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Instructions The second project involves benchmarking the behavior of the Floyd-Warshall's algorithm on directed weighted graphs. You do not have to write the algorithm yourself, you may take it from some source, but you must reference your source. You are to benchmark the algorithm in two different ways, by counting critical operations and by measuring the actual run time. You are to write driver code to determine their efficiency based on these two measurements. Use all of the following examples to complete your assignment: 5 5 14 10 20 4 10 5 20 30 2 5 15 1 10 10 C 25 11 9 7 5 3 5 20 2) 15 5 15 2 17 8 15 5 10 20 - 9 10 12 * (10) Figane 4 - 2 2 A 8 B) c 4 1 3/ 10 10 . 2 - , 5 D 9 E F 2 G 8/ 4 6 6 6 H 8 O C You are to submit a paper written with Microsoft Word that discusses the results of your analysis. It should include the following: A brief introduction of the algorithm(s) that you have selected and how the algorithm(s compare, if any A discussion of the critical operation that you chose to count with an explanation of why you selected it A Big-p analysis of the algorithm(s) used. Make sure that critical operation count is included with this discussion. A discussion of the results of your study, which should include graphs of your results, and discussion of how your results compare to your Big-O analysis Comprehensive Documentation containing Approach, Lessons Learned, and Possible Improvements sub-sections A conclusion that summarizes the important observations of your study

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Brief Introduction of the Used Algorithm

The algorithm used for finding all source shortest path pairs in this project is based on Floyd-Warshall’s method. Among the most-known applications can be highlighted the computation of the transitive closure of a relation, inversion of matrices and optimal routing between any pair of nodes (in directed and weighted graphs with both positive and negative costs).
The present algorithm is used to discover the shortest paths from all the source nodes to the rest of the vertices of the graph. A strong point is represented by the fact that it is usable in a weighted directed graph; although it performs slower than classical Dijkstra’s algorithm (since it also performs more computations), it remains a valuable application in the field.
Floyd-Warshall’s algorithm compares the paths between any pair of vertices and updates the shortest distances progressively. It can also be seen recursively as below:
Shortest_Path(i,j,k+1) =MINIMUM (Shortest_Path(i,j,k), Shortest_Path (i,k+1,k) +Shortest_Path(k+1,j,k)), where the base case is represented by the following formulation: Shortest_Path(i,j,0)=Weight(i,j). This assertion is also applied within the selected algorithm....
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