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单词 Flow
释义

flow


On a digraphMathworldPlanetmath, define a sink to be a vertex without-degree zero and a source to be a vertex within-degree zero.Let G be a digraph with non-negative weights and with exactlyone sink and exactly one source.A flow on G is an assignmentf:E(G)of values to each edge of G satisfying certain rules:

  1. 1.

    For any edge e, we must have0f(e)W(e) (where W(e) is the weight of e).

  2. 2.

    For any vertex v, excluding the source and the sink, let Ein bethe set of edges incidentPlanetmathPlanetmathPlanetmath to v and let Eout be the set of edgesincident from v.Then we must have

    eEinf(e)=eEoutf(e).

Let Esource be the edges incident from the source, and let Esink bethe set of edges incident to the sink.If f is a flow, then

eEsinkf(e)=eEsourcef(e).

We will refer to this quantity as the amount of flow.

Note that a flow given by f(e)=0 trivially satisfies these conditions.We are typically more interested in maximum flows, where the amountof flow is maximized for a particular graph.

We may interpret a flow as a means of transmitting something througha network.Suppose we think of the edges in a graph as pipes, with the weightscorresponding with the capacities of the pipes; we are pouring waterinto the system through the source and draining it through the sink.Then the first rule requires that we do not pump more water througha pipe than is possible, and the second rule requires that any waterentering a junction of pipes must leave.Under this interpretationMathworldPlanetmathPlanetmath, the maximum amount of flow corresponds tothe maximum amount of water we could pump through this network.

Instead of water in pipes, one may think of electric charge in a network of conductors.Rule (2) above is one of Kirchoff’s two laws for such networks; the othersays that the sum of the voltage drops around any circuitMathworldPlanetmath is zero.

Titleflow
Canonical nameFlow
Date of creation2013-03-22 13:00:50
Last modified on2013-03-22 13:00:50
Ownerbgins (4516)
Last modified bybgins (4516)
Numerical id6
Authorbgins (4516)
Entry typeDefinition
Classificationmsc 05C20
Classificationmsc 94C15
Synonymnetwork flow
Related topicMaximumFlowMinimumCutTheorem
Related topicMaximumFlowminimumCutTheorem
Definesmaximum flow
Definessource
Definessink
DefinesKirchoff’s law
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