1. Lecture 11 Topics Application of BFS Shortest Path
Reference: Introduction to Algorithm by Cormen
Chapter 25: Single-Source Shortest Paths
Data Structure and Algorithm

2. Dijkstra's Shortest Path Algorithm
Dijkstra’s algorithm solves the single- source shortest paths problem on a weighted, directed graph G =(V,E) fro the case in which all edge weights are nonnegative.
We assume that w(u,v) >=0 for each edge (u,v)
Data Structure and Algorithm

3. Algorithm Dijkstra(G,w,s) Initialize(G,s) S = 0 Q = V[G]
While Q != 0 do
U = ExtractMin(Q)
S = S union {u}
For each vertex v of adj[u] do
Relax(u,v,w)
Complexty:
Using priority queue:O (V2 +E)
Using heap as priority queue: O( (V+E) lg2V)
cost of building heap is O(V)
cost of ExtracMin is O(lgV) and there are |V| such operations
cost of relax is O(lgV) and there are |E| such operations.
Data Structure and Algorithm

4. Algorithm( Cont.) Initialize(G,s) relax(u,v,w)
for each vertex of V[G] do
d[v] = infinity
Л[v] = NIL
d[s] = 0
relax(u,v,w)
if d[v] >d[u] +w(u,v) then
d[v] = d[u] + w(u,v)
Л[v] = u
Data Structure and Algorithm

5. Complexity Using priority queue:O (V2 +E)
Using heap as priority queue: O( (V+E) lg2V)
cost of building heap is O(V)
cost of ExtracMin is O(lgV) and there are |V| such operations
cost of relax is O(lgV) and there are |E| such operations.
Data Structure and Algorithm

6. Dijkstra's Shortest Path Algorithm
Find shortest path from s to t. 2 23 3 9 s 18 14 2 6 6 4 30 19 11 15 5 5 6 20 16 t 7 44
Data Structure and Algorithm

7. Dijkstra's Shortest Path Algorithm
Q = { s, 2, 3, 4, 5, 6, 7, t }   2 23 3 9 s 18 14  2 6 6  4 30  19 11 15 5 5 6 20 16 t 7 44 
distance label 
Data Structure and Algorithm

8. Dijkstra's Shortest Path Algorithm
Q = { s, 2, 3, 4, 5, 6, 7, t }
ExtractMin()   2 23 3 9 s 18 14  2 6 6  4 30  19 11 15 5 5 6 20 16 t 7 44 
distance label 
Data Structure and Algorithm

9. Dijkstra's Shortest Path Algorithm
Q = { 2, 3, 4, 5, 6, 7, t }
decrease key   X 9 2 23 3 9 s 18 14  X 14 2 6 6   4 30 19 11 15 5 5 6 20 16 t 7 44 
distance label  15 X
Data Structure and Algorithm

10. Dijkstra's Shortest Path Algorithm
Q = { 2, 3, 4, 5, 6, 7, t }
ExtractMin()   X 9 2 23 3 9 s 18 14  X 14 2 6 6   4 30 19 11 15 5 5 6 20 16 t 7 44 
distance label  15 X
Data Structure and Algorithm

11. Dijkstra's Shortest Path Algorithm
Q = { 3, 4, 5, 6, 7, t }   X 9 2 23 3 9 s 18 14  X 14 2 6 6  4 30  19 11 15 5 5 6 20 16 t 7 44   15 X
Data Structure and Algorithm

12. Dijkstra's Shortest Path Algorithm
Q = { 3, 4, 5, 6, 7, t }
decrease key  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6   4 30 19 11 15 5 5 6 20 16 t 7 44   15 X
Data Structure and Algorithm

13. Dijkstra's Shortest Path Algorithm
Q = { 3, 4, 5, 6, 7, t }  X 32  X 9 2 23 3 9
ExtractMin() s 18 14  X 14 2 6 6   4 30 19 11 15 5 5 6 20 16 t 7 44   15 X
Data Structure and Algorithm

14. Dijkstra's Shortest Path Algorithm
Q = { 3, 4, 5, 7, t }  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 30  X 4 19 11 15 5 5 6 20 16 t 7 44   15 X
Data Structure and Algorithm

15. Dijkstra's Shortest Path Algorithm
Q = { 3, 4, 5, 7, t }  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 4 30  X 19 11 15 5 5 6 20 16 t 7 44   15 X
ExtractMin()
Data Structure and Algorithm

16. Dijkstra's Shortest Path Algorithm
Q = { 3, 4, 5, t }  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 X 35  X 4 30 19 11 15 5 5 6 20 16 t 7 44 59   15 X X
Data Structure and Algorithm

17. Dijkstra's Shortest Path Algorithm
Q = { 3, 4, 5, t }
ExtractMin  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 X 35 4 30  X 19 11 15 5 5 6 20 16 t 7 44 59   15 X X
Data Structure and Algorithm

18. Dijkstra's Shortest Path Algorithm
Q = { 4, 5, t }  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 X 35 X 34 30  X 4 19 11 15 5 5 6 20 16 t 7 44 51 59   15 X X X
Data Structure and Algorithm

19. Dijkstra's Shortest Path Algorithm
Q = { 4, 5, t }  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 X 35 X 34 30  X 4 19 11 15 5 5 6 20
ExtractMin 16 t 7 44 51 59   15 X X X
Data Structure and Algorithm

20. Dijkstra's Shortest Path Algorithm
Q = { 4, t }  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 X 35 X 34 45 X 30  X 4 19 11 15 5 5 6 20 16 t 7 44 50 51 59   15 X X X X
Data Structure and Algorithm

21. Dijkstra's Shortest Path Algorithm
Q = { 4, t }  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 X 35 34 45 X X 4 30  X 19 11 15 5
ExtractMin 5 6 20 16 t 7 44 50 51 59   15 X X X X
Data Structure and Algorithm

22. Dijkstra's Shortest Path Algorithm
Q = { t }  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 X 35 X 34 45 X 30  X 4 19 11 15 5 5 6 20 16 t 7 44 50 51 59   15 X X X X
Data Structure and Algorithm

23. Dijkstra's Shortest Path Algorithm
Q = { t }  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 X 35 34 45 X X 4 30  X 19 11 15 5 5 6 20 16 t 7 44
ExtractMin 50 51 59   15 X X X X
Data Structure and Algorithm

24. Dijkstra's Shortest Path Algorithm
S = { s, 2, 3, 4, 5, 6, 7, t }
Q = { }  X 32  X 9 2 23 3 9 s 18 14  X 14 2 6 6  44 X 35 34 45 X X 4 30  X 19 11 15 5 5 6 20 16 t 7 44
ExtractMin 50 51 59   15 X X X X
Data Structure and Algorithm

25. The Bellman-Form Algorithm
BELLMAN-FORD(G,w,s)
Initialize()
for i = 1 to |V[G]| -1 do
for each edge (u,v) of E[G] do
relax(u,v,w)
if d[v] > d[u] + w(u,v) then
return FALSE
return TRUE
Complexity
O(VE)
Data Structure and Algorithm

26. Example:Bellman-Fordz u x v y  7 6 8 9 5 -2 -3 -4 2
Data Structure and Algorithm

27. Example:Bellman-Fordz u x v y 6  7 8 9 5 -2 -3 -4 2
Data Structure and Algorithm

28. Example:Bellman-Fordz u x v y 6 4 2 7 8 9 5 -2 -3 -4
Data Structure and Algorithm

29. Example:Bellman-Fordz u x v y 2 4 7 6 8 9 5 -2 -3 -4
Data Structure and Algorithm

30. Example:Bellman-Fordz u x v y 2 4 -2 7 6 8 9 5 -3 -4
Data Structure and Algorithm