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Lecture 6 Shortest Path Problem. s t Dynamic Programming.

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Presentation on theme: "Lecture 6 Shortest Path Problem. s t Dynamic Programming."— Presentation transcript:

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

8 Counterexample

9 Smart Implementation

10 An Example Initialize 1 0      Select the node with the minimum temporary distance label.

11 Update Step      1

12 Choose u such that N_(u) S   

13 Update Step    The predecessor of node 3 is now node 2

14 Choose u Such That N_(u) S  3

15 Update d(5) is not changed 

16 Choose u s.t. N_(u) S  5

17 Update  5 d(4) is not changed 6

18 Choose u s.t. N_(u) S

19 Update d(6) is not updated

20 Choose u s.t. N_(u) S There is nothing to update

21 End of Algorithm 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

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24 Lemma

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26 Proof of Lemma s u w S T

27 Theorem

28 Counterexample

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37 Dijkstra’s Algorithm

38 An Example Initialize 1 0      Select the node with the minimum temporary distance label.

39 Update Step      1

40 Choose Minimum Temporary Label   

41 Update Step    The predecessor of node 3 is now node 2

42 Choose Minimum Temporary Label  3

43 Update d(5) is not changed 

44 Choose Minimum Temporary Label  5

45 Update  5 d(4) is not changed 6

46 Choose Minimum Temporary Label

47 Update d(6) is not updated

48 Choose Minimum Temporary Label There is nothing to update

49 End of Algorithm 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

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

61 An Example Initialize distance labels 1 0      Select the node with the minimum temporary distance label  Initialize buckets.

62 Update Step       1

63 Choose Minimum Temporary Label  Find Min by starting at the leftmost bucket and scanning right till there is a non-empty bucket   

64 Update Step    

65 Choose Minimum Temporary Label   Find Min by starting at the leftmost bucket and scanning right till there is a non-empty bucket.

66 Update  

67 Choose Minimum Temporary Label   6 45

68 Update  

69 Choose Minimum Temporary Label

70 Update

71 Choose Minimum Temporary Label There is nothing to update

72 End of Algorithm 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

73

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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


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