Shortest Paths that is strictly increasing

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I have a directed weighted graph without negative cycles ( but maybe with negative weights) and a starting vertex s. I know that for every vertex v,the weights of the edges along the shortest path from s to v are strictly increasing. I need to find an efficient way to compute the shortest path from s to every vertex.

I'm kind of stuck on this. I'm think I need to relax the edges along the shortest path but not sure how to do that. Hope someone can help

Thanks!

Ron

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