Tight Bounds for Vertex Connectivity in Dynamic Streams

TL;DR

This paper introduces a nearly optimal streaming algorithm for determining vertex connectivity in dynamic graph streams, matching lower bounds and improving previous algorithms.

Contribution

It provides a space-efficient algorithm for vertex connectivity in dynamic streams, closing gaps between upper and lower bounds and extending prior work.

Findings

Achieves $ ilde{O}(k n)$ space complexity for vertex connectivity

Matches the known $ ilde{ ext{Omega}}(k n)$ lower bound

Provides a model-independent certificate for $k$-vertex-connectivity

Abstract

We present a streaming algorithm for the vertex connectivity problem in dynamic streams with a (nearly) optimal space bound: for any $n$-vertex graph $G$ and any integer $k \geq 1$, our algorithm with high probability outputs whether or not $G$ is $k$-vertex-connected in a single pass using $\widetilde{O}(k n)$ space. Our upper bound matches the known $\Omega(k n)$ lower bound for this problem even in insertion-only streams -- which we extend to multi-pass algorithms in this paper -- and closes one of the last remaining gaps in our understanding of dynamic versus insertion-only streams. Our result is obtained via a novel analysis of the previous best dynamic streaming algorithm of Guha, McGregor, and Tench [PODS 2015] who obtained an $\widetilde{O}(k^2 n)$ space algorithm for this problem. This also gives a model-independent algorithm for computing a "certificate" of $k$-vertex-connectivity as a union of $O(k^2\log{n})$ spanning forests, each on a random subset of $O(n/k)$ vertices, which may be of independent interest.