Suppose P is the shortest path from a starting node s to a destination node v in a graph G. Suppose also that G' is G with all edge weights modified by adding a constant factor C > 0.Then P (with the corresponding edge weight modifications) is also the shortest path between s and v in the modified graph G'.
Question
Suppose P is the shortest path from a starting node s to a destination node v in a graph G. Suppose also that G' is G with all edge weights modified by adding a constant factor C > 0.Then P (with the corresponding edge weight modifications) is also the shortest path between s and v in the modified graph G'.
Solution
The statement is true. The shortest path in graph G remains the shortest path in the modified graph G' even after adding a constant factor C > 0 to all edge weights.
Here's why:
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Let's denote the weight of an edge e in G as w(e) and in G' as w'(e). Then for every edge e in G, we have w'(e) = w(e) + C.
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Suppose P is the shortest path from s to v in G, and let's denote the weight of path P in G as W(P) and in G' as W'(P). Then W'(P) = W(P) + kC, where k is the number of edges in P.
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Now, let's consider any other path P' from s to v in G. The weight of P' in G' is W'(P') = W(P') + k'C, where k' is the number of edges in P'. Since P is the shortest path in G, we have W(P) ≤ W(P').
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Therefore, W'(P) = W(P) + kC ≤ W(P') + kC = W'(P') for any path P' from s to v in G'. This means that P is also the shortest path from s to v in G'.
So, adding a constant factor to all edge weights does not change the shortest path between any pair of nodes.
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