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Dijkstra’s Algorithm

Dijkstra’s Algorithm. Text Read Weiss, § 9.3 Dijkstra’s Algorithm Single Source Multiple Destination Shortest Path Algorithm . Requirements. Works with directed and undirected graphs Works with weighted and unweighted graphs Rare type of algorithm 

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Dijkstra’s Algorithm

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  1. Dijkstra’s Algorithm Text • Read Weiss, § 9.3 Dijkstra’s Algorithm • Single Source Multiple Destination Shortest Path Algorithm

  2. Requirements • Works with directed and undirected graphs • Works with weighted and unweighted graphs • Rare type of algorithm  A greedy algorithm that produces an optimal solution

  3. Walk-Through 2 Initialize array 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  4. 2 Start with G 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  5. 2 Update unselected nodes 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  6. 2 Select minimum distance 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  7. 2 Update unselected nodes 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  8. 2 Select minimum distance 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  9. 2 Update unselected nodes 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  10. 2 Select minimum distance 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  11. 2 Update unselected nodes 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  12. 2 Select minimum distance 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  13. 2 Update unselected nodes 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  14. 2 Select minimum distance 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  15. 2 Update unselected nodes 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  16. 2 Select minimum distance 3 F C 5 A 10 7 3 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  17. 2 Update unselected nodes 3 F C 5 A 10 7 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7

  18. 2 Select minimum distance 3 F C 5 A 10 7 4 8 18 4 B D 9 10 H 9 25 2 3 G E 7 Done

  19. Order of Complexity • Analysis • findMin() takes O(V) time • outer loop iterates (V-1) times  O(V2) time • Optimal for dense graphs, i.e., |E| = O(V2) • Suboptimal for sparse graphs, i.e., |E| = O(V)

  20. Order of Complexity • If the graph is sparse, i.e., |E| = O(V) • maintain distances in a priority queue • insert new (shorter) distance produced by line 10 of Figure 9.32  O(|E| log |V|) complexity

  21. Negative Edge Weights • Read § 9.3.3 • Dijkstra’s algorithm as shown in Figure 9.32 does not work! Why?

  22. Acyclic Graphs • Read § 9.3.4 • Combine topological sort with Dijkstra’s algorithm

  23. All-Pairs Shortest Paths • One option: run Dijktra’s algorithm |V| times  O(V3) time • A more efficient O(V3) time algorithm is discussed in Chapter 10

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