1. BellmanFord

2. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true

3. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true i = 0 6 7 9 2 -4 -3 7 5 -2 8 s z y x t 0

4. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true i = 1 6 7 9 2 -4 -3 7 5 -2 8 s z y x t 0 7 2

5. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true i = 2 6 7 9 2 -4 -3 7 5 -2 8 s z y x t 0 7 2 5 13

6. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true i = 2 6 7 9 2 -4 -3 7 5 -2 8 s z y x t 0 7 2 5 9

7. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true i = 3 6 7 9 2 -4 -3 7 5 -2 8 s z y x t 0 6 2 5 9

8. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true i = 4 6 7 9 2 -4 -3 7 5 -2 8 s z y x t 0 6 2 4 9 Correctness I) If no negative cycle reachable from s: BF returns true, BF finds shortest paths, BF builds predecessor tree II) Otherwise: BF returns false

9. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true

10. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true O(V)

11. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true O(V) O(V*E)

12. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true O(V) O(V*E) O(E)

13. BellmanFord(G,w,s) 1 InitializeSingleSource(G,s) 2 for i 1 to |V[G]| - 1 3 do for each (u,v) E[G] 4 do Relax(u,v,w) 5 for each edge (u,v) E[G] 6 do if d[v] > d[u] + w[u,v] 7 then return false 8 return true O(V) O(V*E) O(E) O(V*E)

14. DAG Shortest Path

15. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w)

16. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) 6 7 2 -3 7 -2 8 s z y x t

17. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) 6 7 2 -3 7 -2 8 s z y x t 1/ 2/ 3/4

18. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) 6 7 2 -3 7 -2 8 s z y x t 1/10 2/7 3/4 5/6 8/9

19. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0

20. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0 72

21. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0 725

22. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0 7259

23. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0 7259 Correct?

24. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0 7259 Correct? Yes, follows directly from L4 and L5

25. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0 7259 Time?

26. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0 7259 O(V+E) Time?

27. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0 7259 O(V+E) Time? O(V)

28. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0 7259 O(V+E) Time? O(V) O(E)

29. DAGshortestPaths(G,w,s) 1 topologically sort the vertices of G 2 InitializeSingleSource(G,s) 3 for each vertex u, taken in topological order 4 do for each vertex v adj[u] 5 do Relax(u,v,w) s z y x t 1/10 2/7 3/4 5/6 8/9 7 2-2 6 7 8 0 7259 O(V+E) Time: O(V+E) – linear in |adj| O(V) O(E)

30. Dijkstra’s Algorithm (no negative edges) Greedy

31. 0

32. 0

33. 0 1 3 4 1 3 4

34. 0 1 3 4 1 3 4 Could these be optimal?

35. 0 1 3 4 1 3 4 Could these be optimal? I don’t know yet 1?

36. 0 1 3 4 1 3 4 optimal?

37. 0 1 3 4 1 3 4 optimal? Yes any path from “3” and “4” will be non-neg, and there is no unexplored paths from “0”

38. 0 1 3 4 1 3 4

39. 0 1 3 4 1 2 4 1 2 4 3 5 4

40. 0 1 3 4 1 2 4 1 2 4 3 5 4 Optimal?

41. 0 1 3 4 1 2 4 1 2 4 3 5 4 Optimal? No, as before there could be frontier edges causing better paths

42. 0 1 3 4 1 2 4 1 2 4 3 5 4 optimal?

43. 0 1 3 4 1 2 4 1 2 4 3 5 4 optimal? Yes! As before no better path from frontier, No better path from explored vertices

44. Dijkstra(G,w,s) 1 InitializeSingleSource(G,s) 2 S Ø 3 Q V[G] 4 while Q Ø 5 do u ExtractMin(Q) 6 S S {u} 7 for each vertex v Adj[u] 8 do Relax(u,v,w)

45. Dijkstra(G,w,s) 1 InitializeSingleSource(G,s) 2 S Ø 3 Q V[G] 4 while Q Ø 5 do u ExtractMin(Q) 6 S S {u} 7 for each vertex v Adj[u] 8 do Relax(u,v,w) O(V)

46. Dijkstra(G,w,s) 1 InitializeSingleSource(G,s) 2 S Ø 3 Q V[G] 4 while Q Ø 5 do u ExtractMin(Q) 6 S S {u} 7 for each vertex v Adj[u] 8 do Relax(u,v,w) O(V) BinH O(V) – Build Heap

47. Dijkstra(G,w,s) 1 InitializeSingleSource(G,s) 2 S Ø 3 Q V[G] 4 while Q Ø 5 do u ExtractMin(Q) 6 S S {u} 7 for each vertex v Adj[u] 8 do Relax(u,v,w) O(V) BinH O(V) – Build Heap O(VlgV)

48. Dijkstra(G,w,s) 1 InitializeSingleSource(G,s) 2 S Ø 3 Q V[G] 4 while Q Ø 5 do u ExtractMin(Q) 6 S S {u} 7 for each vertex v Adj[u] 8 do Relax(u,v,w) O(V) BinH O(V) – Build Heap O(V*lgV) O(E*lgV) – Dec.Key

49. Dijkstra(G,w,s) 1 InitializeSingleSource(G,s) 2 S Ø 3 Q V[G] 4 while Q Ø 5 do u ExtractMin(Q) 6 S S {u} 7 for each vertex v Adj[u] 8 do Relax(u,v,w) O(V) BinH O(V) – Build Heap O(V*lgV) O(E*lgV) – Dec.Key Time: O( (V+E)*lgV )