We provide new tradeoffs between approximation and running time for the decremental all-pairs shortest paths (APSP) problem. For undirected graphs with $m$ edges and $n$ nodes undergoing edge deletions, we provide two new approximate decremental APSP algorithms, one for weighted and one for unweighted graphs. Our first result is an algorithm that supports $(2+ \epsilon)$-approximate all-pairs constant-time distance queries with total update time $\tilde{O}(m^{1/2}n^{3/2})$ when $m= O(n^{5/3})$ (and $m= n^{1+c}$ for any constant $c >0$), or $\tilde{O}(mn^{2/3})$ when $m = \Omega(n^{5/3})$. Prior to our work the fastest algorithm for weighted graphs with approximation at most $3$ had total $\tilde O(mn)$ update time providing a $(1+\epsilon)$-approximation [Bernstein, SICOMP 2016]. Our technique also yields a decremental algorithm with total update time $\tilde{O}(nm^{3/4})$ supporting $(2+\epsilon, W_{u,v})$-approximate queries where the second term is an additional additive term and $W_{u,v}$ is the maximum weight on the shortest path from $u$ to $v$. Our second result is a decremental algorithm that given an \textit{unweighted} graph and a constant integer $k \geq 2 $, supports $(1+\epsilon, 2(k-1))$-approximate queries and has $\tilde{O}(n^{2-1/k}m^{1/k})$ total update time (when $m=n^{1+c}$ for any constant $c >0$). For comparison, in the special case of $(1+\epsilon, 2)$-approximation, this improves over the state-of-the-art algorithm by [Henzinger, Krinninger, Nanongkai, SICOMP 2016] with total update time of $\tilde{O}(n^{2.5})$. All of our results are randomized and work against an oblivious adversary.

翻译：我们为递减全源最短路径（APSP）问题提供了近似精度与运行时间之间的新权衡方案。针对边删除操作下含$m$条边和$n$个节点的无向图，我们提出了两种新的递减近似APSP算法，分别适用于加权图和非加权图。第一个算法支持$(2+\epsilon)$近似的全源常数时间距离查询，当$m=O(n^{5/3})$时（且$m=n^{1+c}$对任意常数$c>0$成立），总更新时间为$\tilde{O}(m^{1/2}n^{3/2})$；当$m=\Omega(n^{5/3})$时，总更新时间为$\tilde{O}(mn^{2/3})$。此前，加权图中近似精度不超过$3$的最快算法总更新时间为$\tilde O(mn)$，提供$(1+\epsilon)$近似[Bernstein, SICOMP 2016]。我们的技术还生成了一个总更新时间为$\tilde{O}(nm^{3/4})$的递减算法，支持$(2+\epsilon, W_{u,v})$近似查询，其中第二项为附加加性项，$W_{u,v}$表示从$u$到$v$最短路径上的最大权重。第二个结果是针对非加权图的递减算法：给定常数整数$k \geq 2$，支持$(1+\epsilon, 2(k-1))$近似查询，总更新时间为$\tilde{O}(n^{2-1/k}m^{1/k})$（当$m=n^{1+c}$对任意常数$c>0$成立时）。作为对比，在$(1+\epsilon, 2)$近似的特例中，本结果改进了[Henzinger, Krinninger, Nanongkai, SICOMP 2016]的最新算法，其总更新时间为$\tilde{O}(n^{2.5})$。本文所有结果均为随机算法，且针对 oblivious 敌手有效。