1. Shortest Path
Graph Theory Basics
Anil Kishore

2. Introduction Weighted Graph – Edges have weights
Shortest Path problem is finding a path between two vertices such that the sum of the weights of edges along the path is minimized 2 B C 20 3 A 4 8 E G 6 5 D 2 F

3. Variations
Single-Source Shortest Path - find shortest paths from a source vertex S to all other vertices
Single-Destination Shortest Path - find shortest paths from all vertices in the directed graph to a single destination vertex D
All-Pairs Shortest Path - find shortest paths between every pair of vertices u, v in the graph.

4. Applications Road Network … many other
Cities in a country can be considered as vertices and the roads connecting them are edges
Edge weight is length of the road ( distance to be travelled )
One-way roads ( Directed Edges )
Find shortest path from city A to city B
… many other

5. Algorithms Dijkstra's algorithm : single-source shortest path problem
Bellman–Ford algorithm : single-source shortest path problem if edge weights may be negative.
Floyd–Warshall algorithm : all pairs shortest paths.

6. Dijkstra’s Algorithm
Dijkstra( S ) set dist[u] = INF for all vertices u set dist[S] = 0 and insert S in a data structure q while( q is not empty ) u = remove the vertex with minimum dist in q for v in nbrs(u) newDist = dist[u] + weight(u,v) if( newDist < dist[v] ) dist[v] = newDist; end-if end-for end-while end-Diijkstra

7. Dijkstra’s Algorithm exampleB C D E F G 20 4 6 2 5 8 3

8. Dijkstra’s Algorithm exampleB C D E F G 20 4 6 2 5 8 3

9. Dijkstra’s Algorithm example20 A B C D E F G 20 4 6 2 5 8 3 4 6

10. Dijkstra’s Algorithm example20 A B C D E F G 20 4 6 2 5 8 3 4 6

11. Dijkstra’s Algorithm example20 A B C D E F G 20 4 6 2 5 8 3 4 12 6 9

12. Dijkstra’s Algorithm example20 A B C D E F G 20 4 6 2 5 8 3 4 12 6 9

13. Dijkstra’s Algorithm example20 A B C D E F G 20 4 6 2 5 8 3 4 12 6 8

14. Dijkstra’s Algorithm example20 A B C D E F G 20 4 6 2 5 8 3 4 12 6 8

15. Dijkstra’s Algorithm example20 A B C D E F G 20 4 6 2 5 8 3 4 12 6 8

16. Dijkstra’s Algorithm example20 A B C D E F G 20 4 6 2 5 8 3 4 12 6 8

17. Dijkstra’s Algorithm example20 15 A B C D E F G 20 4 6 2 5 8 3 4 12 6 8

18. Dijkstra’s Algorithm example20 15 A B C D E F G 20 4 6 2 5 8 3 4 12 6 8

19. Dijkstra’s Algorithm example17 15 A B C D E F G 20 4 6 2 5 8 3 4 12 6 8

20. Dijkstra’s Algorithm example17 15 A B C D E F G 20 4 6 2 5 8 3 4 12 6 8

21. Dijkstra’s Algorithm example17 15 A B C D E F G 4 6 2 8 3 4 12 6 8
Shortest Path Tree

22. Dijkstra’s Algorithm Analysis
Running time depends mainly on the associated data structure q q
Array : O(n2)
Binary Heap : O( (n+m) logn )
Fibonacci Heap : O( m + n logn )
What if the edge weights are negative ?

23. Bellman-Ford Algorithm
Bellman­Ford( S ) set dist[u] = INF for all u dist[S] = 0; Repeat (n-1) times for all edges (u,v) in the graph if(dist[v] > dist[u] + weight(u,v) ) dist[v] = dist[u] + graph[u][v]; end-if end-for end-repeat end-Bellman-Ford // Complexity : O( n m )

24. Bellman-Ford Algorithm exampleC D E F G 20 4 6 2 5 8 3

25. All-Pairs Shortest Path
Repeat Single-Source Shortest Path n times
Matrix Exponentiation – Recursive Squaring
Dynamic Programming – Floyd-Warshall

26. Floyd-Warshall Algorithm
Floyd-Warshall set dist[u][v] = INF for all u,v for each edge (u,v) in the graph dist[u][v] = weight(u,v) end-for // Each stage has shortest paths using intermediate vertices (1..k−1) for k: 1 to n for u : 1 to n for v : 1 to n dist[u][v] = minimum( dist[u][v], dist[u][k] + dist[k][v] ) end-for-v end-for-u end-for-k end-floyd-warshall // Complexity : O( n3 )

27. References http://en.wikipedia.org/wiki/Shortest_path_problem
Introduction to Algorithms
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein