1. Lecture 7 All-Pairs Shortest Paths

2. All-Pairs Shortest Paths

3. Path Counting Problem

4. Adjacency Matrix

5. 1 2 3 1 2 3 123

6. Theorem Proof. We prove it by induction on k.

7. 1 2 3 1 2 3 123 k=1 True!

8. Induction Step

9. All-Pairs Shortest Paths with at most k edges

10. Recursive formula Note: For i≠j,

11. Key Observation

12. Dynamic Program

13. Speed Up dynamic Program

14. Idea

15. Weighted Adjacency Matrix

16. 1 2 3 1 2 3 123 4 6 5

18. A New Multiplication

19. Associative Law

20. Theorem Proof. We prove it by induction on k.

21. 1 2 3 1 2 3 123 4 6 5 k=1 True!

22. Induction Step

23. All-Pairs Shortest Paths Theorem Proof

24. How to Compute

25. Lemma

26. Theorem

28. Floyd-Warshall Algorithm

29. Observation

30. Dynamic Program

32. Theorem

33. What we learnt in this lecture? The relationship between shortest path and matrix multiplication. Faster-All-Pairs-Shortest-Paths algorithm Floyd-Warshall algorithm.

34. Puzzle 1

35. Puzzle 2

36. Puzzle 3