链接版第十四章网络.docVIP

CHAPTER 14 　 Glossary network：网络 flow：：：：：： value of a flow：流量 maximal flow：最大流 minimal cut：： matching：： maximal matching：： matching network： place：：：：：：：：：：：：：：’ problem：：：本章内容及教学要点： .1　教学内容： 14.2　教学内容： 14.3 Petri Nets 教学内容：定理证明及例题解答 14.1 自从克希霍夫运用图论的方法研究电网络以来，网络理论的研究和应用十分引人注目，特别是近几十年来尤其如此. 电路网络、运输网络、通讯网络和计算机网络等与工程和应用密切相关的研究领域受到了高度的重视，有重要的应用价值. 无自回路的有向赋权图就是网络. 最有代表性的网络就是所谓的运输网络，如公路系统、铁路系统、供水系统、供电系统等. 网络理论有代表性的应用就是所谓的网络流及最大化问题. 最大流问题：假设从P地到Q地有一个公路网，公路网中每段公路上每天可以通过的汽车的数量有上限，那么经过该公路网，每天最多可以从P开多少辆车到Q？应该怎样分流？ In this chapter, we will restrict directed graphs as that have no loops and at most one edge between distinct vertices. We call them as simple connected directed graphs(弱连通简单有向图). 定义14..1 A network(网络) is a weighted directed graph G(V,E,c) with a special vertex a, called source(源), another special vertex z, called the sink(汇), and a weight function c: E→N such that (1) indeg(a)=0. (2) outdeg(z)=0. For each edge e of G, the positive integer c(e) is called the capacity of e(边e的容量). 定义14..2 A flow(网络) in a network is a function f: E→N∪{0} such that (1) for all e∈E, 0≤f(e)≤c(e). (2) for all v∈V such that v≠a, z, =(the flow into a vertex is equal to the flow out of the vertex, the conservation of flow(流的守恒)). in(v) is the set of edges for which v is the terminal vertex and out(v) is the set of edges for which v is the initial vertex. 定义14..3 For a fixed flow f, let flow(a)= and flow(z)= . From the conservation of flow, we know that flow(a)=flow(z), that is, the flow out of a is equal to the flow into z. To represent a flow in a network, we assign each edge an ordered pair whose first member is the capacity and second member is the flow through the edge. 例14..1 Network and flows. 解 例14..1 定理.1.1 For any fixed flow f, flow(a)= ==flow(z). 定义14..4 Flow(a)(flow(z)) is called the value of a flow f(流量), denoted by val(f). 定义14..5 Let S be a subset of V and T=V-S. Then (S,T)={e∈E: the initial vertex of e is in