Ford Fulkerson algorithm for Maximum Flow Problem Example

Introduction

This tutorial explains the Ford Fulkerson algorithm, a method used to solve the Maximum Flow Problem in network flow theory. Understanding this algorithm is essential for optimizing flows in networks, such as traffic systems, supply chains, and data networks. We will walk through the algorithm step-by-step using a practical example to illustrate its application.

Step 1: Understand the Maximum Flow Problem

• The Maximum Flow Problem aims to find the greatest flow from a source node to a sink node in a flow network.
• A flow network consists of nodes (vertices) connected by edges (arcs) with specified capacities.
• The goal is to maximize the flow while adhering to the capacity constraints on each edge.

Step 2: Define the Flow Network

• Identify the source (S) and sink (T) nodes in your network.
• List all nodes and edges along with their capacities. For example:
  • Nodes: A, B, C, D, E
  • Edges with capacities:
    • A to B: 10
    • A to C: 5
    • B to C: 15
    • B to D: 10
    • C to D: 10
    • D to T: 10
    • C to T: 5

Step 3: Initialize Flow Values

• Set the initial flow for all edges to zero. This means no flow is passing through any edges at the beginning.

Step 4: Find Augmenting Paths

• Use Depth First Search (DFS) or Breadth First Search (BFS) to find a path from the source (S) to the sink (T) where additional flow can be pushed through.
• An augmenting path is a path where the residual capacity (original capacity minus current flow) is greater than zero.

Step 5: Calculate Residual Capacity

• For each edge in the found augmenting path, determine the residual capacity:
  • Residual Capacity = Capacity - Current Flow
• Keep track of the minimum residual capacity along the augmenting path.

Step 6: Update Flows

• Increase the flow along the augmenting path by the minimum residual capacity found in the previous step.
• For each edge in the path:
  • Update the flow:
    • Flow (u, v) += Minimum Residual Capacity
    • Flow (v, u) -= Minimum Residual Capacity (for the reverse direction)

Step 7: Repeat Until No More Augmenting Paths

• Continue the process of finding augmenting paths and updating flows until no more augmenting paths exist from the source to the sink.
• Once no more paths can be found, the current flow is the maximum flow of the network.

Step 8: Analyze the Results

• Calculate the total flow from the source to the sink. This value represents the maximum flow possible in the network.
• Review the flow values on each edge to understand how the flow has been distributed throughout the network.

Conclusion

The Ford Fulkerson algorithm provides a systematic approach to solve the Maximum Flow Problem in flow networks. By understanding the algorithm's steps—from defining the network and initializing flows to finding augmenting paths and updating flows—you can apply this knowledge to various real-world scenarios. For further exploration, consider studying other algorithms for network flow, such as the Edmonds-Karp algorithm, which is an implementation of Ford Fulkerson using BFS.