How to use Edmonds-Karp algorithm for maximum network flow?

How to use Edmonds-Karp algorithm for maximum network flow?

The Edmonds-Karp algorithm is used to find the maximum flow in a network. It is an implementation of the Ford-Fulkerson algorithm that uses the Breadth-First Search (BFS) algorithm to find the augmenting path with the smallest number of edges.

To use the Edmonds-Karp algorithm, we first need to represent the network as a graph. Each edge in the graph has a capacity, which represents the maximum amount of flow that can pass through that edge. The source node is the node where the flow originates, and the sink node is the node where the flow ends.

The algorithm starts by setting the flow on each edge to zero. Then, it repeatedly finds an augmenting path from the source to the sink using BFS. An augmenting path is a path from the source to the sink that has unused capacity on each edge. The algorithm then increases the flow on each edge in the augmenting path by the minimum capacity of the edges in the path.

The algorithm continues to find augmenting paths and increase the flow until no more augmenting paths can be found. At this point, the maximum flow has been found.

To illustrate the algorithm, let's consider the following network:

```
10
(s)---(1)---(2)
/ | | | \
/ | | | \
20 5 10 5 20
\ | | | /
\ | | | /
(3)---(4)---(t)
10
```

The numbers on the edges represent the capacities. The source node is `s`, and the sink node is `t`.

We start by setting the flow on each edge to zero:

```
0/10
(s)---(1)---(2)
/ | | | \
/ | | | \
0/ 0/5 0/10 0/5 0/20
\ | | | /
\ | | | /
(3)---(4)---(t)
0/10
```

We then find an augmenting path using BFS:

```
0/10
(s)---(1)---(2