Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal Vertex Connectivity Oracles   A $k$-vertex connectivity oracle for undirected $G$ is a data structure that,
given $u,v\in V(G)$, reports $\min\{k,\kappa(u,v)\}$, where $\kappa(u,v)$ 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 shows that a data structure of total size $\tilde{O}(kn)$ can even be
encoded as a $\tilde{O}(k)$-bit labeling scheme so that vertex-connectivity
queries can be answered in $\tilde{O}(k)$ time. The construction time is
polynomial, but unspecified.
  In this paper we address the top three complexity measures: Space, Query
Time, and Construction Time. We give an $\Omega(kn)$-bit lower bound on any
vertex connectivity oracle. We construct an optimal-space connectivity oracle
in max-flow time that answers queries in $O(\log n)$ time, independent of $k$.

32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph