Assume a weighted undirected graph on which we apply Dijkstra's algorithm to find the shortest path between two nodes. If we add 1 to all the edge weights, the shortest path between the two nodes remains the same.

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The question asks about the effect of adding 1 to every edge weight in a weighted undirected graph when running Dijkstra's algorithm to find the shortest path between two nodes. The statement to evaluate is: 'If we add 1 to all the edge weights, the shortest path between the two nodes remains the same.'

First, consider what happens to the length of a given path when we increment every edge by 1. If a path uses k edges, its new length becomes the original length plus k (since each of the k edges contributes an extra 1). Different candidate paths from the source to the destination can have different numbers of edges, so the total added length is not the same for all paths; it depends on how many edges each path contains.

Because the extra distance introduced by the uniform +1 per edge is proportional to the number of edges on the path, comparing two paths with different edge counts can yield a different ordering of their total lengths after the transformation. In particular, a path that originally had more edges could become less favorable after the shift, while a path with fewer edges could become relatively shorter, or vice versa, depending on the original weights.

Consider a simple counterexample to see the issue clearly: suppose there are two s–t paths. Path A is a direct edge with weight 10 (one edge). Path B goes through one intermediate node and has total weight 9 (two edges totaling 9). Before the change, Path B is shorter (9 vs 10). After adding 1 to every edge (each of Path A's 1 edge and Path B's 2 edges increases by 1): Path A becomes 10 + 1 = 11, Path B becomes 9 + 2 = 11. In this particular case, they tie, but with other weights, the ordering can change, and the originally shortest path can cease to be the shortest.

Generalizing, since the path length after the modification equals the original path length plus (#edges in that path), the comparison between any two paths depends on the difference in their edge counts. If two candidate shortest paths have different numbers of edges, adding a constant to every edge can alter which path minimizes the total length. Therefore, the statement is not guaranteed to hold in general.

Thus, the claim is not universally true; there exist graphs where adding 1 to all edge weights changes the shortest path, making the statement false. Note that if all s–t paths happened to have the same number of edges, the relative ordering might be preserved for some cases, but that is a special condition, not a general rule.

CPSC_V 320 201/202/203 2024W2 Reading Quiz #7 (Greedy Algorithms 2)

Assume a weighted undirected graph on which we apply Dijkstra's algorithm to find the shortest path between two nodes. If we add 1 to all the edge weights, the shortest path between the two nodes remains the same.

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