全对最近最小割的最优结构及边插入敏感性预言机
An Optimal Structure for All-Pairs Nearest Mincuts and Sensitivity Oracles for Edge Insertions
- Weizmann Institute of Science（魏茨曼科学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出首个最优空间（O(n)）的全对最近最小割表示，类比Gomory-Hu树，支持O(n)查询，并据此设计O(n)空间、O(n^2)总时间的边插入全对最小割敏感性预言机。
AI中文摘要：
给定一个包含 $n$ 个顶点的无向加权图 $G=(V,E)$，经典的 Gomory-Hu 树是 $G$ 的一种结构，它仅用 $O(n)$ 空间即可编码任意 $s,t\in V$ 之间的最小 $s,t$-割。在本工作中，我们探讨对于自然且结构化的全对最近最小割族，是否也能实现同样的紧凑性。最近最小 $(s,t)$-割是所有包含 $s$ 的最小 $s,t$-割中按包含关系唯一最小的那个。该族已被证明在广泛的应用中非常有用，包括容错可达性、最小割敏感性预言机、仙人掌表示以及快速 Gomory-Hu 树构造。尽管其基础性作用，至今尚无次二次空间表示。$O(n)$ 空间表示仅在单源设置中已知，即给定源点 $s$，可以对任意 $t\in V\setminus \{s\}$ 报告最近最小 $(t,s)$-割。我们通过提出首个全对最近最小割的最优空间表示来填补这一空白，提供了 Gomory-Hu 树的自然类比。我们的主要结果是一个 $O(n)$ 空间结构，它编码了每对顶点之间的最近最小割。此外，给定任意对 $s,t\in V$，它可以在 $O(n)$ 时间内报告最近最小 $s,t$-割。这两个界都与 Gomory-Hu 树匹配，并且是坏情况最优的。作为应用，我们设计了一个用于边插入的全对最小割敏感性预言机：一个占用 $O(n)$ 空间的数据结构，给定任意边 $e$，可以在 $O(n^2)$ 总时间内确定对于所有对 $s,t\in V$，插入 $e$ 后最小 $s,t$-割值是否增加。现有的插入敏感性预言机要么仅限于单源设置，要么使用 $O(n^2)$ 空间用于全对情况 [Baswana, Gupta, and Knollmann, Algorithmica'22; Baswana and Pandey, SODA'22]。
英文摘要：
Given an undirected weighted graph $G=(V,E)$ on $n$ vertices, the classical Gomory-Hu tree of $G$ is a structure that encodes an arbitrary minimum $s,t$-cut for every $s,t\in V$ using just $O(n)$ space. In this work, we ask whether the same compactness is achievable for the natural and structured family of all-pairs \textit{nearest minimum cuts}. The nearest minimum $(s,t)$-cut is the unique inclusion-wise minimal one among all minimum $s,t$-cuts containing $s$. This family has proven useful in a wide range of applications, including fault-tolerant reachability, minimum cut sensitivity oracles, cactus representations, and fast Gomory-Hu tree constructions. Despite its fundamental role, no subquadratic space representation is known for them to date. The $O(n)$ space representations are known only in single-source settings, where given a source $s$, one can report the nearest minimum $(t,s)$-cut for any $t\in V\setminus \{s\}$. We close this gap by presenting the first optimal space representation of all-pairs nearest minimum cuts, providing a natural analogue of the Gomory-Hu tree. Our main result is an $O(n)$ space structure that encodes the nearest minimum cut between every pair of vertices. Furthermore, given any pair $s,t\in V$, it can report the nearest minimum $s,t$-cut in $O(n)$ time. Both bounds match those of the Gomory-Hu tree and are worst-case optimal. As an application, we design an all-pairs minimum cut sensitivity oracle for edge insertion: a data structure that occupies $O(n)$ space and, given any edge $e$, can determine for all pairs $s,t\in V$ whether the minimum $s,t$-cut value increases upon insertion of $e$ in $O(n^2)$ total time. Existing insertion sensitivity oracles were either limited to the single-source setting or used $O(n^2)$ space for all-pairs [Baswana, Gupta, and Knollmann, Algorithmica'22; Baswana and Pandey, SODA'22].
