1. Maximum Flow Neil Tang 4/8/2008
CS223 Advanced Data Structures and Algorithms

2. CS223 Advanced Data Structures and Algorithms
Class Overview
The maximum flow problem
Applications
A greedy algorithm which does not work
The Ford-Fulkerson algorithm
Implementation and time complexity
CS223 Advanced Data Structures and Algorithms

3. The Maximum Flow Problem
The weight of a link (a.k.a link capacity) indicates the maximum amount of flow allowed to pass through this link.
The maximum flow problem: Given a weighted directed graph G, a source node s and sink t, find the maximum amount of flow that can pass from s to t and a corresponding feasible link flow allocation.
Flow feasibility: Both the flow conservation constraint and the capacity constraint must be satisfied.
CS223 Advanced Data Structures and Algorithms

4. The Maximum Flow Problem
CS223 Advanced Data Structures and Algorithms

5. CS223 Advanced Data Structures and Algorithms
Applications
Computer networks: Data traffic routing for throughput maximization.
Transportation networks: Road construction and traffic management.
Graph theory: Matching, assignment problems.
CS223 Advanced Data Structures and Algorithms

6. Flow Graph and Residual Graph
CS223 Advanced Data Structures and Algorithms

7. CS223 Advanced Data Structures and Algorithms
Basic Idea
Keep finding s-t augmenting paths until no such paths can be found in the residual graph.
Update the flow and residual graph according to the augmenting path in each step.
CS223 Advanced Data Structures and Algorithms

8. A Greedy Algorithm which Does Not Work
Find an augmenting path s-a-d-t with flow value 3 and update the flow and residual graphs as follows:
CS223 Advanced Data Structures and Algorithms

9. The Ford-Fulkerson Algorithm
Find an augmenting path s-a-d-t with flow value 3 and update the flow and residual graphs as follows:
CS223 Advanced Data Structures and Algorithms

10. The Ford-Fulkerson Algorithm
Find an augmenting path s-b-d-a-c-t with flow value 2 and update the flow and residual graphs as follows:
CS223 Advanced Data Structures and Algorithms

11. CS223 Advanced Data Structures and Algorithms
A Bad Example
CS223 Advanced Data Structures and Algorithms

12. The Implementation and Time Complexity
If all the link capacities are integers, then the time complexity of the Ford-Fulkerson algorithm is bounded by O(f|E|), where f is the max flow.
In each step, find an augmenting path which allows largest the increase in flow using a modified Dijkstra’s algorithm. It has been proved that it terminates after O(|E|logCapmax) steps, so the time complexity is O(|E|2log|V|logCapmax).
The Edmonds-Karp algorithm: In each step, find an augmenting path with minimum number of edges using BFS. It has been proved that it terminates after O(|E||V|) steps. So the time complexity is O(|E|2|V|).
CS223 Advanced Data Structures and Algorithms