Dijkstra’s Algorithm

1. Dijkstra’s Algorithm Lauren McLaughlin - 11/7/06

2. Quotes • “Object-oriented programming is an exceptionally bad idea which could only have originated in California.” • “Don't compete with me: firstly, I have more experience, and secondly, I have chosen the weapons.”

3. Quotes • “Program testing can be used to show the presence of bugs, but never to show their absence!” • “Perfecting oneself is as much unlearning as it is learning.”

4. Traveling Salesman • Objective: Find the cheapest way of visiting all of the cities and returning to your starting point

5. Dining Philosophers • Objective: To prevent deadlocks and starvation

6. Dining Philosophers

7. Shortest Path Algorithm

8. Ordered pair u = Vertex @ Beginning of path v = Vertex @ End of path Has a value assigned Cost (u,v) Edge

9. Data Structures • Set of vertices whose shortest path has been determined (F) • Set of remaining vertices (R) • Array containing shortest path (length) • Adjacency Array (W) • Array holding current vertex (touch)

10. Basic Steps • Set F to empty set • Initialize arrays • Find shortest path • Place its vertex in F • Remove vertex from R • Relax vertices in R • Change length of shortest vertex to -1

11. Edge Relaxation • After vertex u is placed in F, every edge in R is relaxed • Get current shortest distance (length[v]) • Determine distance by going through the recently placed vertex in R • If the path is shorter when going through u, set length[v] to that amount

12. index i, vnear; char touch[n]; int length[n]; //Empty set F = 0; for(i = 0; i < n; i++) { touch[i] = source; if(W[source][i] == 0) W[source][i] = oo; length[ i ] = W[source][i]; } } repeat(n - 1 times) { int min = oo; for(i = 0; i < n; i++) { if(i != source) { if(0 <= length[i] < min) { min = length[i]; vnear = i; } } Pseudocode

13. Pseudocode add vnear to F; remove vnear from R; for(i = 0; i < n; i++) { if(length[vnear] + W[vnear][i] < length[i]) { length[i] = length[vnear] + W[vnear][i]; touch[i] = vnear; } } length[vnear] = -1; }

14. Example Objective • Find shortest path from source node, A, to all other nodes

16. Shortest Paths

17. Shortest Paths

18. Shortest Paths

20. Shortest Paths

21. Shortest Paths

23. Shortest Paths

24. Shortest Paths

26. Shortest Paths

27. Shortest Paths

29. Running Time • O(V2) • Linear Search • Vertices stored in array or linked list • O((E + V)logV) • Binary heap used as priority queue • Vertices stored in adjacency lists • O(E + V logV) • Fibonacci heap used as priority queue • Vertices stored in adjacency lists

30. Binary Binary Tree More Structure Fibonacci Collection of trees satisfying the minimum-heap priority Maintains pointer to root of tree with smallest priority More flexible Binary vs. Fibonacci

31. “The competent programmer is fully aware of the limited size of his own skull. He therefore approaches his task with full humility, and avoids clever tricks like the plague.”

32. Movie • \\10.1.154.4\Classes\Comp349\Dijkstra