顶点失效下基于简单快速低度数Steiner森林分解的连通性预言机
Connectivity Oracles Under Vertex Failures via a Simple and Fast Low-Degree Steiner Forest Decomposition
- University of Warwick（华威大学）
- University of Michigan（密歇根大学）
- Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究提出一种简单快速的低度数Steiner森林分解算法，构建出性能更优的顶点失效下确定性连通性预言机，消除了现有算法的$n^{o(1)}$因子。
中文摘要 AI 辅助
我们研究低度数Steiner森林分解。给定图$G=(V,E)$和终端集$U\subseteq V$，标准分解返回大小不超过$|U|/2$的顶点集$X\subseteq V$，以及$G-X$中最大度为$\Delta$的森林$T\subseteq G-X$，使得对$G-X$的每个连通分量$C$，$T$的某个连通分量包含所有$U\cap V(C)$中的终端。这是多个顶点失效下连通性预言机[DP20, LS22, LW24]的核心分解方法。现有最优算法要么在度约束为4时耗时$O(mn\log n)$[DP20]，要么在度约束较弱为$O(\log^2n)$时耗时$m^{1+o(1)}$[LW24]。我们证明，若允许$T$包含$X$中的顶点，则可通过一个非常简单的算法在$O(m\alpha(n))$时间内计算出度为4的分解。进一步表明，这种放松后的分解同样适用于构建顶点失效下的连通性预言机。由此，我们得到一个确定性的顶点失效下连通性预言机，其空间为$\tilde{O}(m)$，预处理时间为$\tilde{O}(md_\star)$（$d_\star$是失效顶点数的上界），更新时间为$\tilde{O}(d^2)$，查询时间为$O(d)$。除多对数因子外，该预言机严格优于所有已知预言机；尤其，它消除了[LS22, LW24]预处理和更新时间中的$n^{o(1)}$因子。
英文摘要
We study the low-degree Steiner forest decomposition. Given a graph $G=(V,E)$ and a terminal set $U\subseteq V$, the standard decomposition returns a set $X\subseteq V$ of size at most $|U|/2$ and a forest $T\subseteq G-X$ of maximum degree $Δ$ such that, for every connected component $C$ of $G-X$, some connected component of $T$ contains all terminals in $U\cap V(C)$. This is the central decomposition behind several connectivity oracles under vertex failures [DP20, LS22, LW24]. The state-of-the-art algorithms either take $O(mn\log n)$ time with degree bound $4$ [DP20], or take $m^{1+o(1)}$ time with the weaker degree bound $O(\log^{2}n)$ [LW24]. We show that if $T$ is allowed to contain vertices of $X$, then a degree-$4$ decomposition can be computed by a very simple algorithm in $O(mα(n))$ time. Further, we show that this relaxed decomposition is equally useful for constructing connectivity oracles under vertex failures. As a consequence, we obtain a deterministic connectivity oracle under $d$ vertex failures with $\tilde{O}(m)$ space, $\tilde{O}(md_\star)$ preprocessing time ($d_\star$ is an upper bound on the number of failed vertices), $\tilde{O}(d^{2})$ update time, and $O(d)$ query time. Up to polylogarithmic factors, this oracle strictly improves all known oracles; in particular, it removes the $n^{o(1)}$ factors from the preprocessing and update times of [LS22, LW24].
