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arXiv · 1708.03812

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

Richard Peng·Bryce Sandlund·Daniel D. Sleator
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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.

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Richard Peng, Bryce Sandlund, Daniel D. Sleator. 2017-08-12. Optimal Offline Dynamic $2,3$-Edge/Vertex Connectivity. https://arxiv.org/abs/1708.03812

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