1. Implementation of Dijkstra’s Algorithm

2. Dijkstra’s Algorithm

3. 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

4. An Example      Initialize4 2 4 2 2 1 2 3 1  1 6 4 2 3 3 5   1
Initialize 2 ∞ 3 ∞
Select the node with the minimum temporary distance label. 4 5 6 ∞ ∞ ∞

5. Update Step 2   4 2 4 2 2 1 2 3  1 6 4 2 3 3 5   1 2 ∞ 3 ∞ 4 5 6 ∞ ∞ ∞

6. Update Step 2     Decrease-Key(S,2,2) 4 2 4 2 2 1 2 3 1 6 4 2 3 31 2 3  1 6 4 2 3 3 5   1 2 2 3 ∞
Decrease-Key(S,2,2) 4 5 6 ∞ ∞ ∞

7. Update Step 2     4 Decrease-Key(S,2,2) Decrease-Key(S,3,4) 4 2 41 2 3  1 6 4 2 3 3 5   1 4 2 2 3 4
Decrease-Key(S,2,2)
Decrease-Key(S,3,4) 4 5 6 ∞ ∞ ∞

8. Update Step 2    4 Extract-Min(S) 4 2 4 2 2 1 2 3 1 6 4 2 3 3 5 ∞ 61 2 3  1 6 4 2 3 3 5 4  ∞ 6 2 2 3 4
Extract-Min(S) 4 5 ∞ ∞

9. Choose Minimum Temporary Label2  4 2 4 2 2 1 2 3  1 6 4 2 3 3 5  4 2 2 6 ∞ 3 4 4 5 ∞ ∞

10. Update Step 6 The predecessor of node 3 is now node 2 2    4 4 31 2 3  1 6 4 2 3 3 5  4 2 2 4 3 5 4 3 3
Decrease-Key(S,3,3)
Decrease-Key(S,4,6)
Decrease-Key(S,5,4) 6 4 ∞ 6

11. Choose Minimum Temporary Label2 6 4 2 4 2 2 1 2 3  1 6 4 2 3 3 5 3 4 6 4 5 4 3 3
Extract-Min(S) 6 ∞

12. Choose Minimum Temporary Label2 6 4 2 4 2 2 1 2 3  1 6 4 2 3 3 5 3 4 3 3 5 4 4 6
Extract-Min(S) 6 ∞

13. Update 2 6  3 4 d(5) is not changed. 4 2 4 2 2 1 2 3 1 6 4 2 3 3 5 31 2 3  1 6 4 2 3 3 5 3 4 3 3 5 4 4 6
d(5) is not changed. 6 ∞

14. Choose Minimum Temporary Label2 6 4 2 4 2 2 1 2 3  1 6 4 2 3 3 5 3 4 6 ∞ 5 4 4 6
Extract-Min(S)

15. Choose Minimum Temporary Label2 6 4 2 4 2 2 1 2 3  1 6 4 2 3 3 5 3 4 5 4 6 4 6
Extract-Min(S) ∞

16. Update d(4) is not changed 2 6 6  3 4 Decrease-Key(S,6,6) 4 2 4 2 2 16 1 2 3  1 6 4 2 3 3 5 3 4 5 4 6 4 6
Decrease-Key(S,6,6) 6

17. Choose Minimum Temporary Label2 6 4 2 4 2 2 1 2 3 6 1 6 4 2 3 3 5 3 4 4 6 6 6

18. Update 2 6 6 3 4 d(6) is not updated 4 2 4 2 2 1 2 3 1 6 4 2 3 3 5 4 61 2 3 6 1 6 4 2 3 3 5 3 4 4 6 6 6
d(6) is not updated

19. Choose Minimum Temporary Label2 6 4 2 4 2 2 1 2 3 6 1 6 4 2 3 3 5 3 4 6 6
There is nothing to update

20. End of Algorithm 2 6 6 3 4 All nodes are now permanent1 2 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

21. Implementations Decrease-Key(S,x,key): Call at most m times.
Extract-Min(S): Call n times.
With min-priority queue, Dijkstra algorithm can be
implemented in time