Maximum flow of network flow

I. Residual networks and Augmented roads

Residual networks, augmented paths and cuts are the three basic concepts that make up the maximum flow minimum cut theorem, which skillfully uses the smallest cut in the network to describe the maximum flow value.

1. Residual network

For network g= (V,E,C), set stream F is the stream in G. The residual network is intuitively composed of edges that can also hold more streams. For each edge in G <u,v>, you can define the residual capacity as additional network traffic that can pass without exceeding the capacity limit

CF (U,V) =c (U,v)-F (u,v)

In fact, the residual network is also a network, the capacity is given by CF. Set F is a stream of network g= (V,E,C), F1 is a stream of residual network GF, then F+F1 is still a stream of network G. This lemma gives the relationship between the residual network and the original network, and provides a precondition for the proposed augmented Road 1.

2. Augmented Road

　　definition: the augmented path p is a simple path to the source s to sink T on the residual network GF. The residual capacity of this path is the maximum additional traffic that can be added along the path: CF (p) =min{cf (u,v) | <u,v>∈p}. This value is strictly greater than 0.

From the above description can be pushed to export the augmented nature of the path, the definition of flow FP to

Given a stream f in Network G, F+FP remains a stream of network G.

The maximum flow theorem of the minimum cut

1. Cut

The network g= (v,e,c) cut [s,t] divides the point set V into S, T (s=v-t) two parts, making s∈s and T∈t. The symbol [S,t] represents the Edge collection {<u,v>| <u,v>∈e, U∈s, v∈t}. The net flow through the cut [s,t] is defined as f (s,t), and the capacity of the cut [s,t] is defined as C (s,t). The smallest cut of a network is the smallest cut in the network.

2. Minimum cut maximum flow theorem

The maximum flow theorem for the minimum cut can be fully described as follows:

If f is a stream in network G={v,e,c} with source S and Sink T, the following conditions are equivalent:

(1) F is a maximum flow of g;

(2) The residual network GF does not include an augmented path;

(3) for a certain cut of G [s,t], there is |f|=c (s,t).

Three, maximum flow algorithm

The basic idea of 1.ford-fulkson method

The Ford-fulkson (FF) method was invented by Ford and Fulkson, a two-bit mathematician. Making full use of the minimum cutting maximum flow theorem and creatively inventing the fallback edge, the augmentation becomes a dynamic modification process, and the correctness of the final result is ensured.

The specific steps of the FF method:

(1) Initialize the capacity of all sides in the network,c<u,v> inherit the capacity of the edge,c<v,u> initialized to 0, where Edge <v,u> is the fallback edge. Initializes a maximum of 0 streams.

(2) Find an augmented path p from source s to sink T in the residue network. If found, go to step (3); If not found, go to step (5).

(3) Find the so-called "bottleneck" edge in the Add-on road p, which is the smallest edge in the path, record this value x, and add it to the maximum flow, to step (4).

(4) Subtract x from all c<u,v> in the augmented path and all c<v,u> plus x to form a new residue network. Step (2).

(5) Get the maximum stream of the network and exit.

In the introduction to algorithms, the FF method is strictly defined as a method rather than an algorithm, perhaps because step (2) of the FF method does not give a specific way to find the augmented path. The uncertainty leading to the search for an augmented path method

Maximum flow of network flow