Optimal Offline Dynamic $2,3$-Edge/Vertex Connectivity

TL;DR

This paper introduces efficient offline algorithms for processing multiple 2- and 3-edge/vertex connectivity queries in dynamic graphs, achieving optimal per-operation time through a divide-and-conquer approach that preserves connectivity information.

Contribution

It presents the first offline algorithms with $O( ext{log} n)$ per-operation time for 2- and 3-connectivity queries, surpassing existing online methods.

Findings

Achieves optimal $O( ext{log} n)$ per-operation complexity.

Develops a divide-and-conquer scheme for graph transformation.

Connects the approach to vertex sparsifiers and dynamic graph data structures.

Abstract

We give offline algorithms for processing a sequence of $2$ and $3$ edge and vertex connectivity queries in a fully-dynamic undirected graph. While the current best fully-dynamic online data structures for $3$-edge and $3$-vertex connectivity require $O(n^{2/3})$ and $O(n)$ time per update, respectively, our per-operation cost is only $O(\log n)$, optimal due to the dynamic connectivity lower bound of Patrascu and Demaine. Our approach utilizes a divide and conquer scheme that transforms a graph into smaller equivalents that preserve connectivity information. This construction of equivalents is closely-related to the development of vertex sparsifiers, and shares important connections to several upcoming results in dynamic graph data structures, outside of just the offline model.