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 $\epsilon$ where $\epsilon\in[1/\log^{c} n,1]$ and $c>0$ is a small constant, we can deterministically maintain a data structure with $n^{\epsilon}$ worst-case update time that, given any pair of vertices $(u,v)$, returns a $2^{{\rm poly}(1/\epsilon)}$-approximate distance between $u$ and $v$ in ${\rm poly}(1/\epsilon)\log\log n$ 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 $n^{2-\Omega(1)}$ update and $n^{o(1)}$ query time, while our algorithm offers a constant $O_{\epsilon}(1)$-approximation with $n^{\epsilon}$ update time and $O_{\epsilon}(\log \log n)$ query time. Second, even if amortized update time is allowed, it is the first deterministic constant-approximation algorithm with $n^{1-\Omega(1)}$ 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 $(\log\log n)^{2^{O(1/\epsilon^{3})}}$ with amortized update time of $n^{\epsilon}$ and query time of $2^{{\rm poly}(1/\epsilon)}\log n\log\log n$. We obtain the result by dynamizing tools related to length-constrained expanders [Haeupler-R\"acke-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.

翻译：我们提出了一种全新动态距离预言机：给定一个包含$n$个顶点的加权无向图$G=(V,E)$，支持边插入和删除操作，并给定任意参数$\epsilon \in [1/\log^c n, 1]$（其中$c>0$为小常数），我们能够确定性地维护一个最坏情况更新时间为$n^\epsilon$的数据结构。该结构可在${\rm poly}(1/\epsilon)\log\log n$查询时间内，对任意顶点对$(u,v)$返回$2^{{\rm poly}(1/\epsilon)}$-近似距离。我们的算法在两方面显著推进了现有技术水平——既适用于全动态算法，也适用于减量算法。首先，现有算法在最坏情况更新时间内，无法同时保证$o(n)$-近似比与$n^{2-\Omega(1)}$更新时间及$n^{o(1)}$查询时间，而我们的算法能以$n^\epsilon$更新时间和$O_\epsilon(\log\log n)$查询时间，实现常数$O_\epsilon(1)$-近似比。其次，即使允许分摊更新时间，这也是首个具有$n^{1-\Omega(1)}$更新与查询时间的确定性常数近似算法。该领域的最佳结果来自Chuzhoy与Zhang的最新确定性距离预言机[STOC 2023]，其近似比为$(\log\log n)^{2^{O(1/\epsilon^3)}}$，分摊更新时间为$n^\epsilon$，查询时间为$2^{{\rm poly}(1/\epsilon)}\log n\log\log n$。我们通过动态化与长度受限扩展图相关的工具[见Haeupler-Räckle-Ghaffari, STOC 2022; Haeupler-Hershkowitz-Tan, 2023; Haeupler-Huebotter-Ghaffari, 2022]获得该结果。该技术完全绕过了存在40年之久的Even-Shiloach树——该工具在该领域长期最为普遍，但其本质是分摊的。