Maximum Flow-min cut maxflow-mincut ISAP

There is no need for a simple narrative.

For a graph, we want to find the maximum flow, for the algorithm based on the augmented path, we must first establish a reverse edge.

Correctness of the opposite side:

I tried to find a lot of information and found no theoretical proof of the correctness of the opposite side.

But in fact, it is not difficult to understand that for each of the opposite sides, we flow through it equivalent to withdrawing a positive edge of the flow.

And it is necessary:

　　　　

And theoretically, we get the maximum flow after we join the opposite side, we consider from the residual network.

We have to realize that the reverse side does not reduce the maximum flow flow, which is obvious. Have Flow<=flow '.

Next we consider whether the flow of all points can only be obtained with positive edges.

And we examine the meeting point, because the meeting point does not have the edge, all the opposite side is from it to leave, then obviously does not have the influence to the meeting point.

For other points, we consider that the capacity of a reverse edge depends on the flow of the forward edge, and there is always a flow_positive=capacity_negative,

Or in other words, flow_positive=-flow_negative.

That is, if the flow from this point is going out through the opposite side of the u->v, this flow flows to K, then there must be a flow of the same flow from the V->u,

So it's better to go straight from v->k. (If the u->v->k contains other reverse edges, it is discussed recursively, but the problem size is reduced.) ）

So that the proof of the stream of consciousness is done.

Maximum flow minimum cut theorem:

There is a universal method of proof, the flow of the first arbitrary feasible flow can not be greater than any one cut. That is, there is a minimum cut >= any feasible flow.

And due to the maximum flow algorithm, the maximum flow of residual network source points, sinks must not be connected, otherwise the maximum flow can be greater (along the road runoff in the Unicom past can be greater).

This time the maximum flow algorithm gets one of the cuts.

That is, the maximum flow >= minimum cut.

So there is the maximum flow = minimum cut.

To find the maximum flow, we have a very intuitive idea, constantly in the residual network to find the path of s-t, if found on the flow of the past, the flow of natural +1, these will reduce traffic.

This is the simple Fold-fulkerson algorithm, which is correct.

Fold-fulkerson FF algorithm correctness proves that:

First the FF algorithm can get a cut, above we prove that the capacity of the cut is always greater than any flow, then the FF gets the flow is clearly the maximum flow.

On the FF algorithm, according to the characteristics of the shortest path, it can be proved that the feasible road runoff can be achieved by the algorithm complexity of O (v^2e) each time according to the minimum route of the residual network.

--ek algorithm

The EK algorithm can be said to be the first SAP (shortest Augment path shortest augmented path) algorithm, after which the dinic is just an extension.

In fact, after studying, it can be found that dinic and ISAP are actually very similar.

Theoretically, the ISAP is added more optimized.

After using pre-labeled technology, ISAP should not be inferior to dinic. But the fact that the Poi Kos and NOI elevation all made Isap kneel down (heard?) ）

But I think it's probably a twist.

It is important to note that the pre-marking is required on a hierarchical chart.

For example bzoj1001 Wolf catch Rabbit (i isap ran 1200ms), if not add pre-marking, must be run.

But after it was added, it was 2/3 faster than dinic, and faster than some of the algorithms that turned the shortest path.

Reference:

1. https://tadvent.wordpress.com/2009/04/07/usaco-4-2-1-ditch-%E7%BD%91%E7%BB%9C%E6%9C%80%E5%A4%A7%E6%B5%81%E9%97 %ae%e9%a2%98%e7%ae%97%e6%b3%95%e5%b0%8f%e7%bb%93/
2. Http://www.cs.yale.edu/homes/aspnes/pinewiki/MaxFlow.html
3. Https://en.wikipedia.org/wiki/Maximum_flow_problem
4. http://blog.csdn.net/qq_21110267/article/details/43540483

Maximum Flow-min cut maxflow-mincut ISAP