Dynamic Deterministic Constant-Approximate Distance Oracles with n^ϵ Worst-Case Update Time

We present a new distance oracle in the fully dynamic setting: given a weighted undirected graph G=(V,E) with n vertices undergoing both edge insertions and deletions, and an arbitrary parameter ϵ where 1/log^c n<ϵ<1 and c>0 is a small constant, we can deterministically maintain a data structure with n^ϵ worst-case update time that, given any pair of vertices (u,v), returns a 2^ poly(1/ϵ)-approximate distance between u and v in poly(1/ϵ)loglog n query time. Our algorithm significantly advances the state-of-the-art in two aspects, both for fully dynamic algorithms and even decremental algorithms. First, no existing algorithm with worst-case update time guarantees a o(n)-approximation while also achieving an n^2-Ω(1) update and n^o(1) query time, while our algorithm offers a constant O_ϵ(1)-approximation with n^ϵ update time and O_ϵ(loglog n) query time. Second, even if amortized update time is allowed, it is the first deterministic constant-approximation algorithm with n^1-Ω(1) update and query time. The best result in this direction is the recent deterministic distance oracle by Chuzhoy and Zhang [STOC 2023] which achieves an approximation of (loglog n)^2^O(1/ϵ^3) with amortized update time of n^ϵ and query time of 2^ poly(1/ϵ)log nloglog n. We obtain the result by dynamizing tools related to length-constrained expanders [Haeupler-Räcke-Ghaffari, STOC 2022; Haeupler-Hershkowitz-Tan, 2023; Haeupler-Huebotter-Ghaffari, 2022]. Our technique completely bypasses the 40-year-old Even-Shiloach tree, which has remained the most pervasive tool in the area but is inherently amortized.