Given a weighted undirected graph $G=(V,E,w)$, a hopset $H$ of hopbound $\beta$ and stretch $(1+\epsilon)$ is a set of edges such that for any pair of nodes $u, v \in V$, there is a path in $G \cup H$ of at most $\beta$ hops, whose length is within a $(1+\epsilon)$ factor from the distance between $u$ and $v$ in $G$. We provide a decremental algorithm for maintaining hopsets with a polylogarithmic hopbound, with a total update time that matches the best known static algorithm up to polylogarithmic factors. Previously, the best known decremental hopset algorithm had a hopbound of $2^{\tilde{O}(\log^{3/4} n)}$[HKN, FOCS'14]. Our decremental hopset algorithm allows us to obtain the following improved decremental algorithms for maintaining shortest paths. -$(1+\epsilon)$-approximate single source shortest paths in amortized update time of $2^{\tilde{O}(\sqrt{\log n})}$. This improves super-polynomially over the best known amortized update time of $2^{\tilde{O}(\log^{3/4} n)}$ by [HKN, FOCS'14]. -$(1+\epsilon)$-approximate shortest paths from a set of $s$ sources in $\tilde{O}(s)$ amortized update time, assuming that $s= n^{\Omega(1)}$, and $|E|= n^{1+\Omega(1)}$. In this regime, we give the first decremental algorithm, whose running time matches, up to polylogarithmic factors, the best known static algorithm. -$(2k-1)(1+\epsilon)$-approximate all-pairs shortest paths (for any constant $k \geq 2)$, in $\tilde{O}(n^{1/k})$ amortized update time and $O(k)$ query time. This improves over the best-known amortized update time of $\tilde{O}(n^{1/k})\cdot(1/\epsilon)^{O(\sqrt{\log n})}$ [Chechik, FOCS'18]. Moreover, we reduce the query time from $O(\log \log (nW))$ to a constant $O(k)$, and hence eliminate the dependence on $n$ and the aspect ratio $W$.

中文翻译：

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更快的递减近似最短路径通过具有低跳界的 Hopsets

给定一个加权无向图 $G=(V,E,w)$，hopbound $\beta$ 和拉伸 $(1+\epsilon)$ 的 hopset $H$ 是一组边，使得对于任何节点对$u, v \in V$，在$G \cup H$ 中存在最多$\beta$ 跳的路径，其长度在$u$ 之间的距离的$(1+\epsilon)$ 因子内和 $v$ 中的 $G$。我们提供了一种递减算法，用于维护具有多对数跳界的 hopset，总更新时间与最知名的静态算法相匹配，直到多对数因子。以前，最著名的递减跳集算法的跳界为 $2^{\tilde{O}(\log^{3/4} n)}$[HKN, FOCS'14]。我们的递减跳跃集算法允许我们获得以下改进的递减算法来维护最短路径。-$(1+\epsilon)$-在$2^{\tilde{O}(\sqrt{\log n})}$ 的摊销更新时间内近似单源最短路径。这通过 [HKN, FOCS'14] 在 $2^{\tilde{O}(\log^{3/4} n)}$ 的最著名摊销更新时间上进行了超多项式改进。-$(1+\epsilon)$-$\tilde{O}(s)$ 中一组$s$ 源的近似最短路径分摊更新时间，假设$s= n^{\Omega(1)} $, 和 $|E|= n^{1+\Omega(1)}$。在这种情况下，我们给出了第一个递减算法，它的运行时间与多对数因子相匹配，这是最著名的静态算法。-$(2k-1)(1+\epsilon)$-近似所有对最短路径（对于任何常数 $k \geq 2）$，在 $\tilde{O}(n^{1/k})$摊销更新时间和 $O(k)$ 查询时间。这比 $\tilde{O}(n^{1/k})\cdot(1/\epsilon)^{O(\sqrt{\log n})}$ [Chechik , FOCS'18]。此外，我们将查询时间从 $O(\log \log (nW))$ 减少到一个常数 $O(k)$，从而消除了对 $n$ 和纵横比 $W$ 的依赖。