Faster Deterministic Worst-Case Fully Dynamic All-Pairs Shortest Paths via Decremental Hop-Restricted Shortest Paths

Dynamic all-pairs shortest paths is a well-studied problem in the field of dynamic graph algorithms. More specifically, given a directed weighted graph G = (V, E, omega) on n vertices which undergoes a sequence of vertex or edge updates, the goal is to maintain distances between any pair of vertices in V. In a classical work by [Demetrscu and Italiano, 2004], the authors showed that all-pairs shortest paths can be maintained deterministically in amortized (O) over tilde (n(2)) time(1), which is nearly optimal. For worst-case update time guarantees, so far the best randomized algorithm has (O) over tilde (n(3-1/3)) time [Abraham, Chechik, Krinninger, 2017], and the best deterministic algorithm needs (O) over tilde (n(3-2/7)) time [Probst Gutenberg, Wulff-Nilsen, 2020]. We provide a faster deterministic worst-case update time of (O) over tilde (n(3-20/61)) for fully dynamic all-pairs shortest paths. To achieve this improvement, we study a natural variant of this problem where a hop constraint is imposed on shortest paths between vertices, that is, given a parameter h, the h-hop shortest path between any pair of vertices s, t is an element of V is a path from s to t with at most h edges whose total weight is minimized. As a result which might be of independent interest, we give a deterministic algorithm that maintains all-pairs h-hop shortest paths under vertex deletions in total update time (O) over tilde (n(3)h +Kn(2) h(2)), where K bounds the total number of vertex deletions.