Designing approximate all-pairs distance oracles in the fully dynamic setting is one of the central problems in dynamic graph algorithms. Despite extensive research on this topic, the first result breaking the $O(\sqrt{n})$ barrier on the update time for any non-trivial approximation was introduced only recently by Forster, Goranci and Henzinger [SODA'21] who achieved $m^{1/\rho+o(1)}$ amortized update time with a $O(\log n)^{3\rho-2}$ factor in the approximation ratio, for any parameter $\rho \geq 1$. In this paper, we give the first constant-stretch fully dynamic distance oracle with a small polynomial update and query time. Prior work required either at least a poly-logarithmic approximation or much larger update time. Our result gives a more fine-grained trade-off between stretch and update time, for instance we can achieve constant stretch of $O(\frac{1}{\rho^2})^{4/\rho}$ in amortized update time $\tilde{O}(n^{\rho})$, and query time $\tilde{O}(n^{\rho/8})$ for a constant parameter $\rho <1$. Our algorithm is randomized and assumes an oblivious adversary. A core technical idea underlying our construction is to design a black-box reduction from decremental approximate hub-labeling schemes to fully dynamic distance oracles, which may be of independent interest. We then apply this reduction repeatedly to an existing decremental algorithm to bootstrap our fully dynamic solution.

翻译：设计全动态环境下的近似全对距离预言机是动态图算法中的核心问题之一。尽管该方向已有大量研究，但首个打破非平凡近似更新复杂度$O(\sqrt{n})$屏障的结果近期才由Forster、Goranci和Henzinger提出[SODA'21]，他们实现了$m^{1/\rho+o(1)}$的均摊更新复杂度，并附带$O(\log n)^{3\rho-2}$倍的近似比因子（其中$\rho \geq 1$为任意参数）。本文首次给出具有小多项式级更新和查询复杂度的常数伸缩比全动态距离预言机。此前工作要么需要至少多对数近似比，要么需更高更新复杂度。我们的结果在伸缩比与更新复杂度之间提供了更细粒度的权衡，例如可在均摊更新复杂度$\tilde{O}(n^{\rho})$和查询复杂度$\tilde{O}(n^{\rho/8})$下（常数参数$\rho<1$）实现$O(\frac{1}{\rho^2})^{4/\rho}$的常数伸缩比。本算法为随机化算法，假设对手为弱适应型。构建的核心技术思想是设计一种黑箱规约方法，将递减近似枢纽标记方案转换为全动态距离预言机，这一方法或具独立研究价值。我们随后将该规约反复应用于现有递减算法，通过自举过程构建出全动态解决方案。