Optimal Vertex Connectivity Oracles

A k-vertex connectivity oracle for undirected G is a data structure that, given u, upsilon is an element of V (G), reports min{k, kappa (u,upsilon)}, where kappa (u,upsilon) is the pairwise vertex connectivity between u,upsilon. 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 (kn log n), which can even be encoded as a O (k log(3) n)-bit labeling scheme, can answer vertex-connectivity queries in O (k logn) 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 Omega(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 Omega(k logn) 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 k(1/2-O(1)) and k(1-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 m(4/3+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.