【COMP282 LEC8 The Maximum Flow ProblemBipartite Matchings】

Digraph

1. Edges all directed

2. A fundamental issue with directed graphs is the notion of reachability, which deals with determining where we can get to in a directed graph.

 3. Given two vertices u and v of a digraph G, we say that u reaches v (or v is reachable from u) if G has a directed path from u to v.

4. Digraph is Strongly connected if we have that u reaches v, and v reaches u.

5. Acyclic if it has no directed cycles.

Weight digraph

1. A weighted graph is a graph that has a numerical label w(e) associated with each edge e, called the weight of e.

Network Flow - The basics 

Ford-Fulkerson algorithm

The Ford-Fulkerson algorithm depends on three important ideas :

1. residual networks

2. augmenting paths

3. cuts

These three ideas are essential for the important Max-Flow/Min-Cut theorem.

Residual Networks 

1. A residual network consists of edges that can admit more net flow.

2. Given the flflow network G, and a flflow f in that network, we defifine the residual network Gf  (So this depends upon the given flflow f.)

Augmenting Paths 

Update the flow

  

Cuts in Network

其实就是一次一次的列举 找出minimum capacity of the cuts

Max-Flow/Min-Cut Theorem

The maximum flow is a network is equal to capacity of a minimum cut in the network

Complexity of the Ford-Fulkerson algorithm

。。。。有点难

不是很懂

Bipartite graphs

Matching

A matching is a subset of the edges of a bipartite graph where each vertex appears in at most one edge (i.e. edges in the matching share no common endpoints). 一个顶点最多使用一次

1. determining the size of the largest matching in a bipartite graph

2. To use the algorithm, 我们需要先处理一下bipartite graph ：a : adding two new vertices, a source vertex s and a sink vertex t  b: Join all vertices in X to s and all vertices in Y to t. Direct all edges from s to X, from X to Y, and from Y to t. c : Give each edge a capacity of 1.