All-Pairs Shortest Path         Theory and Algorithms   Carlos Andres Theran SuarezProgram Mathematics and Scientific Comp...
Introduction
Recall
What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
Predecessor Matrix
Predecessor Matrix
Outline1. Present a dynamic programming algorithms based on   matrix multiplication to solve the problem.2. Dynamic progra...
Shortest path and matrix multiplication
Shortest path and matrix multiplication (cont.)
Shortest path and matrix multiplication (cont.)
Shortest path and matrix multiplication (cont.)
Shortest path and matrix multiplication (cont.)
Shortest path and matrix multiplication (cont.)
Shortest path and matrix multiplication (cont.)
Shortest path and matrix multiplication (cont.)
The Floyd-Warshall algorithm
The Floyd-Warshall algorithm (cont)
The Floyd-Warshall algorithm (cont)
The Floyd-Warshall algorithm (cont)
The Floyd-Warshall algorithm (cont)
The Floyd-Warshall algorithm (cont)
Johnsons algorithm for sparse graphs.• It is asymtoticaly better than repeated squaring of matrices  or the Floyd-Warshall...
Johnsons algorithm for sparse graphs (cont.).
Johnsons algorithm for sparse graphs (cont.).
Johnsons algorithm for sparse graphs (cont.).• Producing no negative weight by reweighting
Johnsons algorithm for sparse graphs (cont.).
Johnsons algorithm for sparse graphs (cont.).
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My presentation all shortestpath

  1. 1. All-Pairs Shortest Path Theory and Algorithms Carlos Andres Theran SuarezProgram Mathematics and Scientific Computing University of Puerto Rico Carlos.theran@upr.edu October – 2011 Mayaguez-Puerto Rico Dr Marko Schütz
  2. 2. Introduction
  3. 3. Recall
  4. 4. What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
  5. 5. What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
  6. 6. What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
  7. 7. What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
  8. 8. What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
  9. 9. What do you think?Can we solve all-pair shortest paths by running asingle source-paths algorithms?
  10. 10. Predecessor Matrix
  11. 11. Predecessor Matrix
  12. 12. Outline1. Present a dynamic programming algorithms based on matrix multiplication to solve the problem.2. Dynamic programming algorithms called Floyd-Warshall algorithms.3. Unlike the others algorithms, Johnsons algorithms used adjacency-list representation of a graph.
  13. 13. Shortest path and matrix multiplication
  14. 14. Shortest path and matrix multiplication (cont.)
  15. 15. Shortest path and matrix multiplication (cont.)
  16. 16. Shortest path and matrix multiplication (cont.)
  17. 17. Shortest path and matrix multiplication (cont.)
  18. 18. Shortest path and matrix multiplication (cont.)
  19. 19. Shortest path and matrix multiplication (cont.)
  20. 20. Shortest path and matrix multiplication (cont.)
  21. 21. The Floyd-Warshall algorithm
  22. 22. The Floyd-Warshall algorithm (cont)
  23. 23. The Floyd-Warshall algorithm (cont)
  24. 24. The Floyd-Warshall algorithm (cont)
  25. 25. The Floyd-Warshall algorithm (cont)
  26. 26. The Floyd-Warshall algorithm (cont)
  27. 27. Johnsons algorithm for sparse 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.• Johnsons algorithm use the technique of reweighting.
  28. 28. Johnsons algorithm for sparse graphs (cont.).
  29. 29. Johnsons algorithm for sparse graphs (cont.).
  30. 30. Johnsons algorithm for sparse graphs (cont.).• Producing no negative weight by reweighting
  31. 31. Johnsons algorithm for sparse graphs (cont.).
  32. 32. Johnsons algorithm for sparse graphs (cont.).
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