Ford-Fulkerson in 5 minutes

Introduction

This tutorial provides a step-by-step guide on implementing the Ford-Fulkerson algorithm for calculating the maximum flow in a flow network. Understanding this algorithm is crucial for solving various problems in network theory, optimization, and operations research.

Step 1: Understand the Basics of Flow Networks

Before diving into the algorithm, familiarize yourself with the core concepts:

• Flow Network: A directed graph where each edge has a capacity, and flow must satisfy two conditions: it cannot exceed capacity and must conserve flow at nodes.
• Source and Sink: Identify the source (starting point for flow) and sink (endpoint for flow) in your network.

Step 2: Set Up the Flow Network

Prepare your flow network for analysis:

1. Define Nodes and Edges: List all nodes and the directed edges with their respective capacities.
2. Create an Adjacency Matrix: Represent the network using a matrix where the entry at row i and column j indicates the capacity of the edge from node i to node j.

Step 3: Initialize Flow Values

Start with zero flow:

• Create a flow matrix where all values are initialized to zero. This will represent the current flow in the network.

Step 4: Implement the Ford-Fulkerson Algorithm

Follow these steps to execute the algorithm:

1. Find Augmenting Path: Use Depth-First Search (DFS) or Breadth-First Search (BFS) to locate a path from the source to the sink where additional flow can be pushed through.
2. Calculate Residual Capacity: For the found path, determine the minimum capacity available (the bottleneck capacity).
3. Update Flow:
   • Increase the flow along the path by the bottleneck capacity.
   • Update the residual graph by decreasing the capacity of the forward edges and increasing the capacity of the backward edges.

Step 5: Repeat Until No Augmenting Path Exists

Continue to search for augmenting paths and update flows until no further paths can be found:

• When no augmenting paths are found, the maximum flow has been reached.

Step 6: Retrieve Maximum Flow Value

Once the algorithm concludes:

• Sum the flow values exiting from the source or entering the sink to obtain the maximum flow value.

Practical Tips

• Ensure that you thoroughly test your implementation with different network configurations to validate its correctness.
• If working with larger networks, consider using optimized data structures to improve performance.

Common Pitfalls

• Failing to account for flow conservation at nodes can lead to incorrect results.
• Overlooking the need for a residual graph can complicate the implementation.

Conclusion

You have now learned how to implement the Ford-Fulkerson algorithm to find the maximum flow in a flow network. To further enhance your understanding, consider reviewing the provided code on GitHub and experimenting with additional flow network scenarios. For more in-depth study, refer to the linked resources that discuss network flow theory in detail.