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Floyd Warshall Algorithm

Overview on Floyd-Warshall Algorithm with procedure to finding shortest path.

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Floyd Warshall Algorithm

  1. 1. WELCOME TO My SESSION Presented By: MD. Saidur Rahman Kohinoor DIU Student E-mail: saidur95@gmail.com Social Network: www.fb.com/kohinoor11
  2. 2. Presentation Topic: Floyd Warshall’s Algorithm
  3. 3. Floyd Warshall Algorithm - what? An example of dynamic programming An algorithm for finding shortest paths in a weighted graph with positive or negative edge weights no negative cycles find the lengths of the shortest paths between all pairs of vertices
  4. 4. History and naming - how?  Bernard Roy in 1959 Robert Floyd in 1962 Stephen Warshall in 1962 Peter Ingerman in 1962
  5. 5. The algorithm is also known as History and naming - how? The Floyd's algorithm  the Roy–Warshall algorithm  the Roy–Floyd algorithm, or  the WFI algorithm The Floyd's algorithm  the Roy–Warshall algorithm  the Roy–Floyd algorithm, or  the WFI algorithm
  6. 6. Shortest paths – mean? Path 1: A -> B -> D = 7 Path 2: A -> C -> D = 7 Path 3: A -> B -> C -> D = 6 There are several paths between A and D: 5 4 312
  7. 7. There are several things to notice here: There can be more then one route between two nodes. The number of nodes in the route isn’t important (Path 3 has 4 nodes but is shorter than Path 1 or 2, which has 3 nodes). There can be more than one path of minimal length. Shortest paths – mean?
  8. 8. Floyd Warshall Algorithm- programs Distance Table Sequence Table Iteration is N-1 here, N= number of node = 4 so, 4-1 = 3 iteration. According to this algorithm, we need-
  9. 9. Distance Table by D0, D1, D2, ……. ,Dn Sequence Table by S0, S1, S2,……. ,Sn Iteration by K Here we denoted- Floyd Warshall Algorithm- programs
  10. 10. D0 A B C D A - 2 4 B 2 - 1 5 C 4 1 - 3 D 5 3 - S0 A B C D A - 2 3 4 B 1 - 3 4 C 1 2 - 4 D 1 2 3 - Iteration = 0 K = 0 All Diagonal = null Floyd Warshall Algorithm- programs
  11. 11. D1 A B C D A - 2 4 B 2 - 1 5 C 4 1 - 3 D 5 3 - S1 A B C D A - 2 3 4 B 1 - 3 4 C 1 2 - 4 D 1 2 3 - 1st row unchanged 1st Colum unchanged Iteration = 1 K = 1 if (dij > dik + dkj ) D1(ij) = dik+dkj else D1(ij) = dij Floyd Warshall Algorithm- programs
  12. 12. D2 A B C D A - 2 3 B 2 - 1 5 C 3 1 - 3 D 5 3 - S2 A B C D A - 2 2 4 B 1 - 3 4 C 2 2 - 4 D 1 2 3 - Iteration = 2 K = 2 2nd row unchanged 2nd Colum unchanged if (dij > dik + dkj ) D1(ij) = dik+dkj else D1(ij) = dij Floyd Warshall Algorithm- programs
  13. 13. D3 A B C D A - 2 3 6 B 2 - 1 4 C 3 1 - 3 D 6 4 3 - S3 A B C D A - 2 2 3 B 1 - 3 3 C 2 2 - 4 D 3 3 3 - Iteration = 3 K = 3 3rd row unchanged 3rd Colum unchanged if (dij > dik + dkj ) D1(ij) = dik+dkj else D1(ij) = dij Floyd Warshall Algorithm- programs
  14. 14. Shortest Path A B C D A - 2 3 6 B 2 - 1 4 C 3 1 - 3 D 6 4 3 - A B C D A - 2 2 3 B 1 - 3 3 C 2 2 - 4 D 3 3 3 - A >> C i=1, j=3 Distance: d13 = 3 Path: S13 = 2 A >> B >> C S12 = 2 A >> B >> C 2+1 = 3
  15. 15. A B C D A - 2 3 6 B 2 - 1 4 C 3 1 - 3 D 6 4 3 - A B C D A - 2 2 3 B 1 - 3 3 C 2 2 - 4 D 3 3 3 - A >> D i=1, j=4 Distance: d14 = 6 Path: S14 = 3 A >> C >> D S13 = 2 A >> B >> C >> D S12 = 2 A >> B >> C >> D Shortest Path
  16. 16.  The running time is O(n3 ).  The space requirements are O(n2 ) 16 Time and Space Requirements
  17. 17. Shortest paths in directed graphs Transitive closure of directed graphs. Inversion of real matrices Optimal routing. Maximum bandwidth paths Computing canonical form of difference bound matrices Applications and generalizations
  18. 18. My Complete Code C Programming http://pastebin.com/s3vBx3KD
  19. 19. References https://en.wikipedia.org/wiki/Floyd %E2%80%93Warshall_algorithm https://compprog.wordpress.com/200 7/11/15/all-sources-shortest-path-the- floyd-warshall-algorithm/
  20. 20. Thanks to All

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