发表机构
University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出顶点故障距离预言机和标记方案的新算法，实现常数近似、近线性空间和高效查询，并首次给出非平凡标记方案。
AI 中文摘要
我们提出了在无向加权图中顶点故障距离预言机和标记方案问题的新算法。顶点故障距离预言机是一种数据结构，给定两个顶点 $x$ 和 $y$ 以及一个大小至多为 $f$ 的故障顶点集合 $F$，返回 $G \setminus F$ 中 $x$ 和 $y$ 之间距离的近似值。在标记方案设置中，数据结构需要以顶点上的标签形式分布式存储，每个查询 $(x,y,F)$ 必须仅通过访问 $F \cup \{x,y\}$ 中顶点的标签来回答。对于任意 $f\geq 1$ 和 $k \ge 1$，我们获得了一个具有 $O(k^{6})$ 近似比、空间 $\tilde{O}(f^{2}n^{1+1/k})$、查询时间 $\tilde{O}(f^{5}n^{1/k})$ 以及多项式预处理时间的顶点故障距离预言机。特别地，这是第一个针对多个顶点故障且空间接近线性的高效时间预言机，也是第一个在容忍 $\Omega(\log n)$ 个顶点故障时具有多项式空间和常数近似比的预言机。先前的结果，由 [Duan-Gu-Ren, SODA'21] 给出，提供了两种替代方案：对于任意常数 $c \ge 1$ 和 $\epsilon>0$，一个预言机具有 $\mathrm{poly}(\log n,f)$ 近似比、空间 $n^{2+1/c}\mathrm{poly}(\log n,f)$ 和查询时间 $\mathrm{poly}(\log n,f^{c})$，而另一个具有 $(1+\epsilon)$ 近似比、空间 $n^{2+1/c}(\log n/\epsilon)^{O(f)}$ 和查询时间 $\mathrm{poly}(\log n,f^{c},1/\epsilon)$。我们还获得了一个具有 $O(k^{6})$ 近似比和标签大小 $f^{3}n^{1/k}\log^{O(k)} n$ 的顶点故障距离标记方案。这是第一个非平凡的顶点故障距离标记方案。我们的技术建立在最近与长度约束顶点扩展器相关的工具之上，并引入了一种新的基于扩展器的捷径稀疏化方法。后者还导致了一个大小为 $\tilde{O}(f^{2})$ 的确定性顶点故障连通性标记方案。
英文摘要
We present new algorithms for the vertex-failure distance oracles and labeling schemes problems in undirected weighted graphs. A vertex-failure distance oracle is a data structure that, given two vertices $x$ and $y$ and a failed vertex set $F$ of size at most $f$, returns an approximation to the distance between $x$ and $y$ in $G \setminus F$. In the labeling-scheme setting, the data structure needs to be stored distributively as labels on the vertices, and each query $(x,y,F)$ must be answered by accessing only the labels of the vertices in $F \cup \{x,y\}$. For any $f\geq 1$ and $k \ge 1$, we obtain a vertex-failure distance oracle with $O(k^{6})$ approximation, space $\tilde{O}(f^{2}n^{1+1/k})$, query time $\tilde{O}(f^{5}n^{1/k})$, and polynomial preprocessing time. In particular, this is the first time-efficient oracle for multiple vertex failures with space close to linear, as well as the first constant-approximation oracle with polynomial space when tolerating $Ω(\log n)$ vertex failures. The previous results, due to [Duan-Gu-Ren, SODA'21], gave two alternatives: for any constant $c \ge 1$ and $ε>0$, one oracle has $\mathrm{poly}(\log n,f)$ approximation, space $n^{2+1/c}\mathrm{poly}(\log n,f)$, and query time $\mathrm{poly}(\log n,f^{c})$, while the other has $(1+ε)$ approximation, space $n^{2+1/c}(\log n/ε)^{O(f)}$, and query time $\mathrm{poly}(\log n,f^{c},1/ε)$. We also obtain a vertex-failure distance labeling scheme with $O(k^{6})$ approximation and label size $f^{3}n^{1/k}\log^{O(k)} n$. This is the first nontrivial distance labeling scheme for vertex failures. Our techniques build on recent tools related to length-constrained vertex expanders and also introduce a new expander-based shortcut sparsification. The latter also leads to a deterministic vertex-failure connectivity labeling scheme of size $\tilde{O}(f^{2})$.
Comments80 pages, to appear in FOCS 2026
