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Chapter 25: All-Pairs Shortest Paths

Comp 750, Fall 2009. All-pairs SPs - 2. ?Repeated Squaring" Algorithm. A dynamic-programming algorithm.Assume input graph is given by an adjacency matrix.W = (wij)Let dij(m) = minimum weight of any path from vertex i to vertex j, containing at most m edges.dij(0) =dij

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Chapter 25: All-Pairs Shortest Paths

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    1. Comp 750, Fall 2009 All-pairs SPs - 1 Chapter 25: All-Pairs Shortest Paths

    2. Comp 750, Fall 2009 All-pairs SPs - 2 “Repeated Squaring” Algorithm

    3. Comp 750, Fall 2009 All-pairs SPs - 3 “Repeated Squaring” (Continued)

    4. Comp 750, Fall 2009 All-pairs SPs - 4 “Repeated Squaring” and Matrix Mult.

    5. Comp 750, Fall 2009 All-pairs SPs - 5 Improving the Running Time

    6. Comp 750, Fall 2009 All-pairs SPs - 6 Floyd-Warshall Algorithm

    7. Comp 750, Fall 2009 All-pairs SPs - 7 FW (Continued)

    8. Comp 750, Fall 2009 All-pairs SPs - 8 Predecessor Matrix

    9. Comp 750, Fall 2009 All-pairs SPs - 9 Transitive Closure

    10. Comp 750, Fall 2009 All-pairs SPs - 10 Example

    11. Comp 750, Fall 2009 All-pairs SPs - 11 Another O(V3) T.C. Algorithm

    12. Comp 750, Fall 2009 All-pairs SPs - 12 Code

    13. Comp 750, Fall 2009 All-pairs SPs - 13 Johnson’s Algorithm An O(V2 lg V + VE) algorithm. Good for sparse graphs. Uses Dijkstra and Bellman-Ford as subroutines. Basic Idea: Reweight edges to be nonnegative. Then, run Dijkstra’s algorithm once per vertex. Use Bellman-Ford to compute new edge weights w. Must have: For all u and v, a SP from u to v using w is also a SP form u to v using w. For all u and v, w(u,v) is nonnegative.

    14. Comp 750, Fall 2009 All-pairs SPs - 14 A General Result about Reweighting

    15. Comp 750, Fall 2009 All-pairs SPs - 15 Proof of (ii)

    16. Comp 750, Fall 2009 All-pairs SPs - 16 Reweighting in Johnson’s Algorithm

    17. Comp 750, Fall 2009 All-pairs SPs - 17 Code for Johnson’s Algorithm

    18. Comp 750, Fall 2009 All-pairs SPs - 18 Example

    19. Comp 750, Fall 2009 All-pairs SPs - 19 Example

    20. Comp 750, Fall 2009 All-pairs SPs - 20 Example

    21. Comp 750, Fall 2009 All-pairs SPs - 21 Example

    22. Comp 750, Fall 2009 All-pairs SPs - 22 Example

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