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![Floyd-Warshall Algorithm
Consider a graph, G = {V, E} where V is the set of all vertices present in the graph and E is
the set of all the edges in the graph. The graph, G, is represented in the form of an
adjacency matrix, A, that contains all the weights of every edge connecting two vertices.
Algorithm:
Step 1 − Construct an adjacency matrix A with all the costs of edges present in the graph.
If there is no path between two vertices, mark the value as ∞.
Step 2 − Derive another adjacency matrix A1 from A keeping the first row and first
column of the original adjacency matrix intact in A1. And for the remaining values, say
A1[i,j], if A[i,j]>A[i,k]+A[k,j] then replace A1[i,j] with A[i,k]+A[k,j]. Otherwise, do not
change the values. Here, in this step, k = 1 (first vertex acting as pivot).
Step 3 − Repeat Step 2 for all the vertices in the graph by changing the k value for every
pivot vertex until the final matrix is achieved.
Step 4 − The final adjacency matrix obtained is the final solution with all the shortest
paths.
2](https://image.slidesharecdn.com/floydalgorithminnetworkanalysis-250810144157-5ac3983c/75/Floyd-algorithm-in-network-analysis-pptx-4-2048.jpg)
![solution
Step 2:
Considering the above adjacency matrix as the input, derive another matrix
A0 by keeping only first rows and columns intact. Take k = 1, and replace
all the other values by A[i,k]+A[k,j].
3](https://image.slidesharecdn.com/floydalgorithminnetworkanalysis-250810144157-5ac3983c/75/Floyd-algorithm-in-network-analysis-pptx-7-2048.jpg)
![solution
Step 3:
Considering the above adjacency matrix as the input, derive another matrix
A0 by keeping only first rows and columns intact. Take k = 1, and replace
all the other values by A[i,k]+A[k,j].
3](https://image.slidesharecdn.com/floydalgorithminnetworkanalysis-250810144157-5ac3983c/75/Floyd-algorithm-in-network-analysis-pptx-8-2048.jpg)
![solution
Step 4:
Considering the above adjacency matrix as the input, derive another matrix
A0 by keeping only first rows and columns intact. Take k = 1, and replace
all the other values by A[i,k]+A[k,j].
3](https://image.slidesharecdn.com/floydalgorithminnetworkanalysis-250810144157-5ac3983c/75/Floyd-algorithm-in-network-analysis-pptx-9-2048.jpg)
![solution
Step 5:
Considering the above adjacency matrix as the input, derive another matrix
A0 by keeping only first rows and columns intact. Take k = 1, and replace
all the other values by A[i,k]+A[k,j].
3](https://image.slidesharecdn.com/floydalgorithminnetworkanalysis-250810144157-5ac3983c/75/Floyd-algorithm-in-network-analysis-pptx-10-2048.jpg)
![solution
Step 6:
Considering the above adjacency matrix as the input, derive another matrix
A0 by keeping only first rows and columns intact. Take k = 1, and replace
all the other values by A[i,k]+A[k,j].
3](https://image.slidesharecdn.com/floydalgorithminnetworkanalysis-250810144157-5ac3983c/75/Floyd-algorithm-in-network-analysis-pptx-11-2048.jpg)
Floyd algorithm in network analysis.pptx
- 1. Floyd algorithm innetwork analysis Prepared By: Bimal Ghosh CSE 4th Year (7th Sem) Roll No: 25700123014 Paper Code: OECCS701A
- 2. Agenda What is theFloyd Warshall Algorithm? 1 Algorithm 2 Example & solution 3 Applications of Floyd Warshall Algorithm 4 Advantages & Disadvantages of Floyd Warshall Algorithm 5
- 3. What is theFloyd Warshall Algorithm? • The Floyd Warshall algorithm is a graph analysis algorithm that can be used to find the shortest paths between all pairs of nodes in a graph. It is named after Robert Floyd, who published the algorithm in 1962. • The Floyd-Warshall algorithm is designed to find the shortest paths between every pair of nodes in a graph, making it a powerful tool for network analysis. The algorithm works by systematically considering all possible paths through intermediate nodes and updating the shortest path estimates accordingly. 3 1
- 4. Floyd-Warshall Algorithm Consider agraph, G = {V, E} where V is the set of all vertices present in the graph and E is the set of all the edges in the graph. The graph, G, is represented in the form of an adjacency matrix, A, that contains all the weights of every edge connecting two vertices. Algorithm: Step 1 − Construct an adjacency matrix A with all the costs of edges present in the graph. If there is no path between two vertices, mark the value as ∞. Step 2 − Derive another adjacency matrix A1 from A keeping the first row and first column of the original adjacency matrix intact in A1. And for the remaining values, say A1[i,j], if A[i,j]>A[i,k]+A[k,j] then replace A1[i,j] with A[i,k]+A[k,j]. Otherwise, do not change the values. Here, in this step, k = 1 (first vertex acting as pivot). Step 3 − Repeat Step 2 for all the vertices in the graph by changing the k value for every pivot vertex until the final matrix is achieved. Step 4 − The final adjacency matrix obtained is the final solution with all the shortest paths. 2
- 5. Example Consider the followingdirected weighted graph G = {V, E}. Find the shortest paths between all the vertices of the graphs using the Floyd- Warshall algorithm. 3
- 6. solution Step 1: Construct anadjacency matrix A with all the distances as values. 3
- 7. solution Step 2: Considering theabove adjacency matrix as the input, derive another matrix A0 by keeping only first rows and columns intact. Take k = 1, and replace all the other values by A[i,k]+A[k,j]. 3
- 8. solution Step 3: Considering theabove adjacency matrix as the input, derive another matrix A0 by keeping only first rows and columns intact. Take k = 1, and replace all the other values by A[i,k]+A[k,j]. 3
- 9. solution Step 4: Considering theabove adjacency matrix as the input, derive another matrix A0 by keeping only first rows and columns intact. Take k = 1, and replace all the other values by A[i,k]+A[k,j]. 3
- 10. solution Step 5: Considering theabove adjacency matrix as the input, derive another matrix A0 by keeping only first rows and columns intact. Take k = 1, and replace all the other values by A[i,k]+A[k,j]. 3
- 11. solution Step 6: Considering theabove adjacency matrix as the input, derive another matrix A0 by keeping only first rows and columns intact. Take k = 1, and replace all the other values by A[i,k]+A[k,j]. 3
- 12.
- 13. Applications of FloydWarshall Algorithm There are many applications of the Floyd Warshall algorithm. Some of the most popular applications are finding the shortest path between two vertices in a graph, detecting negative cycles in a graph, and computing the transitive closure of a graph. The Floyd Warshall algorithm can also be used for other purposes such as solving the all-pairs shortest path problem in weighted graphs, finding the closest pairs of vertices in a graph, and computing the diameter of a graph. 4
- 14. Advantages & Disadvantagesof Floyd Warshall Algorithm Advantages: One of the biggest advantages of the Floyd Warshall algorithm is its versatility. The algorithm can be used to solve a wide range of problems, including finding the shortest path between two nodes in a graph, calculating the transitive closure of a graph, and detecting negative cycles in a graph. Another advantage is its simplicity. Unlike some other algorithms, the Floyd Warshall algorithm is relatively easy to understand and implement. This makes it an ideal choice for students and professionals who are just starting out with graph algorithms. The Floyd Warshall algorithm is very efficient. It has a time complexity of O(n^3), which means it can handle large graphs with ease. Additionally, the algorithm is parallelisable, meaning it can be run on multiple processors to further improve its efficiency. 5
- 15. Resources https://byjus.com/gate/floyd-warshall-algorithm-notes/ https://www.geeksforgeeks.org/dsa/floyd-warshall-algorith m-in-python/ https://www.tutorialspoint.com/data_structures_algorithms /floyd_warshall_algorithm.htm 5
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