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Lecture 6

Lecture 6. Shortest Path Problem. Quiz Sample. True or False Every dynamic programming can be analyzed with formula: Run-time = (table size) x (computation time of recursive formula). Answer: False A counterexample can be seen in study of the shortest path problem.

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Lecture 6

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  1. Lecture 6 Shortest Path Problem

  2. Quiz Sample • True or False • Every dynamic programming can be analyzed with formula: • Run-time = (table size) • x (computation time of recursive formula). • Answer: False • A counterexample can be seen in study of the shortest path problem.

  3. s t

  4. Dynamic Programming

  5. Dynamic Programming

  6. Dynamic Programming

  7. Dijkstra’s Algorithm Find the shortest path from a network with nonnegative edge weights.

  8. Dijkstra’s Algorithm

  9. Lemma

  10. Lemma

  11. Proof of Lemma T w u S s

  12. Dijkstra’s Algorithm

  13. 1 An Example   4 2 4 2 2 0 2 1 3  1 6 4 2 3 3 5   Initialize Select the node with the minimum temporary distance label.

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

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

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

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

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

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

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

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

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

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

  24. End of Algorithm 2 6 4 2 4 2 2 0 2 1 3 6 1 6 4 2 3 3 5 3 4 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

  25. Theorem

  26. Counterexample 4 -2 -2 2 1

  27. Counterexample 3 -2 -2 1 2

  28. Counterexample 4 -2 -2 1 2

  29. Counterexample 4 -2 -2 1 2

  30. Counterexample 4 -2 -2 1 2

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

  32.  0  1    2 0 1 2 3 4 5 6 7 3 4 5 6 An Example Initialize distance labels 4 2 4 2 2 Initialize buckets. 2 1 3 1 6 4 2 3 Select the node with the minimum temporary distance label. 3 5 1

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