Fully Dynamic Connectivity in $O(\log n(\log\log n)^2)$ Amortized Expected Time

TL;DR

This paper introduces a randomized data structure for dynamic graph connectivity that achieves near-optimal amortized update times and efficient query times, advancing the theoretical limits of dynamic graph algorithms.

Contribution

The paper presents a novel randomized Las Vegas data structure for dynamic connectivity with near-optimal amortized update time and improved query efficiency.

Findings

Achieves $O( ext{log} n ( ext{log} ext{log} n)^2)$ amortized expected update time.

Provides $O( ext{log} n / ext{log} ext{log} ext{log} n)$ worst-case query time.

Approaches the cell probe lower bounds for dynamic connectivity.

Abstract

Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a randomized Las Vegas dynamic connectivity data structure with $O(\log n(\log\log n)^2)$ amortized expected update time and $O(\log n/\log\log\log n)$ worst case query time, which comes very close to the cell probe lower bounds of Patrascu and Demaine (2006) and Patrascu and Thorup (2011).