All-Hops Shortest Paths

Let G = (V, E, w) be a weighted directed graph without negative cycles. For two vertices s, t ∈ V , we let d_≤h(s, t) be the minimum, according to the weight function w, of a path from s to t that uses at most h edges, or hops. We consider algorithms for computing d_≤h(s, t) for every 1 ≤ h ≤ n, where n = |V |, in various settings. We consider the single-pair, single-source and all-pairs versions of the problem. We also consider a distance oracle version of the problem in which we are not required to explicitly compute all distances d_≤h(s, t), but rather return each one of these distances upon request. We consider both the case in which the edge weights are arbitrary, and in which they are small integers in the range {−M, . . ., M}. For some of our results we obtain matching conditional lower bounds.