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 $1/\log^{c} n<\epsilon<1$ 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：我们提出了一种全动态环境下的新型距离预言：给定一个包含$n$个顶点、支持边插入和删除操作的加权无向图$G=(V,E)$，以及任意参数$\epsilon$（其中$1/\log^{c} n<\epsilon<1$，$c>0$为小常数），我们能够确定性维护一种数据结构，其最坏情况更新时间为$n^{\epsilon}$。对于任意顶点对$(u,v)$，该结构可在${\rm poly}(1/\epsilon)\log\log n$查询时间内返回$u$与$v$之间的$2^{{\rm poly}(1/\epsilon)}$近似距离。我们的算法在全动态算法乃至递减算法领域实现了两方面重大突破。首先，现有最坏情况更新时间算法均无法在实现$n^{2-\Omega(1)}$更新时间和$n^{o(1)}$查询时间的同时保证$o(n)$近似比，而我们的算法以$n^{\epsilon}$更新时间和$O_{\epsilon}(\log \log n)$查询时间提供了常数$O_{\epsilon}(1)$近似比。其次，即使允许平摊更新时间，该算法也是首个实现$n^{1-\Omega(1)}$更新与查询时间的确定性常数近似算法。该方向的最新成果是Chuzhoy和Zhang [STOC 2023]提出的确定性距离预言，其近似比为$(\log\log n)^{2^{O(1/\epsilon^{3})}}$，平摊更新时间为$n^{\epsilon}$，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约束扩展图相关的工具[Haeupler-R\"acke-Ghaffari, STOC 2022; Haeupler-Hershkowitz-Tan, 2023; Haeupler-Huebotter-Ghaffari, 2022]获得了该成果。我们的技术完全绕开了具有40年历史的Even-Shiloach树——该树虽是该领域最普遍使用的工具，但本质上只能实现平摊性能。