1. 1 Advanced Algorithms All-pairs SPs DP algorithm Floyd-Warshall alg.

2. 2 APSP Problem Given a directed graph G = (V, E), weight function w : E → R, |V| = n determine the length of the shortest path between all pairs of vertices in G. In other words, we want to create an n × n matrix of shortest- path distances δ(u, v). Here we assume that there are no cycles with zero or negative cost. Could run BELLMAN-FORD once from each vertex: O(V 2 E) - which is O(V 4 ) if the graph is dense (E = Ө(V 2 )).

3. 3 APSP (cont.) If no negative-weight edges, could run Dijkstra.s algorithm once from each vertex: O(V E lg V) with binary heap - O(V 3 lg V) if dense, O(V 2 lg V + V E) with Fibonacci heap - O(V 3 ) if dense. We will see how to do in O(V 3 ) in all cases, with no fancy data structure.

4. 4 Terminologies G = (V, E) where V are numbered 1, 2,.. n=|V| Input :, n  n matrix Output:, n  n matrix

5. 5 Dynamic Programming Approach Optimal substructure property holds Define subproblem: : minimum weight of any path from vertex i to j that contains at most m edges. Recursive solution Base case : m = 0

6. 6 Recursive Solution cont. Goal: Example:

7. 7 Build DP-table Compute in order = W Go from

8. 8 Pseudo-code cont. Time complexity: Extend: Slow-Apsp: Can we do better ?

9. 9 Matrix Multiplication Extend procedure same as matrix multiplication L’ = L W ! = L (m) W Goal: compute W (n-1) Only need to compute,,, …, W 2 W 1 W (n-1) W 4

10. 10 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

11. 11 Faster Algorithm Time complexity:

12. 12 Floyd-Warshall Algorithm Structure of a shortest path

13. 13 Structure of a shortest path (cont.)

14. 14 Recursive solution Recursive solution: : the weight of shortest path from vertex i to j where all intermediate vertices are from set {1, 2, …, k}. Goal: To compute = D (n)

15. 15 Pseudo-code Time complexity:

16. 16 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

17. 17 Constructing a shortest Path Compute a sequence of matrices Π (0), Π (1), Π (2),…, Π (n), where Π= Π (n).

18. 18 Construct a shortest path Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

19. 19 Thank you.