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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Ford–Fulkerson algorithm ： ウィキペディア英語版 Ford–Fulkerson algorithm The Ford–Fulkerson method or Ford–Fulkerson algorithm (FFA) is an algorithm that computes the maximum flow in a flow network. It is called a "method" instead of an "algorithm" as the approach to finding augmenting paths in a residual graph is not fully specified or it is specified in several implementations with different running times. It was published in 1956 by L. R. Ford, Jr. and D. R. Fulkerson. The name "Ford–Fulkerson" is often also used for the Edmonds–Karp algorithm, which is a specialization of Ford–Fulkerson. The idea behind the algorithm is as follows: as long as there is a path from the source (start node) to the sink (end node), with available capacity on all edges in the path, we send flow along one of the paths. Then we find another path, and so on. A path with available capacity is called an augmenting path. ==Algorithm== Let $G(V,E)$ be a graph, and for each edge from $u$ to $v$, let $c(u,v)$ be the capacity and $f(u,v)$ be the flow. We want to find the maximum flow from the source $s$ to the sink $t$. After every step in the algorithm the following is maintained: : f(u,w) = 0 || That is, unless $u$ is $s$ or $t$. The net flow to a node is zero, except for the source, which "produces" flow, and the sink, which "consumes" flow. |- | Value(f): || $\backslash sum\_\; f(s,\; u)\; =\; \backslash sum\_\; f(v,\; t)$ || That is, the flow leaving from $s$ must be equal to the flow arriving at $t$. |- |} This means that the flow through the network is a ''legal flow'' after each round in the algorithm. We define the residual network $G\_f(V,E\_f)$ to be the network with capacity $c\_f(u,v)\; =\; c(u,v)\; -\; f(u,v)$ and no flow. Notice that it can happen that a flow from $v$ to $u$ is allowed in the residual network, though disallowed in the original network: if $f(u,v)>0$ and $c(v,u)=0$ then $c\_f(v,u)=c(v,u)-f(v,u)=f(u,v)>0$. Algorithm Ford–Fulkerson :Inputs Given a Network $G\; =\; (V,E)$ with flow capacity $c$, a source node $s$, and a sink node $t$ :Output Compute a flow $f$ from $s$ to $t$ of maximum value :# $f(u,v)\; \backslash leftarrow\; 0$ for all edges $(u,v)$ :# While there is a path $p$ from $s$ to $t$ in $G\_f$, such that $c\_f(u,v)\; >\; 0$ for all edges $(u,v)\; \backslash in\; p$: :## Find $c\_f(p)\; =\; \backslash min\backslash$ :## For each edge $(u,v)\; \backslash in\; p$ :### $f(u,v)\; \backslash leftarrow\; f(u,v)\; +\; c\_f(p)$ (''Send flow along the path'') :### $f(v,u)\; \backslash leftarrow\; f(v,u)\; -\; c\_f(p)$ (''The flow might be "returned" later'') The path in step 2 can be found with for example a breadth-first search or a depth-first search in $G\_f(V,E\_f)$. If you use the former, the algorithm is called Edmonds–Karp. When no more paths in step 2 can be found, $s$ will not be able to reach $t$ in the residual network. If $S$ is the set of nodes reachable by $s$ in the residual network, then the total capacity in the original network of edges from $S$ to the remainder of $V$ is on the one hand equal to the total flow we found from $s$ to $t$, and on the other hand serves as an upper bound for all such flows. This proves that the flow we found is maximal. See also Max-flow Min-cut theorem. If the graph $G(V,E)$ has multiple sources and sinks, we act as follows: Suppose that $T=\backslash$ and $S=\backslash$. Add a new source $s^$ * with an edge $(s^$ *,s) from $s^$ * to every node $s\backslash in\; S$, with capacity $c(s^$ *,s)=d_s\;(d_s=\sum_c(s,u)). And add a new sink $t^$ * with an edge $(t,\; t^$ *) from every node $t\backslash in\; T$ to $t^$ * , with capacity $c(t,\; t^$ *)=d_t\;(d_t=\sum_c(v,t)). Then apply the Ford–Fulkerson algorithm. Also, if a node $u$ has capacity constraint $d\_u$, we replace this node with two nodes $u\_,u\_$, and an edge $(u\_,u\_)$, with capacity $c(u\_,u\_)=d\_u$. Then apply the Ford–Fulkerson algorithm. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ford–Fulkerson algorithm」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.