AI 中文总结
给定带权无向图及源顶点，设计单源双容错距离预言机。核心方法是利用特定算法，主要贡献是在处理最多两条故障边时，以\(\tilde{O}(n\sqrt{n})\)空间和\(\tilde{O}(1)\)查询时间给出\((1 + O(\epsilon))\)近似最短路径权重。
AI 中文摘要
给定一个具有\(n\)个顶点和\(m\)条边、边权重在\([1, W]\)的无向加权图\(G\)以及一个指定源顶点\(s\)。我们为\(G\)设计了一个单源双容错距离预言机。给定目标顶点\(t\)和最多两条故障边的集合\(F\)，该预言机返回从源\(s\)到\(t\)避开\(F\)的最短路径权重的\((1 + O(\epsilon))\)近似值。我们的预言机使用\(\tilde{O}(n\sqrt{n})\)空间且具有\(\tilde{O}(1)\)查询时间。在我们的结果之前，已知单源单容错预言机使用\(\tilde{O}(n)\)空间和\(O(1)\)查询时间返回最短路径权重的\((1+\epsilon)\)近似值。然而，将这些方法扩展到多个故障仍然是一个开放问题。实际上，所有处理多个故障的\((1+\epsilon)\)近似距离预言机都需要\(\Omega(n^2)\)空间。我们通过提出第一个具有\(o(n^2)\)空间的双容错距离预言机打破了这个界限。
英文摘要
We are given an undirected weighted graph $G$ with $n$ vertices and $m$ edges, edge weights in $[1, W]$, and a designated source vertex $s$. We design a single source dual fault tolerant distance oracle for $G$. Given a destination vertex $t$ and a set $F$ of at most two faulty edges, the oracle returns a $(1 + O(ε))$-approximation of the weight of the shortest path from the source $s$ to $t$ avoiding $F$. Our oracle uses $\tilde{O}(n\sqrt{n})$ space and has $\tilde{O}(1)$ query time. Prior to our result, single source single fault tolerant oracles were known to return a $(1+ε)$ approximation of the weight of the shortest path using $\tilde{O}(n)$ space and $O(1)$ query time. However, extending these approaches to multiple faults remained an open problem. Indeed, all $(1+ε)$-approximate distance oracles that handle multiple faults require $Ω(n^2)$ space. We break this bound by presenting the first dual fault tolerant distance oracle with $o(n^2)$ space.
Comments30 Pages, 15 figures, Accepted at ESA 2026
