1. Lecture 6 Shortest Path Problem

2. s t

3. Dynamic Programming

4. Dijkstra’s Algorithm is motivated from a way to implement of this dynamic programming.

5. Dynamic Programming

6. Lemma Proof

7. Theorem 2 -1 2 1

8. Counterexample

9. Smart Implementation

10. An Example 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 Initialize 1 0      Select the node with the minimum temporary distance label.

11. Update Step 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0      1

12. Choose u such that N_(u) S 1 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0 2   

13. Update Step 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 6 4 3 0    The predecessor of node 3 is now node 2

14. Choose u Such That N_(u) S 1 24 5 6 2 4 2 1 3 4 2 3 2 2 3 6 4 0  3

15. Update 1 24 5 6 2 4 2 1 3 4 2 3 2 0 d(5) is not changed. 3 2 3 6 4 

16. Choose u s.t. N_(u) S 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4  5

17. Update 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4  5 d(4) is not changed 6

18. Choose u s.t. N_(u) S 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4

19. Update 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 d(6) is not updated

20. Choose u s.t. N_(u) S 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 There is nothing to update

21. End of Algorithm 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 All nodes are now permanent The predecessors form a tree The shortest path from node 1 to node 6 can be found by tracing back predecessors

22. Dijkstra’s Algorithm

24. Lemma

26. Proof of Lemma s u w S T

27. Theorem

28. Counterexample 3 -1 -2 2 1

37. Dijkstra’s Algorithm

38. An Example 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 Initialize 1 0      Select the node with the minimum temporary distance label.

39. Update Step 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0      1

40. Choose Minimum Temporary Label 1 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0 2   

41. Update Step 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 6 4 3 0    The predecessor of node 3 is now node 2

42. Choose Minimum Temporary Label 1 24 5 6 2 4 2 1 3 4 2 3 2 2 3 6 4 0  3

43. Update 1 24 5 6 2 4 2 1 3 4 2 3 2 0 d(5) is not changed. 3 2 3 6 4 

44. Choose Minimum Temporary Label 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4  5

45. Update 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4  5 d(4) is not changed 6

46. Choose Minimum Temporary Label 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4

47. Update 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 d(6) is not updated

48. Choose Minimum Temporary Label 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 There is nothing to update

49. End of Algorithm 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 All nodes are now permanent The predecessors form a tree The shortest path from node 1 to node 6 can be found by tracing back predecessors

60. Dijkstra’s Algorithm with simple buckets (also known as Dial’s algorithm)

61. An Example 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 Initialize distance labels 1 0      Select the node with the minimum temporary distance label. 01234567 1  2 3 4 5 6 Initialize buckets.

62. Update Step 01234567 1  2 3 4 5 6 23 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0      1

63. Choose Minimum Temporary Label 01234567  4 5 6 23 Find Min by starting at the leftmost bucket and scanning right till there is a non-empty bucket. 1 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0 2   

64. Update Step 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 6 4 3 0    01234567  4 5 6 23 3 4 5

65. Choose Minimum Temporary Label 1 24 5 6 2 4 2 1 3 4 2 3 2 2 3 6 4 0  3 01234567  6 3 45 Find Min by starting at the leftmost bucket and scanning right till there is a non-empty bucket.

66. Update 1 24 5 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4  01234567  6 3 45

67. Choose Minimum Temporary Label 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4  5 01234567  6 45

68. Update 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4  5 6 01234567  6 45 6

69. Choose Minimum Temporary Label 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 01234567 4 6

70. Update 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 01234567 4 6

71. Choose Minimum Temporary Label 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 01234567 6 There is nothing to update

72. End of Algorithm 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 All nodes are now permanent The predecessors form a tree The shortest path from node 1 to node 6 can be found by tracing back predecessors

76. Implementations With min-priority queue, Dijkstra algorithm can be implemented in time With Fibonacci heap, Dijkstra algorithm can be implemented in time With Radix heap, Dijkstra algorithm can be implemented in time