Decremental all-pairs shortest paths in deterministic near-linear time

We study the decremental All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. The input to the problem is an undirected n-vertex m-edge graph G with non-negative lengths on edges, that undergoes an online sequence of edge deletions. The goal is to support approximate shortest-paths queries: given a pair x, y of vertices of G, return a path P connecting x to y, whose length is within factor α of the length of the shortest x-y path, in time Õ(|E(P )|), where α is the approximation factor of the algorithm. APSP is one of the most basic and extensively studied dynamic graph problems. A long line of work culminated in the algorithm of [Chechik, FOCS 2018] with near optimal guarantees: for any constant 0 < ǫ ≤ 1 and parameter k ≥ 1, the algorithm achieves approximation factor (2 + ǫ)k -1, and total update time O(mn 1/k+o(1) log(nL)), where L is the ratio of longest to shortest edge lengths. Unfortunately, as much of prior work, the algorithm is randomized and needs to assume an oblivious adversary; that is, the input edge-deletion sequence is fixed in advance and may not depend on the algorithm's behavior. In many real-world scenarios, and in applications of APSP to static graph problems, it is crucial that the algorithm works against an adaptive adversary, where the edge deletion sequence may depend on the algorithm's past behavior arbitrarily; ideally, such an algorithm should be deterministic. Unfortunately, unlike the oblivious-adversary setting, its adaptive-adversary counterpart is still poorly understood. For unweighted graphs, the algorithm of [Henzinger, Krinninger and Nanongkai, FOCS '13, SICOMP '16] achieves a (1+ǫ)-approximation with total update time Õ(mn/ǫ); the best current total update time guarantee of n 2. 5+O(ǫ) is achieved by the recent deterministic algorithm of [Chuzhoy, Saranurak, SODA'21], with 2 O(1/ǫ) -multiplicative and 2 O(log 3/4 n/ǫ) -additive approximation. To the best of our knowledge, for arbitrary non-negative edge weights, the fastest current adaptive-update algorithm has total update time O(n 3 log L/ǫ), achieving a (1 + ǫ)-approximation. Even if we are willing to settle for any o(n)-approximation factor, no currently known algorithm has a better than Θ(n 3 ) total update time in weighted graphs and better than Θ(n 2.5 ) total update time in unweighted graphs. Several conditional lower bounds suggest that no algorithm with a sufficiently small approximation factor can achieve an o(n 3 ) total update time. Our main result is a deterministic algorithm for decremental APSP in undirected edge-weighted graphs, that, for any Ω(1/ log log m) ≤ ǫ < 1, achieves approximation factor (log m) 2 O(1/ǫ) , with total update time O m 1+O(ǫ) • (log m) O(1/ǫ 2 ) • log L . In particular, we obtain a (poly log m)approximation in time O(m 1+ǫ ) for any constant ǫ, and, for any slowly growing function f (m), we obtain (log m) f (m) -approximation in time m 1+o (1) . We also provide an algorithm with similar guarantees for decremental Sparse Neighborhood Covers.