Can Floyd's Algorithm for the Shortest Paths Problem 2 (Algorithm 3.4 ) be modificd to give just the shortest path from a given vertex to another specificd vertex in a graph? Justify your answer.

Floyd's Algorithm computes the shortest paths between all pairs of vertices in a graph. It does so by gradually improving an initial solution (with paths passing through only a subset of vertices) until it finds the final shortest paths using all vertices. The improvement is performed via a sequence of iterations. In the kth iteration, the algorithm finds shortest paths between all pairs of vertices that pass through only the first k vertices.

The requirement is to modify the algorithm to find the shortest path from a specific source vertex to a particular target vertex. To achieve this, we might immediately suggest to stop the algorithm once the shortest path from source to target has been computed. However, this isn't possible because the distances between the vertices change with each iteration and the final shortest distance is only confirmed after all iterations are complete.

Theoretically, Floyd's Algorithm cannot be optimized for a single source-target pair because it's designed and optimized to work with every pair of vertices by involving each vertex progressively. The shortest path for one particular pair of vertices can only be accurately determined after the algorithm has iterated through all the vertices. Therefore, it's not effective to modify Floyd's algorithm to determine the shortest path between two specific vertices, because it would have to run the algorithm fully, regardless of the vertices in question. It means that if a query is for a single source and target, it would be more efficient to use algorithms that are specially intended for single-source shortest path problems, like Dijkstra's algorithm.