This document discusses algorithms for solving the all-pairs shortest path problem. It presents the Floyd-Warshall algorithm, which uses dynamic programming and matrix multiplication to solve this problem. The algorithm computes the shortest paths between all pairs of vertices in a graph. It also describes Johnson's algorithm, which is more efficient for sparse graphs and uses Dijkstra's and Bellman-Ford algorithms as subroutines. It works by reweighting the graph to eliminate negative edges before computing single-source shortest paths.

1. All-Pairs Shortest Path
Theory and Algorithms
Carlos Andres Theran Suarez
Program Mathematics and Scientific Computing
University of Puerto Rico
Carlos.theran@upr.edu
October – 2011
Mayaguez-Puerto Rico
Dr Marko Schütz

2. Introduction

3. Recall

4. What do you think?
Can we solve all-pair shortest paths by running a
single source-paths algorithms?

5. What do you think?
Can we solve all-pair shortest paths by running a
single source-paths algorithms?

6. What do you think?
Can we solve all-pair shortest paths by running a
single source-paths algorithms?

7. What do you think?
Can we solve all-pair shortest paths by running a
single source-paths algorithms?

8. What do you think?
Can we solve all-pair shortest paths by running a
single source-paths algorithms?

9. What do you think?
Can we solve all-pair shortest paths by running a
single source-paths algorithms?

10. Predecessor Matrix

11. Predecessor Matrix

12. Outline
1. Present a dynamic programming algorithms based on
matrix multiplication to solve the problem.
2. Dynamic programming algorithms called Floyd-Warshall
algorithms.
3. Unlike the others algorithms, Johnson's algorithms used
adjacency-list representation of a graph.

13. Shortest path and matrix multiplication

14. Shortest path and matrix multiplication (cont.)

15. Shortest path and matrix multiplication (cont.)

16. Shortest path and matrix multiplication (cont.)

17. Shortest path and matrix multiplication (cont.)

18. Shortest path and matrix multiplication (cont.)

19. Shortest path and matrix multiplication (cont.)

20. Shortest path and matrix multiplication (cont.)

21. The Floyd-Warshall algorithm

22. The Floyd-Warshall algorithm (cont)

23. The Floyd-Warshall algorithm (cont)

24. The Floyd-Warshall algorithm (cont)

25. The Floyd-Warshall algorithm (cont)

26. The Floyd-Warshall algorithm (cont)

27. Johnson's algorithm for sparse graphs.
• It is asymtoticaly better than repeated squaring of matrices
or the Floyd-Warshall algoritm.
• It use a subroutine both Dijkstra’s algorithm and Bellman-
Ford algorithm.
• Johnson's algorithm use the technique of reweighting.

28. Johnson's algorithm for sparse graphs (cont.).

29. Johnson's algorithm for sparse graphs (cont.).

30. Johnson's algorithm for sparse graphs (cont.).
• Producing no negative weight by reweighting

31. Johnson's algorithm for sparse graphs (cont.).

32. Johnson's algorithm for sparse graphs (cont.).