Mastering All-Pairs Shortest Paths: Theorem Proof & Dynamic Programming Ideas

1. Lecture 7 All-Pairs Shortest Paths

2. All-Pairs Shortest Paths

3. Path Counting Problem

4. Adjacency Matrix

5. 1 3 2 1 2 3 1 2 3

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

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

8. Induction Step

9. All-Pairs Shortest Pathswith at most k edges

10. Recursive formula Proof.

11. Case 1. The path with length contains at most edges. Case 2. the path with length contains exactly edges.

12. Key Observation

13. Dynamic Program

14. Speed Up dynamic Program

15. Idea

16. Weighted Adjacency Matrix

17. 1 4 5 6 3 2 1 2 3 1 2 3

19. A New Multiplication

20. Associative Law

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

22. 1 k=1 True! 4 5 6 3 2 1 2 3 1 2 3

23. Induction Step

24. All-Pairs Shortest Paths Theorem Proof

25. How to Compute

26. Lemma

27. Theorem

28. Theorem

29. Floyd-Warshall Algorithm

30. Observation

31. Dynamic Program

33. Theorem

34. Theorem

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

36. Puzzle 1

37. Puzzle 2

38. Puzzle 3