1 / 17

Shortest Path Problems:

Shortest Path Problems: . Floyd-Warshall Algorithm for the All-Pairs Shortest Path Problem with Arbitrary Arc Costs Updated 18 February 2008. Floyd-Warshall Algorithm (AMO pg 148). begin for all node pairs [ i , j ]  N x N do d [ i , j ] :=  , pred[ i , j ] := 0;

yahto
Download Presentation

Shortest Path Problems:

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Shortest Path Problems: Floyd-Warshall Algorithm for the All-Pairs Shortest Path Problem with Arbitrary Arc Costs Updated 18 February 2008

  2. Floyd-Warshall Algorithm (AMO pg 148) begin for all node pairs [i,j] N x N do d[i,j] := , pred[i,j] := 0; for all nodes i N do d[i,i] := 0; for all arcs (i, j) A do d[i, j] := cij and pred[i, j] := i; for k = 1 to n do for all node pairs [i, j] N x N do if d[i, j] > d[i, k] + d[k, j]then d[i, j] := d[i, k] + d[k, j]and pred[i, j] := pred[k, j]; end;

  3. Interpretation of d and pred Matrices • At the end of iteration k, d[i, j] is the length of a shortest path from i to j that uses only nodes in the set {1, 2, …, k} as internal nodes. • pred[i, j] is the node prior to node j on the (current) shortest path from i to j.

  4. The Triangle Operation: Iteration k i k j Check if d[i, j] > d[i, k] + d[k, j]

  5. The Triangle Operation: Update pred[i,j] k i pred[k, j] old pred[i, j] j new pred[i, j] = pred[k, j]

  6. Floyd-Warshall (FW) Example 1 12 1 2 1 2 1 4 3 4 3

  7. FW Example 1: End of Iteration 1 12 1 2 1 2 1 4 3 4 3

  8. FW Example 1: End of Iteration 2 12 1 2 1 2 1 4 3 4 3

  9. FW Example 1: End of Iteration 3 12 1 2 1 2 1 4 3 4 3

  10. FW Example 1: Solution 12 1 2 1 2 1 4 3 4 3

  11. FW Example 1: Shortest Path from 1 to 2 12 1 2 1 2 1 4 3 4 3 pred[1,2] = 3 pred[1,3] = 4 pred[1,4] = 1

  12. Complexity of Floyd-Warshall • Each triangle operation is O(1) • Each iteration does n2 triangle operations • There are n iterations • Complexity is O(n3)

  13. Testing for Negative-Cost Cycles for k = 1 to n do for all node pairs [i, j] N x N do if d[i, j] > d[i, k] + d[k, j]then begin d[i, j] := d[i, k] + d[k, j]and pred[i, j] := pred[k, j]; if i = j and d[i, i] < 0 then exit; (G has a negative-cost cycle) end;

  14. 1 2 -4 1 1 3 4 3 FW Example 2 (From Papadimitriou and Steiglitz) 2

  15. 1 2 -4 1 1 3 4 3 FW Example 2 (after k = 1) 2

  16. 1 2 -4 1 1 3 4 3 FW Example 2 (after k = 2) 2

  17. FW Example 2: d[4, 4] = 1 2 1 2 -4 1 1 3 4 3 pred[4,4] = 1 pred[4,1] = 2 pred[4,2] = 4

More Related