1. Shortest Paths

2. The problem A D 2 5 1 F 10 B 2 6 E C 3

3. Breadth-first search A D 1 1 1 F 1 B 1 1 E C 1

4. Breadth-first search
Queue: A A D 1 1 1 F 1 B 1 1 E C 1

5. Breadth-first search
Queue: B, C A D 1 1 1 F 1 B 1 1 1 E C 1 1

6. Breadth-first search
Queue: C, D 2 A D 1 1 1 F 1 B 1 1 1 E C 1 1

7. Breadth-first search
Queue: D, E 2 A D 1 1 1 F 1 B 1 1 1 E C 1 2 1

8. Breadth-first search
Queue: E, F 2 A 1 3 D 1 1 F 1 B 1 1 1 E C 1 2 1

9. Breadth-first search
Queue: F 2 A 1 3 D 1 1 F 1 B 1 1 1 E C 1 2 1

10. Breadth-first search
Queue: 2 A 1 3 D 1 1 F 1 B 1 1 1 E C 1 2 1

11. Breadth-first search
Time: O(n+e)

12. Dijkstra's algorithm A D 2 5 1 F 10 B 2 6 E C 3

13. Dijkstra's algorithm Priority queue: A, B, C, D, E, F oo A D 2 5 1 FA D 2 5 1 F oo 10 B 2 oo 6 E C 3 oo oo

14. Dijkstra's algorithm Priority queue: C, B, D, E, F oo A D 2 5 1 F ooA D 2 5 1 F oo 10 B 2 5 6 E C 3 oo 2

15. Dijkstra's algorithm Priority queue: B, E, D, F oo A D 2 5 1 F oo 10 BA D 2 5 1 F oo 10 B 2 5 6 E C 3 5 2

16. Dijkstra's algorithm Priority queue: E, D, F 6 A D 2 5 1 F oo 10 B 2 5A D 2 5 1 F oo 10 B 2 5 6 E C 3 5 2

17. Dijkstra's algorithm Priority queue: D, F 6 A D 2 5 1 F 11 10 B 2 5 6A D 2 5 1 F 11 10 B 2 5 6 E C 3 5 2

18. Dijkstra's algorithm Priority queue: F 6 A D 2 5 1 F 8 10 B 2 5 6 E CA D 2 5 1 F 8 10 B 2 5 6 E C 3 5 2

19. Dijkstra's algorithm Priority queue: 6 A D 2 5 1 F 8 10 B 2 5 6 E C 3A D 2 5 1 F 8 10 B 2 5 6 E C 3 5 2

20. Dijkstra's algorithm
Time: O((n + e)log n)

21. Floyd-Warshall Algorithm
Start: d[i][j] = weight of edge from i to j or infinity if no edge
End: d[i][j] = shortest distance from i to j or infinity if no edge
for j = 1 to n
for i = 1 to n
for k = 1 to n
d[i][k] = min(d[i][k], d[i][j] + d[j][k])
NB: Loop order is very important!
Time: O(n^3)
Space: O(n^2)