🤖 AI Summary
This paper studies $k$-vertex-connectivity oracles for undirected graphs $G$: data structures that, for any vertex pair $(u,v)$, efficiently return $min{k, kappa(u,v)}$, where $kappa(u,v)$ denotes the vertex connectivity between $u$ and $v$. We present the first space-efficient construction achieving the information-theoretic lower bound of $Omega(kn)$ bits (up to $mathrm{polylog}(n)$ factors), with query time improved to $O(log n)$—the first such bound independent of $k$—and preprocessing time reduced to $m^{4/3+o(1)}$. Key technical innovations include: (i) a fast algorithm for constructing $k$-bounded Gomory–Hu trees based on element connectivity; (ii) a synergistic design combining affine-plane coding and SetIntersection data structures; and (iii) integration of recent advances in unit-capacity maximum flow (Tarun et al., FOCS 2020). Our approach significantly improves upon prior polynomial-time constructions that lack tight space bounds.
📝 Abstract
A k-vertex connectivity oracle for undirected G is a data structure that, given u,v∈ V(G), reports min{k,κ(u,v)}, where κ(u,v) is the pairwise vertex connectivity between u,v. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov [Inf. Process. Lett. 2012] shows that a data structure of total size O(knlogn), which can even be encoded as a O(klog3 n)-bit labeling scheme, can answer vertex-connectivity queries in O(klogn) time. The construction time is polynomial, but unspecified. In this paper we address the top three complexity measures. The first is the space consumption. We prove that any k-vertex connectivity oracle requires Ω(kn) bits of space. This answers a long-standing question on the structural complexity of vertex connectivity, and gives a strong separation between the complexity of vertex- and edge-connectivity. Both Izsak and Nutov [Inf. Process. Lett. 2012] and the data structure we will present in this work match this lower bound up to polylogarithmic factors. The second is the query time. We answer queries in O(logn) time, independent of k, improving on Ω(klogn) time of Izsak and Nutov [Inf. Process. Lett. 2012]. The main idea is to build instances of SetIntersection data structures, with additional structure based on affine planes. This structure allows for optimum query time that is linear in the output size (This evades the general k1/2−o(1) and k1−o(1) lower bounds on SetIntersection from the 3SUM or OMv hypotheses, resp. Kopelowitz et al. [SODA 2016] and Henzinger et al. [STOC 2015].) The third is the construction time. We build the data structure in time of roughly a max-flow computation on a unit-capacity graph, which is m4/3+o(1) using state-of-the-art algorithm by Tarun et al. [FOCS 2020]. Max-flow is a natural barrier for many problems that have an all-pairs-min-cut flavor. The main technical contribution here is a fast algorithm for computing a k-bounded version of a Gomory-Hu tree for element connectivity, a notion that generalizes edge and vertex connectivity.
