In the decremental single-source shortest paths (SSSP) problem, the input is an undirected graph $G=(V,E)$ with $n$ vertices and $m$ edges undergoing edge deletions, together with a fixed source vertex $s\in V$. The goal is to maintain a data structure that supports shortest-path queries: given a vertex $v\in V$, quickly return an (approximate) shortest path from $s$ to $v$. The decremental all-pairs shortest paths (APSP) problem is defined similarly, but now the shortest-path queries are allowed between any pair of vertices of $V$. Both problems have been studied extensively since the 80's, and algorithms with near-optimal total update time and query time have been discovered for them. Unfortunately, all these algorithms are randomized and, more importantly, they need to assume an oblivious adversary. Our first result is a deterministic algorithm for the decremental SSSP problem on weighted graphs with $O(n^{2+o(1)})$ total update time, that supports $(1+\epsilon)$-approximate shortest-path queries, with query time $O(|P|\cdot n^{o(1)})$, where $P$ is the returned path. This is the first $(1+\epsilon)$-approximation algorithm against an adaptive adversary that supports shortest-path queries in time below $O(n)$, that breaks the $O(mn)$ total update time bound of the classical algorithm of Even and Shiloah from 1981. Our second result is a deterministic algorithm for the decremental APSP problem on unweighted graphs that achieves total update time $O(n^{2.5+\delta})$, for any constant $\delta>0$, supports approximate distance queries in $O(\log\log n)$ time; the algorithm achieves an $O(1)$-multiplicative and $n^{o(1)}$-additive approximation on the path length. All previous algorithms for APSP either assume an oblivious adversary or have an $\Omega(n^{3})$ total update time when $m=\Omega(n^{2})$.

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基于分层核心分解的递减最短路径的确定性算法

在递减单源最短路径 (SSSP) 问题中，输入是一个无向图 $G=(V,E)$，其中 $n$ 个顶点和 $m$ 个边进行边删除，以及一个固定的源顶点 $s \in V$。目标是维护支持最短路径查询的数据结构：给定顶点 $v\in V$，快速返回从 $s$ 到 $v$ 的（近似）最短路径。递减所有对最短路径 (APSP) 问题的定义类似，但现在允许在 $V$ 的任何顶点对之间进行最短路径查询。自 80 年代以来，这两个问题都得到了广泛的研究，并且已经为它们发现了具有接近最佳总更新时间和查询时间的算法。不幸的是，所有这些算法都是随机的，更重要的是，它们需要假设一个不经意的对手。我们的第一个结果是加权图上递减 SSSP 问题的确定性算法，总更新时间为 $O(n^{2+o(1)})$，支持 $(1+\epsilon)$-approximate shortest-path查询，查询时间为 $O(|P|\cdot n^{o(1)})$，其中 $P$ 是返回的路径。这是针对自适应对手的第一个 $(1+\epsilon)$ 近似算法，该算法支持时间低于 $O(n)$ 的最短路径查询，打破了 $O(mn)$ 的总更新时间界限Even 和 Shiloah 的经典算法从 1981 年开始。我们的第二个结果是一个确定性算法，用于解决未加权图上的递减 APSP 问题，对于任何常数 $\delta> 实现总更新时间 $O(n^{2.5+\delta})$ 0$，支持$O(\log\log n)$时间内的近似距离查询；该算法在路径长度上实现了 $O(1)$-multiplicative 和 $n^{o(1)}$-additive 近似。所有以前的 APSP 算法要么假设对手不知情，要么在 $m=\Omega(n^{2})$ 时具有 $\Omega(n^{3})$ 总更新时间。