Shortest-Paths Trees

1. Shortest-Paths Trees Kun-Mao Chao (趙坤茂) Department of Computer Science and Information Engineering National Taiwan University, Taiwan E-mail: kmchao@csie.ntu.edu.tw WWW: http://www.csie.ntu.edu.tw/~kmchao

2. Shortest-Paths Trees • The objective is to find the set of edges connecting all nodes such that the sum of the edge lengths from the source to each node is minimized. • In order to minimize the total path lengths, the path from the source to each node must be a shortest path connecting them.

3. Shortest-Paths Trees

4. Negative edges in an undirected graph

5. Directed graphs

6. Dijkstra's Algorithm

7. Choose a

8. Relax (a, b) and (a, g)

9. Choose b; Add (a, b) to T

10. Relax (b, c) and (b, d)

11. Choose g; Add (a, g) to T

12. Relax (g, e) and (g, h)

13. Choose d; Add (b, d) to T

14. Relax (d, e)

15. Choose h; Add (g, h) to T

16. Choose e; Add (d, e) to T

17. Relax (e, f)

18. Choose c; Add (b, c) to T

19. Relax (c, h)

20. Choose f; Add (e, f) to T

21. Relax (f, d) and (f, h)

22. The resulting SPT

23. Negative edge

24. Choose a

25. Choose b; Add (a, b) to T

26. Choose d; Add (b, d) to T

27. Choose c; Add (b, c) to T

28. Choose e; Add (d, e) to T

29. Something went wrong

30. A wrong SPT

31. A correct SPT

32. The Bellman-Ford Algorithm

33. δ[b] and δ[g] modified

34. δ[c], δ[d], δ[e] and δ[h] modified

35. δ[f] modified

36. δ[h] modified

37. A correct SPT

38. Try this in class