 13
Análise de Complexidade
O tempo de execução depende de como os caminhos sãodeterminados.
S t
u
v
1.000.000
1.000.000999.999
11
999.999
1
P.A.V. Miranda – p. 13
Análise de Complexidade
O tempo de execução depende de como os caminhos sãodeterminados.
S t
u
v
999.999
999.999
11
999.999
11
999.999
1
após 2 iterações.
Se as capacidades forem valores inteiros então o algoritmoexecuta em O(|E|.|f∗|), onde |f∗| é o valor do fluxo máximo.
P.A.V. Miranda – p. 13
Problemas do Min-Cut/Max-Flow
No contexto de segmentação de imagens, existem duaspreocupações na literatura sobre o uso do algoritmo de GCoriginal (min-cut/max-flow):
metrication error (“blockiness”), e
viés de encolhimento (“shrinking bias”).
P.A.V. Miranda – p. 14
Problemas do Min-Cut/Max-Flow
O problema métrico (“blockiness”) surge quandocalculamos o fluxo máximo em um grafo de imagem comvizinhança-4. As figuras abaixo mostram o problema.
(a) (b) (c)
Claramente, o GC em vizinhança-4 está produzindo umcorte irregular (Figura b), em vez da borda suaveesperada. Um melhor resultado pode ser obtido usandoGC com vizinhança-8 (Figura c).
P.A.V. Miranda – p. 15
Problemas do Min-Cut/Max-Flow
Isso acontece porque, em um grafo de vizinhança-4,uma borda em diagonal corta o mesmo número dearcos que uma borda em formato de “L” (formando umcanto de 90 graus).
Isto contradiz a nossa expectativa, visto que a diagonalé o caminho mais curto entre os pontos considerados,o que levaria (intuitivamente) para um corte de menorvalor.
P.A.V. Miranda – p. 16
Problemas do Min-Cut/Max-Flow
A fim de compreender melhor esse fenômeno, a figuraabaixo ilustra o mesmo problema numericamente.
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(a) (b)
(a) Um grafo de vizinhança-4 onde os números indicam ascapacidades das arestas, com sementes de fundo empreto, e sementes de objeto em branco. P.A.V. Miranda – p. 17
Problemas do Min-Cut/Max-Flow
A fim de compreender melhor esse fenômeno, a figuraabaixo ilustra o mesmo problema numericamente.
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(a) (b)
(b) A segmentação por GC apresenta um efeito de“blockiness”, formando um canto (corte irregular) ao invésda fronteira suave esperada. P.A.V. Miranda – p. 18
Problemas do Min-Cut/Max-Flow
Outros exemplos:
(a) (b) (c) (d)
(a) Imagem original, (b) imagem das capacidades dasarestas, (c) GC com vizinhança-4, (d) GC comvizinhança-8.
P.A.V. Miranda – p. 19
Problemas do Min-Cut/Max-Flow
Outros exemplos:
(a) (b) (c) (d)
(a) Imagem original, (b) imagem das capacidades dasarestas, (c) GC com vizinhança-4, (d) GC comvizinhança-8.
P.A.V. Miranda – p. 20
