1. All-Pairs Shortest Paths HONG-MING CHU 2013/10/31

2. Refernece Lecture slides from Prof. Hsin-Mu Tsai’s course slides and Prof. Ya- Yuin Su course slides.

3. Today’s Goal Quick recap of single source shortest path Floyd-Warshall algorithm Johson algorithm

4. Things we have learned so far Single-source shortest paths problem Two algorithms Bellman Ford algorithm Dijakstra’s algorithm Several properties about shorthest path

5. Optimal Substructure V0V0 ViVi VjVj VkVk

6. Triangle inequality

7. Algortihms we have learned Graph typeAlgorithmRunnging Time Unweighted graphBFSO(V+E) Non-negative edge weight graph DijkstraO(E+VlgV) General graphBellman-FordO(VE) DAGBellman-FordO(V+E)

8. Algortihms we have learned

9. Algorithms we have learned Graph typeAlgorithmRunnging Time Unweighted graph BFSO(V+E)O(V) Non-negative edge weight graph DijsktraO(E+VlgV)O(VlgV) General graphBellman-FordO(VE) DAGBellman-FordO(V+E)O(V)

10. All-pair shortest paths

11. But how…….? Recall that shortest paths has some useful properties. One of them is that a shortest path has an optimal substructure. As soon as we realize this……. Dynamic programming may be a good choice to solve this problem!

12. Floyd-Warshall algorithm

13. The transition function

14. The transition function(Cont.)

15. Pseudo code

16. An Alternative: Transitive closure of a directed graph

17. An Alternative: Transitive closure of a directed graph(Cont.)

18. Another idea

19. Johnson’s algorithm

20. Theorem

21. Theorem(Cont.) The same for every path

22. Collorary

23. Difference constraints

24. Difference constraints(Cont.)

25. V1V1 V2V2 V3V3 2 3 -2 S 0 0 0

26. Difference constraints(End)

27. Return to Johnson Algorthim