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# How to perform Edmonds-Karp on a graph

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Get ready to dive into the exciting world of network flow problems! If you're looking for a powerful algorithm that can help you calculate the maximum flow between two points in a graph, look no further than Edmonds-Karp.

This clever variation of the Ford-Fulkerson algorithm has proven to be incredibly useful for solving scheduling and routing problems, as well as finding the shortest path between two vertices.

In this article, we'll take a deep dive into Edmonds-Karp, exploring its inner workings and sharing some tips to help you get the most out of this powerful algorithm.

## What is Edmonds-Karp?

Edmonds-Karp algorithm is a variation of the Ford-Fulkerson algorithm that can help you find the maximum flow between two points in a graph. This powerful algorithm is perfect for optimizing routes, scheduling and routing problems, and even finding the shortest path between two vertices.

How does it work? Edmonds-Karp algorithm repeatedly searches for paths that have not yet been explored and adds their flow to the total, allowing you to find the most efficient flow between two points.

But that's not all! Edmonds-Karp algorithm is not only efficient but also incredibly fast, thanks to its polynomial time complexity. This means that no matter how large your graph is, the algorithm will always find the optimal solution in a reasonable amount of time. Plus, it's relatively easy to implement, making it a popular choice for solving network flow problems.

## Applying the Edmonds-Karp Algorithm to a Graph

The Edmonds-Karp algorithm is a fantastic way to solve one of the most exciting problems in graph theory: finding the maximum flow in a flow network. Whether you're a computer science student, a math enthusiast, or a problem-solving lover, the Edmonds-Karp algorithm is an excellent tool to have in your arsenal.

Here are the steps to apply the Edmonds-Karp algorithm to a graph:

The time complexity of the Edmonds-Karp algorithm is O(E^2 * V), where E is the number of edges and V is the number of nodes in the flow network. However, with the use of advanced data structures such as a priority queue, the time complexity can be reduced to O(E * V^2 * log(V)).

## Example of performing Edmonds-Karp on a graph

Let's work through an example of performing the Edmonds-Karp algorithm on a graph to find the maximum flow.

Consider the following graph:

```				```
10
/     \
s /       \ t
/         \
A --4--> B
\         /
c \       / d
\     /
15

```
```

Here, s is the source node, t is the sink node, and the edges are labeled with their capacities.

1. Initialization: Set the flow f to 0, and find an augmenting path using BFS.

Starting from node s, the algorithm looks for an edge with available capacity, and finds the edge (s, A) with a capacity of 10. This becomes the first edge in the augmenting path.

```				```
f = 0
Path = []
Queue = [s]

```
```

2. BFS: Perform BFS to find the augmenting path.

The algorithm performs a BFS search to find the shortest augmenting path from s to t. Since the edge (s, A) has available capacity, it is added to the augmenting path.

```				```
f = 0
Path = [(s, A)]
Queue = [A]

```
```

From A, the algorithm looks for an adjacent node with available capacity, and finds the edge (A, B) with a capacity of 4. This is added to the augmenting path.

```				```
f = 0
Path = [(s, A), (A, B)]
Queue = [B]

```
```

From B, the algorithm looks for an adjacent node with available capacity, and finds the edge (B, t) with a capacity of 10. This is added to the augmenting path.

```				```
f = 0
Path = [(s, A), (A, B), (B, t)]
Queue = []

```
```

The augmenting path from s to t has been found.

3. Updating flow: Update the flow along the augmenting path.

The algorithm updates the flow along the augmenting path from s to t by adding the minimum capacity of the edges in the path, which is 4.

```				```
f = 4
Path = [(s, A), (A, B), (B, t)]
Queue = []

```
```

The residual graph is updated by subtracting the flow from the capacities of the forward edges, and adding the flow to the capacities of the backward edges.

```				```
6
/     \
s /       \ t
/         \
A --0--> B
\         /
c \       / d
\     /
15

```
```

4. Repeat: Repeat the above steps until there are no more augmenting paths.

The algorithm repeats the above steps until there are no more augmenting paths.

From s, the algorithm looks for an edge with available capacity, and finds the edge (s, A) with a capacity of 6. This becomes the first edge in the augmenting path.

```				```
f = 4
Path = []
Queue = [s]

```
```

Performing BFS, the algorithm finds the augmenting path (s, A, B, t). The minimum capacity along the path is 6, so the flow is updated accordingly.

```				```
f = 10
Path = [(s, A), (A, B), (B, t)]
Queue = []

```
```

The residual graph is updated again.

```				```
0
/     \
s /       \ t
/         \
A --2--> B
\         /
c \       / d
\     /
15

```
```

Thereare no more augmenting paths from s to t, so the algorithm terminates.

5. Final result: The maximum flow is the sum of the flow along all the augmenting paths.

In this case, the maximum flow is 10, which is the flow along the two augmenting paths found by the algorithm.

The final flow network looks like this:

```				```
4
/   \
s   /     \  t
/       \
A -4-> B -6->
|       |
\     /
15   10

```
```

Here, the edges are labeled with their current flow and capacity values. The dashed lines represent the edges in the original graph that have been saturated.

And that's how you can perform the Edmonds-Karp algorithm on a graph to find the maximum flow!

## Benefits of Using the Edmonds-Karp Algorithm

The Edmonds-karp algorithm offers several benefits that make it an excellent choice for solving maximum flow problems in a variety of applications. Here are some of the benefits of using the Edmonds-Karp algorithm:

## Common Pitfalls When Implementing Edmonds-Karp

While the Edmonds-Karp algorithm is a powerful tool for solving maximum flow problems, there are several common pitfalls that can arise when implementing it. Here are some of the most common pitfalls to watch out for:

## Tips for Optimizing Performance with Edmonds-Karp

When using Edmonds-Karp, there are several tips that can be used to optimize performance.

## Conclusion:

Edmonds-Karp is a powerful algorithm for finding the maximum flow between two vertices in a graph. It can be applied quickly and efficiently, and can handle large graphs with a small amount of memory.

By understanding how the algorithm works and following best practices for implementation, developers can optimize its performance and ensure that issues are quickly addressed.

The Edmonds-Karp algorithm is a great tool for developers who need to find the maximum flow between two vertices in a graph. It is easy to implement and can be used to solve a variety of problems.

Additionally, it is a reliable algorithm that can be used to quickly and accurately find the maximum flow between two vertices in a graph.

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