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Comment on Undirected SS Shortest Paths with Positive Integer Weights in O(n) (1999) [pdf]

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(requires constant-time multiplication) from https://en.wikipedia.org/wiki/Shortest_path_problem#Single-s...

The $n$ in the title refers to the amount of edges.

You're talking past the parent's point. If the integer weights grow faster than n*, then this algorithm will grow faster than O(n)

* integer weights measured in bit-count; log factors from multiplication time ignored

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