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 [Bernstein, SICOMP'16]. Our technique also yields a decremental algorithm with total update time $\tilde{O}(nm^{3/4})$ supporting $(2+\epsilon, W_{u,v})$-approximate queries. Our second result is a decremental algorithm that given an 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 by [Henzinger et al., SICOMP'16] with total update time of $\tilde{O}(n^{2.5})$. All of our results are randomized and work against an oblivious adversary. Our approach also leads to a new static distance oracle construction. In particular, we construct a distance oracle in $\tilde O(mn^{2/3})$ time that supports constant time $2$-approximate queries. For sparse graphs, the preprocessing time of the algorithm matches conditional lower bounds [Patrascu et al., FOCS'12; Abboud et al., STOC'23]. To the best of our knowledge, this is the first 2-approximate distance oracle that has subquadratic preprocessing time in sparse graphs.

翻译：我们针对递减全对最短路径（APSP）问题提供了近似精度与运行时间之间的新权衡。对于经历边删除操作、包含$m$条边和$n$个节点的无向图，我们提出了两种新的近似递减APSP算法，分别适用于加权图和非加权图。第一个成果是：当$m= O(n^{5/3})$（且$m= n^{1+c}$，其中$c>0$为任意常数）时，该算法支持$(2+ \epsilon)$-近似全对常数时间距离查询，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$\tilde{O}(mn^{2/3})$。在此之前，针对近似精度不超过3的加权图，最快算法的总更新时间为$\tilde O(mn)$ [Bernstein, SICOMP'16]。我们的技术还提出了一种递减算法，其总更新时间为$\tilde{O}(nm^{3/4})$，支持$(2+\epsilon, W_{u,v})$-近似查询。第二个成果是：对于非加权图及任意常数整数$k \geq 2$，该递减算法支持$(1+\epsilon, 2(k-1))$-近似查询，总更新时间为$\tilde{O}(n^{2-1/k}m^{1/k})$（当$m=n^{1+c}$且$c>0$为任意常数时）。作为对比，在$(1+\epsilon, 2)$-近似的特殊情形下，本算法改进了由[Henzinger等，SICOMP'16]提出的总更新时间为$\tilde{O}(n^{2.5})$的现有最优方法。所有结果均为随机化算法，且适用于对抗性对手场景。我们的方法还衍生出一种新的静态距离预计算结构：在$\tilde O(mn^{2/3})$时间内可构建一个支持常数时间$2$-近似查询的距离预计算结构。对于稀疏图，该算法的预处理时间与条件性下界相匹配[Patrascu等，FOCS'12；Abboud等，STOC'23]。据我们所知，这是首个在稀疏图中实现亚二次预处理时间的$2$-近似距离预计算结构。