Introduction to Algorithms

1. Introduction to Algorithms All-Pairs Shortest Paths My T. Thai @ UF

2. Single-Source Shortest Paths Problem • Input: A weighted, directed graph G = (V, E) • Output: An n × n matrix of shortest-path distances δ. δ(i, j) is the weight of a shortest path from i to j. My T. Thai mythai@cise.ufl.edu

3. Can use algorithms for Single-Source Shortest Paths • Run BELLMAN-FORD once from each vertex • Time: • If there are no negative-weight edges, could run Dijkstra’s algorithm once from each vertex • Time: My T. Thai mythai@cise.ufl.edu

4. Outline • Shortest paths and matrix multiplication • Floyd-Warshall algorithm • Johnson’s algorithm My T. Thai mythai@cise.ufl.edu

5. Recursive solution • Optimal substructure: subpaths of shortest paths are shortest paths • Recursive solution: Let = weight of shortest path from i to j that contains ≤ m edges. Where wij: My T. Thai mythai@cise.ufl.edu

6. Computing the shortest-path weights bottom up • All simple shortest paths contain ≤ n − 1 edges • Compute from bottom up: L(1), L(2), . . . , L(n-1). • Compute L(i+1) from L(i)by extending one more edge Time: My T. Thai mythai@cise.ufl.edu

7. Time: My T. Thai mythai@cise.ufl.edu

8. Shortest paths and matrix multiplication • Extending shortest paths by one more edge likes matrix product: L(i+1)= L(i).W • Compute L(1), L(2), L(4) . . . , L(r)with Time: My T. Thai mythai@cise.ufl.edu

9. Example My T. Thai mythai@cise.ufl.edu

10. Floyd-Warshall algorithm • For path p = <v1, v2, . . . , vl> , v2 … vl-1areintermediate vertices fromv1to vl • Define = shortest-path weight of any path from i to j with all intermediate vertices in {1, 2, . . . , k} • Consider a shortest path with all intermediate vertices in{1, 2, . . . , k}: • If k is not an intermediate vertex, all intermediate vertices in{1, 2, . . . , k -1} • If k is an intermediate vertex: mythai@cise.ufl.edu

11. Floyd-Warshall algorithm • Recursive formula: • Time: Note:since we have at most n vertices, return My T. Thai mythai@cise.ufl.edu

12. Constructing a shortest path • is the predecessor of vertex j on a shortest path from vertex i with all intermediate vertices in the set {1, 2, . . . , k} (Use vertex k) My T. Thai mythai@cise.ufl.edu

13. Example My T. Thai mythai@cise.ufl.edu

14. My T. Thai mythai@cise.ufl.edu

15. My T. Thai mythai@cise.ufl.edu

16. Johnson’s algorithm • Reweighting edges to get non-negative weight edges: • For all u, v ∈ V, p is a shortest path using w if and only if p is a shortest path using • For all (u, v) ∈ E, • Run Dijkstra’s algorithm once from each vertex My T. Thai mythai@cise.ufl.edu

17. Reweighting My T. Thai mythai@cise.ufl.edu

18. Proof of lemma 25.1 • First, we prove • With cycle , My T. Thai mythai@cise.ufl.edu

19. Producing nonnegative weights • Construct • Since no edges enter s, has the same set of cycles as G • has a negative-weight cycle if and only if G does • Define: • Claim: • Proof: Triangle inequality of shortest paths My T. Thai mythai@cise.ufl.edu

20. Johnson’s algorithm Time: My T. Thai mythai@cise.ufl.edu

21. Example My T. Thai mythai@cise.ufl.edu

22. My T. Thai mythai@cise.ufl.edu

23. Summary • Dynamic-programming algorithm based on matrix multiplication • Define sub-optimal solutions based on the length of paths • Use the technique of “repeated squaring” • Time: • Floyd-Warshall algorithm • Define sub-optimal solutions based on the set of allowed intermediate vertices • Time: My T. Thai mythai@cise.ufl.edu

24. Johnson’s algorithm • Reweight edges to non-negative weight edges • Run Dijkstra’s algorithm once from each vertex • Time: • Faster than Floyd-Warshall algorithm when the graph is dense E = o(V2) My T. Thai mythai@cise.ufl.edu