Ford-Fulkerson Algorithm is an algorithm used for finding the maximum flow in a network. It is a greedy algorithm that takes an iterative approach to find the maximum flow by finding the augmenting path in a residual graph until no more path can be found.
The algorithm follows these steps:
Initialize the flow f to 0.
While there exists an augmenting path in the residual graph:
a. Find an augmenting path in the residual graph.
b. Calculate its bottleneck capacity.
c. Update the flow f with the bottleneck capacity.
d. Update the residual graph.
Return the flow f.
Example:
Consider the following network with source node s and sink node t:
s ----10--→ A ----5--→ B ----12--→ t
/ \ | | |
8 5 4 9 2
/ \ ↓ ↓ ↓
C D —-7–→ E —-6–→ F
The numbers on edges represent their capacities.
To find the maximum flow from s to t using the Ford-Fulkerson algorithm, we first initialize the flow f to 0. Then, we find an augmenting path in the residual graph:
s → A → B → t, with a bottleneck capacity of 5.
We update the flow f to 5 and update the residual graph by subtracting 5 from the forward edges and adding 5 to the backward edges.
The residual graph becomes:
s ----5--→ A ----0--→ B ----7--→ t
/ \ ↓ ↓ |
8 5 4 9 2
/ \ ↓ ↓ ↓
C D —-7–→ E —-6–→ F
We continue finding augmenting paths and updating the flow and residual graph until no more augmenting path can be found.
The final flow is 13, and the residual graph is:
s ----0--→ A ----0--→ B ----10--→ t
/ \ ↓ ↓ |
8 2 4 4 2
/ \ ↓ ↓ ↓
C D —-0–→ E —-0–→ F
Therefore, the maximum flow from s to t is 13.
The Ford-Fulkerson Algorithm is an algorithm for finding the maximum flow in a flow network.
The algorithm works by starting with an initial feasible flow and then iteratively finding augmenting paths that increase the flow.
An augmenting path is a path from the source to the sink that has available capacity, which can be used to increase the flow.
The algorithm terminates when there are no more augmenting paths to be found, at which point the flow is at maximum.
The algorithm uses a residual graph to keep track of available capacity and flow along the edges.
The residual graph is created by subtracting the current flow from the capacity of each edge.
The algorithm can use either the Depth-First Search or Breadth-First Search algorithm to find augmenting paths.
The maximum flow can be found by summing the flows along all the edges leaving the source.
The time complexity of the algorithm is O(E*max flow), where E is the number of edges in the graph.
The Ford-Fulkerson Algorithm is often used as the basis for more advanced algorithms, such as the Dinic’s Algorithm and the Edmonds-Karp Algorithm.
What is the goal of the Ford-Fulkerson Algorithm?
Answer: The goal of the Ford-Fulkerson Algorithm is to find the maximum flow in a network.
What is the Ford-Fulkerson Algorithm based on?
Answer: The Ford-Fulkerson Algorithm is based on the concept of augmenting paths.
Does the Ford-Fulkerson Algorithm always find the maximum flow in a network?
Answer: The Ford-Fulkerson Algorithm may not always find the maximum flow in a network. It depends on the choice of the initial flow network and the order in which the augmenting paths are chosen.
How does the Ford-Fulkerson Algorithm handle negative weights in a network?
Answer: The Ford-Fulkerson Algorithm cannot handle negative weights in a network. In such a case, other algorithms such as the Bellman-Ford Algorithm need to be used to find the maximum flow.
Is the Ford-Fulkerson Algorithm guaranteed to terminate when applied to a finite network?
Answer: Yes, the Ford-Fulkerson Algorithm is guaranteed to terminate when applied to a finite network, as long as there are no negative weight cycles.