This document presents an overview of the Ford-Fulkerson algorithm for computing maximum flow in a flow network. It defines the Ford-Fulkerson algorithm and provides pseudocode. The algorithm works by finding augmenting paths in the residual graph to incrementally increase the flow from the source to the sink. Examples are given to demonstrate computing the maximum flow and residual capacities. Applications of maximum flow problems include modeling traffic networks, fluid flows, electrical circuits, and computer networks.

1. Presented by:
Saruar Chowdhury
ID:182-115-013
CSE 45th
Presented to
Ranit Debnath Akash
Lecturer
Department of CSE
Course: CSE 231; Algorithm

2.  Definition
 Ford Fulkerson Algorithm
 Flow examples
 Residual Graph

3. A minimum cost maximum flow of a
network G = (V, E) is a maximum flow with
the smallest possible cost. This problem
combines maximum flow (getting as much
flow as possible from the source to the
sink) with shortest path (reaching from the
source to the sink with minimum cost).

4. The Ford-Fulkerson method or ford Fulkerson
algorithm is a greedy algorithm that
computes the maximum flow in a flow
network.

5. Ford - Fulkerson(G; s; t)
1. for each edge (u; v) 2 E(G)
2. f(u; v) = 0
3. while there exists a path p from s to t in the
residual network Gf
4. cf (p) = minfcf (u; v) : (u; v) is in pg
5. for each edge (u; v) in p
6. if (u; v) 2 E
7. f(u; v) = f(u; v) + cf (p)
8. else f(v; u) = f(v; u) 􀀀 cf (p)

6. Example:

7. Residual graph (Capacities) Graph (Flow/Capacity)

8. Residual graph (Capacities) Graph (Flow/Capacity)

9. Residual graph (Capacities) Graph (Flow/Capacity)

10. Residual graph (Capacities) Graph (Flow/Capacity)

11. Final network flow:
Now it has no augmenting paths. So, the maximum flow shown
in (d) is 23 is a maximum flow.

12.  1 iteration of FF takes O(E+V ) time
(breadth-first search plus book- keeping).
 Each iteration sends at least one unit of flow.
 Total time O(f*E) .
 This algorithm is only Psuedo-polynomial.

13. 1. Traffic in a road system.
2. Fluids in pipes.
3. Currents in an electrical circuit.
4. Packet traffic in computer network. Etc.

14. Residual Graph of a flow network is
a graph which indicates additional
possible flow. If there is a path from source
to sink in residual graph, then it is possible to
add flow. Every edge of a residual graph has a
value called residual capacity which is equal
to original capacity of the edge minus
current flow.

15. Thank you