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

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Let n be |V|, the number of vertices. To find all n2 of shortestPath(i,j,k) (for all i and j) from those of shortestPath(i,j,k−1) requires 2n2 operations. Since we begin with shortestPath(i,j,0) = edgeCost(i,j) and compute the sequence of n matrices shortestPath(i,j,1), shortestPath(i,j,2), …, shortestPath(i,j,n), the total number of operations used is n · 2n2 = 2n3. Therefore, the complexity of the algorithm is Θ(n3).

In mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R+ on set X such that R+ contains R and R+ is minimal

- 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. Presentation Topic: Floyd Warshall’s Algorithm
- 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. History and naming - how? Bernard Roy in 1959 Robert Floyd in 1962 Stephen Warshall in 1962 Peter Ingerman in 1962
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. The running time is O(n3 ). The space requirements are O(n2 ) 16 Time and Space Requirements
- 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. My Complete Code C Programming http://pastebin.com/s3vBx3KD
- 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. Thanks to All

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