5. Consider the modified version of Dijkstra\'s algorithm for finding the shorte

Question

5. Consider the modified version of Dijkstra's algorithm for finding the shortest path between two nodes in a graph with any number of negative edges. Let -r be the largest negative edge weight in the graph, and add x+e to each edge of the graph so that every edge now has a positive weight. We then execute Dijkstra's algorithm as normal, starting from the source node and return the shortest path to any desired node. Would this modification work? Either prove that it does work, or provide a counterexample to show that it doesn't

Explanation / Answer

Solution:

Explanation:

This algorithm will surely work because adding a constant weight to all the edges in the graph won't affect the overall result of the Dijkstra's algorithm, Single source shortest path algorithm will find the shortest path from source to all the other nodes and then we can subtract e from the shortest path computed at all the vertices from the graph which will give us the correct result:

Reason:

The reason for why this will work is that this added implementation will remove the possibility of negative edged weight cycle from the directed graph which means there won't be any chance of infinite computation of less weight because of any negative edged weight cycle. This is why this solution will perfectly fine.

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