Algorithme de Dijkstra

1. Algorithme de Dijkstra • Référence: • John W. Byers. DijkstraDemo, Course notes CS 330 introduction to Algorithms, DostonUniuversity, Spring 2013, [available at] http://www.cs.bu.edu/fac/byers/cs330.html • Modifié par: Vincent Roberge

2. Pseudocode dist[s] ←0 (distance to source vertex is zero)for  all v ∈ V–{s}        do  dist[v] ←∞ (set all other distances to infinity) S←∅ (S, the set of visited vertices is initially empty) Q←V  (Q, the queue initially contains all vertices) while Q ≠∅ (while the queue is not empty) do   u ← mindistance(Q,dist) (select the element of Q with the min. distance)   S←S∪{u} (add u to list of visited vertices)        for all v ∈ neighbors[u]               do  if   dist[v] > dist[u] + w(u, v) (if new shortest path found)                         then      d[v] ←d[u] + w(u, v) (set new value of shortest path) (if desired, add traceback code) return dist

3. Dijkstra's Shortest Path Algorithm • Find shortest path from s to t. 24 2 3 9 s 18 14 6 2 6 4 19 30 11 5 15 5 6 16 20 t 7 44

4. Dijkstra's Shortest Path Algorithm Visités= { } Non-visités= { s, 2, 3, 4, 5, 6, 7, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ ]   24 2 3 0 9 s 18  14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44  distance label 

5. Dijkstra's Shortest Path Algorithm Visités= { } Non-visités= { s, 2, 3, 4, 5, 6, 7, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ ] delmin   24 2 3 0 9 s 18  14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44  distance label 

6. Dijkstra's Shortest Path Algorithm Visités= { s } Non-visités= { 2, 3, 4, 5, 6, 7, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s ss ] decrease key   9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44  distance label  15 X

7. Dijkstra's Shortest Path Algorithm Visités= { s } Non-visités= { 2, 3, 4, 5, 6, 7, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s ss ] delmin   9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44  distance label  15 X

8. Dijkstra's Shortest Path Algorithm Visités= { s, 2 } Non-visités= { 3, 4, 5, 6, 7, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s ss ]   9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44   15 X

9. Dijkstra's Shortest Path Algorithm Visités= { s, 2 } Non-visités= { 3, 4, 5, 6, 7, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 2 s s ] decrease key  33 X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44   15 X

10. Dijkstra's Shortest Path Algorithm Visités= { s, 2 } Non-visités= { 3, 4, 5, 6, 7, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 2 s s ]  33 X  9 X 24 2 3 0 9 delmin s 18  14 X 14 6 2 6  4  19 30 11 5 15 5 6 16 20 t 7 44   15 X

11. Dijkstra's Shortest Path Algorithm Visités= { s, 2, 6 } Non-visités= { 3, 4, 5, 7, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 6 s s ] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  44 4  X 19 30 11 5 15 5 6 16 20 t 7 44   15 X

12. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 6 } Non-visités = { 3, 4, 5, 7, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 6 s s ] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  44 4  X 19 30 11 5 15 5 6 16 20 t 7 44  delmin  15 X

13. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 6, 7 } Non-visités = { 3, 4, 5, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 7 s s 7] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  35 44 X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  59 X  15 X

14. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 6, 7 } Non-visités = { 3, 4, 5, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 7 s s 7] delmin 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  35 44 X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  59 X  15 X

15. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 3, 6, 7 } Non-visités = { 4, 5, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 3 s s 3] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  35 34 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  51 59 X X  15 X

16. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 3, 6, 7 } Non-visités = { 4, 5, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 3 s s 3] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  35 34 44 X X 4  X 19 30 11 5 15 5 6 16 delmin 20 t 7 44  51 59 X X  15 X

17. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 3, 5, 6, 7 } Non-visités = { 4, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 5 3 s s 5] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  50 51 59 X X X  15 X

18. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 3, 5, 6, 7 } Non-visités = { 4, t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 5 3 s s 5] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 delmin 15 5 6 16 20 t 7 44  50 51 59 X X X  15 X

19. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 3, 4, 5, 6, 7 } Non-visités = { t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 5 3 s s 5] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  50 51 59 X X X  15 X

20. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 3, 4, 5, 6, 7 } Non-visités = { t } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 5 3 s s 5] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  delmin 50 51 59 X X X  15 X

21. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 3, 4, 5, 6, 7, t } Non-visités = { } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 5 3 s s 5] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  50 51 59 X X X  15 X

22. Dijkstra's Shortest Path Algorithm Visités = { s, 2, 3, 4, 5, 6, 7, t } Non-visités = { } suivants = [s 2 3 4 5 6 7 t] precedents = [ s 6 5 3 s s 5] 32  33 X X  9 X 24 2 3 0 9 s 18  14 X 14 6 2 6  45 35 34 X 44 X X 4  X 19 30 11 5 15 5 6 16 20 t 7 44  50 51 59 X X X  15 X