Data structure for node connectivity and cut queries

TL;DR

This paper develops a compact data structure for efficiently answering node connectivity and cut queries in graphs, improving space and time complexity over previous methods, and extends to specialized cases with laminar cut families.

Contribution

It introduces a more space-efficient data structure for node connectivity queries with constant query time, and provides a simplified, generalized approach for cut queries and laminar cut families.

Findings

Achieves $O(1)$ query time for node connectivity with smaller space for $k< ext{log} n$.

Shows that $O(kn)$ cuts suffice to find small $st$-cuts.

Provides a space-efficient data structure for specialized node subsets with laminar cut families.

Abstract

Let $\kappa(s,t)$ denote the maximum number of internally disjoint $st$-paths in an undirected graph $G$. We consider designing a compact data structure that answers $k$-bounded node connectivity queries: given $s,t \in V$ return $\min\{\kappa(s,t),k+1\}$. A trivial data structure has space $O(n^2)$ and query time $O(1)$. A data structure of Hsu and Lu has space $O(k^2n)$ and query time $O(\log k)$,and a randomized data structure of Iszak and Nutov has space $O(kn\log n)$ and query time $O(k \log n)$. We extend the Hsu-Lu data structure to answer queries in time $O(1)$. In parallel to our work, Pettie, Saranurak and Yin extended the Iszak-Nutov data structure to answer queries in time $O(\log n)$. Our data structure is more compact for $k<\log n$, and our query time is always better. We then augment our data structure by a list of cuts that enables to return a pointer to a minimum $st$-cut in the list (or to a cut of size $\leq k$) whenever $\kappa(s,t) \leq k$. A trivial data structure has cut list size $n(n-1)/2$, and cut query time $O(1)$, while the Pettie, Saranurak and Yin data structure has list size $O(kn \log n)$ and cut query time $O(\log n)$. We show that $O(kn)$ cuts suffice to return an $st$-cut of size $\leq k$, and a list of $O(k^2 n)$ cuts contains a minimum $st$-cut for every $s,t \in V$. In the case when $S$ is a node subset with $\kappa(s,t) \geq k$ for all $s,t \in V$, we show that $3|S|$ cuts suffice, and that these cuts can be partitioned into $O(k)$ laminar families. Thus using space $O(kn)$ we can answers each connectivity and cut queries for $s,t \in S$ in $O(1)$ time, generalizing and substantially simplifying the proof of a result of Pettie and Yin for the case $|S|=V$.