Dynamic Bridge-Finding in $\tilde{O}(\log ^2 n)$ Amortized Time

TL;DR

This paper introduces a new deterministic data structure for fully-dynamic bridge-finding in graphs with improved amortized update time of O((log n)^2), matching the best bounds for dynamic connectivity.

Contribution

It presents a combinatorial approach that improves the update time for dynamic bridge-finding, combining it with existing techniques to achieve optimal query times.

Findings

Supports updates in O((log n)^2) amortized time.

Finds bridges in O(log n / log log n) worst case time.

Reduces space complexity to linear space.

Abstract

We present a deterministic fully-dynamic data structure for maintaining information about the bridges in a graph. We support updates in $\tilde{O}((\log n)^2)$ amortized time, and can find a bridge in the component of any given vertex, or a bridge separating any two given vertices, in $O(\log n / \log \log n)$ worst case time. Our bounds match the current best for bounds for deterministic fully-dynamic connectivity up to $\log\log n$ factors. The previous best dynamic bridge finding was an $\tilde{O}((\log n)^3)$ amortized time algorithm by Thorup [STOC2000], which was a bittrick-based improvement on the $O((\log n)^4)$ amortized time algorithm by Holm et al.[STOC98, JACM2001]. Our approach is based on a different and purely combinatorial improvement of the algorithm of Holm et al., which by itself gives a new combinatorial $\tilde{O}((\log n)^3)$ amortized time algorithm. Combining it with Thorup's bittrick, we get down to the claimed $\tilde{O}((\log n)^2)$ amortized time. Essentially the same new trick can be applied to the biconnectivity data structure from [STOC98, JACM2001], improving the amortized update time to $\tilde{O}((\log n)^3)$. We also offer improvements in space. We describe a general trick which applies to both of our new algorithms, and to the old ones, to get down to linear space, where the previous best use $O(m + n\log n\log\log n)$. Finally, we show how to obtain $O(\log n/\log \log n)$ query time, matching the optimal trade-off between update and query time. Our result yields an improved running time for deciding whether a unique perfect matching exists in a static graph.