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Shortest Path Problem: Dijkstra's Algorithm Quiz

Test your knowledge on Dijkstra's Algorithm and the Shortest Path Problem. Determine if statements about dynamic programming and the algorithm are true or false. Find the shortest path using Dijkstra's Algorithm in a network with nonnegative edge weights.

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Shortest Path Problem: Dijkstra's Algorithm Quiz

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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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