Ford-Fulkerson Algorithm in Python

The Ford-Fulkerson algorithm is a widely used algorithm to solve the maximum flow problem in a flow network. The maximum flow problem involves determining the maximum amount of flow that can be sent from a source vertex to a sink vertex in a directed weighted graph, subject to capacity constraints on the edges.

The algorithm works by iteratively finding an augmenting path, which is a path from the source to the sink in the residual graph, i.e., the graph obtained by subtracting the current flow from the capacity of each edge. The algorithm then increases the flow along this path by the maximum possible amount, which is the minimum capacity of the edges along the path.

Problem:

Given a graph which represents a flow network where every edge has a capacity. Also, given two vertices source ‘s’ and sink ‘t’ in the graph, find the maximum possible flow from s to t with the following constraints:

• Flow on an edge doesn’t exceed the given capacity of the edge.
• Incoming flow is equal to outgoing flow for every vertex except s and t.

For example, consider the following graph from the CLRS book. 

The maximum possible flow in the above graph is 23. 

Ford-Fulkerson Algorithm in Python

 The following is simple idea of Ford-Fulkerson algorithm:

1. Start with initial flow as 0.
2. While there exists an augmenting path from the source to the sink:  
   • Find an augmenting path using any path-finding algorithm, such as breadth-first search or depth-first search.
   • Determine the amount of flow that can be sent along the augmenting path, which is the minimum residual capacity along the edges of the path.
   • Increase the flow along the augmenting path by the determined amount.
3. Return the maximum flow.

Below is the implementation of Ford-Fulkerson algorithm. To keep things simple, graph is represented as a 2D matrix.

Python3

# Python program for implementation 
# of Ford Fulkerson algorithm
from collections import defaultdict

# This class represents a directed graph 
# using adjacency matrix representation
class Graph:

    def __init__(self, graph):
        self.graph = graph # residual graph
        self. ROW = len(graph)
        # self.COL = len(gr[0])

    '''Returns true if there is a path from source 's' to sink 't' in
    residual graph. Also fills parent[] to store the path '''

    def BFS(self, s, t, parent):

        # Mark all the vertices as not visited
        visited = [False]*(self.ROW)

        # Create a queue for BFS
        queue = []

        # Mark the source node as visited and enqueue it
        queue.append(s)
        visited[s] = True

        # Standard BFS Loop
        while queue:

            # Dequeue a vertex from queue and print it
            u = queue.pop(0)

            # Get all adjacent vertices of the dequeued vertex u
            # If a adjacent has not been visited, then mark it
            # visited and enqueue it
            for ind, val in enumerate(self.graph[u]):
                if visited[ind] == False and val > 0:
                    # If we find a connection to the sink node, 
                    # then there is no point in BFS anymore
                    # We just have to set its parent and can return true
                    queue.append(ind)
                    visited[ind] = True
                    parent[ind] = u
                    if ind == t:
                        return True

        # We didn't reach sink in BFS starting 
        # from source, so return false
        return False
            
    
    # Returns the maximum flow from s to t in the given graph
    def FordFulkerson(self, source, sink):

        # This array is filled by BFS and to store path
        parent = [-1]*(self.ROW)

        max_flow = 0 # There is no flow initially

        # Augment the flow while there is path from source to sink
        while self.BFS(source, sink, parent) :

            # Find minimum residual capacity of the edges along the
            # path filled by BFS. Or we can say find the maximum flow
            # through the path found.
            path_flow = float("Inf")
            s = sink
            while(s != source):
                path_flow = min (path_flow, self.graph[parent[s]][s])
                s = parent[s]

            # Add path flow to overall flow
            max_flow += path_flow

            # update residual capacities of the edges and reverse edges
            # along the path
            v = sink
            while(v != source):
                u = parent[v]
                self.graph[u][v] -= path_flow
                self.graph[v][u] += path_flow
                v = parent[v]

        return max_flow


# Create a graph given in the above diagram

graph = [[0, 16, 13, 0, 0, 0],
        [0, 0, 10, 12, 0, 0],
        [0, 4, 0, 0, 14, 0],
        [0, 0, 9, 0, 0, 20],
        [0, 0, 0, 7, 0, 4],
        [0, 0, 0, 0, 0, 0]]

g = Graph(graph)

source = 0; sink = 5

print ("The maximum possible flow is %d " % g.FordFulkerson(source, sink))

Output

The maximum possible flow is 23 

Time Complexity: Time complexity of the above algorithm is O(max_flow * E). We run a loop while there is an augmenting path. In worst case, we may add 1 unit flow in every iteration. Therefore the time complexity becomes O(max_flow * E).