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My presentation all shortestpath
- 1. All-Pairs Shortest Path Theory and Algorithms Carlos Andres Theran Suarez Program Mathematics and Scientific Computing University of Puerto Rico Carlos.theran@upr.edu October – 2011 Mayaguez-Puerto Rico Dr Marko Schütz
- 2. Introduction • In thissection we consider de problem of finding shortest path between all pair of vertices in a directed graph = (, ). – With a weight function : → ℝ = − ( , + ) where ∈ ∈ ℕ. = For this goal we are going to use the adjacency-matrix of a graph. • The input: is a = which is adjacency-matrix of a = , . • The output: is the matrix of SPL , , ∀ ∈ .
- 3. Recall • Single-Source shortestpaths. • Nonnegative edge weight. – Dijkstra’s algorithm: Running time Array Running time Binary heap ( + log ) Running time Fibonacci heap ( + log ) • General. – Bellman-Ford: Running time • UDG. – Breadth for search: Running time +
- 4. What do youthink? Can we solve all-pair shortest paths by running a single source-paths algorithms?
- 5. What do youthink? Can we solve all-pair shortest paths by running a single source-paths algorithms? All-pair shortest paths. • Nonnegative edge weight Dijkstra’s algorithm: Running time Array
- 6. What do youthink? Can we solve all-pair shortest paths by running a single source-paths algorithms? All-pair shortest paths. • Nonnegative edge weight Dijkstra’s algorithm: Running time Array Running time Binary heap ( + log )
- 7. What do youthink? Can we solve all-pair shortest paths by running a single source-paths algorithms? All-pair shortest paths. • Nonnegative edge weight Dijkstra’s algorithm: Running time Array Running time Binary heap ( + log ) Running time Binary heap + log
- 8. What do youthink? Can we solve all-pair shortest paths by running a single source-paths algorithms? All-pair shortest paths. • Nonnegative edge weight Dijkstra’s algorithm: Running time Array Running time Binary heap ( + log ) Running time Binary heap + log • General. Bellman-Ford: Running time
- 9. What do youthink? Can we solve all-pair shortest paths by running a single source-paths algorithms? All-pair shortest paths. • Nonnegative edge weight Dijkstra’s algorithm: Running time Array Running time Binary heap ( + log ) Running time Binary heap + log • General. Bellman-Ford: Running time In a dense graph
- 10. Predecessor Matrix Let = the predecessor Matrix, where = = . . Now we define the predecessor subgraph of G for as , = (, , , ). Where , = ∈ : ≠ ∪ , = , : ∈ , − {}
- 11.
- 12. Outline 1. Present adynamic programming algorithms based on matrix multiplication to solve the problem. 2. Dynamic programming algorithms called Floyd-Warshall algorithms. 3. Unlike the others algorithms, Johnson's algorithms used adjacency-list representation of a graph.
- 13. Shortest path andmatrix multiplication 1. The structure of shortest path. Let suppose that we have a shortest path form vertix to vertex , and suppose that have at most < ∞ edge. • If = then have weight 0. • If ≠ then ↝′ → , where ′ has at most − edge, by lemma 24.1 ′ is a shortest path from . So , = , + .
- 14. Shortest path andmatrix multiplication (cont.) 2. A recursive solution. Let () be the minimum weight of any path from vertex to vertex that contains at most edge. () = • = ∞ ≠ • () = min( − , min (−) + ) 1≤≤ (−) = min + 1≤≤ , = (−) = () = (+) = ⋯
- 15. Shortest path andmatrix multiplication (cont.) 3. Computing shortest-path weight bottom up Input = ( ). We compute , , … , − . = ( () ) = , , … , − .
- 16. Shortest path andmatrix multiplication (cont.) • Now we can see the relation to the matrix multiplication. Let = ∗ the matrix product of . For , = , … , . We have = ∗ . = If we set; (−) → → → → + +→∗ ← + ∗
- 17. Shortest path andmatrix multiplication (cont.) Computing the sequence of − matrix () = () ∗ = () = () ∗ = ⋮ (−) = (−) ∗ = −
- 18. Shortest path andmatrix multiplication (cont.) • Improving the running time. Our goal, is to compute − matrices, let go to see that we can compute − with only log( − ) matrix product.
- 19. Shortest path andmatrix multiplication (cont.) () = () = () = ∗ () = () = ∗ () = () = ∗ ⋮ log − ) log − ) log − − log − − ( = ( = ∗
- 20. Shortest path andmatrix multiplication (cont.)
- 21. The Floyd-Warshall algorithm Thealgorithm consider a intermediate vertices of a shortest path. 1. The structure of a shortest path. Intermediate vertex =< , , … , > in a any vertex of other than or , so it can be the set , … , − . Let assume that the vertex of are = , , … , and a subset , , … , for some . • If ∉ of path , then all the vertices intermediate are in the set , , … , − . Thus, a shortest path from vertex to vertex with all intermediate vertices in the set , , … , − is also a shortest path form to with all in the set , , … , .
- 22. The Floyd-Warshall algorithm(cont) • If ∈ of path , we break down into ↝ ↝ . is a shortest path from to , so ∉ of , thus is a shortest path form to with all in the set , , … , − . Similarly is a shortest path form to with all in the set , , … , − .
- 23. The Floyd-Warshall algorithm(cont) 3. A recursive solution. = () = − − − min , + ≥ Since for every path, all intermediate vertices are in the set , , … , , matriz () = ( () ) gives the final answer: () = , .
- 24. The Floyd-Warshall algorithm(cont) • input: A matrix • output: A matrix () of shortest path weight. • () = min − , − + −
- 25.
- 26. The Floyd-Warshall algorithm(cont) 4. Constructing a Shortest path We compute the predecessor matrix just as the Floyd-warshall algorithm compute the matrices () . so = () = ( () ). Recursive formulation. − − − () ≤ + () = − − − () > + () = = ∞. = ≠ < ∞.
- 27. Johnson's algorithm forsparse graphs. • It is asymtoticaly better than repeated squaring of matrices or the Floyd-Warshall algoritm. • It use a subroutine both Dijkstra’s algorithm and Bellman- Ford algorithm. • Johnson's algorithm use the technique of reweighting.
- 28. Johnson's algorithm forsparse graphs (cont.). Reweighting If has a negative weight edge but no negative weight cycle, we compute a new set of nonnegative edge weight that allow as to use Dijkstra’s algorithm. The new set of edge must satisfy two condition. 1. () is a shortest path form ⇔ () is a shortest path form . 2. For all edges (, ), the new weight (, ) is nonnegative.
- 29. Johnson's algorithm forsparse graphs (cont.). • Lemma Give a weighted, directed graph = (, ) with weight funtion : → ℝ be any funtion mapping vertices to real numbers. For each edge (, ) ∈ , define. , = , + − .
- 30. Johnson's algorithm forsparse graphs (cont.). • Producing no negative weight by reweighting
- 31. Johnson's algorithm forsparse graphs (cont.).
- 32. Johnson's algorithm forsparse graphs (cont.).