Dynamic Minimum Spanning Forest with Subpolynomial Worst-case Update Time

TL;DR

This paper introduces a Las Vegas algorithm for dynamically maintaining a minimum spanning forest with subpolynomial worst-case update time, significantly improving previous algorithms and employing novel improvements to existing techniques.

Contribution

It presents a new algorithm achieving $O(n^{o(1)})$ worst-case update time, improving upon prior work by combining and enhancing existing frameworks and techniques.

Findings

Achieves $O(n^{o(1)})$ worst-case update time with high probability.

Improves decremental low-conductance cut removal to subpolynomial time.

Develops a new approach for maintaining minimum spanning forests in sparse graphs.

Abstract

We present a Las Vegas algorithm for dynamically maintaining a minimum spanning forest of an $n$-node graph undergoing edge insertions and deletions. Our algorithm guarantees an $O(n^{o(1)})$ worst-case update time with high probability. This significantly improves the two recent Las Vegas algorithms by Wulff-Nilsen [STOC'17] with update time $O(n^{0.5-\epsilon})$ for some constant $\epsilon>0$ and, independently, by Nanongkai and Saranurak [STOC'17] with update time $O(n^{0.494})$ (the latter works only for maintaining a spanning forest). Our result is obtained by identifying the common framework that both two previous algorithms rely on, and then improve and combine the ideas from both works. There are two main algorithmic components of the framework that are newly improved and critical for obtaining our result. First, we improve the update time from $O(n^{0.5-\epsilon})$ in Wulff-Nilsen [STOC'17] to $O(n^{o(1)})$ for decrementally removing all low-conductance cuts in an expander undergoing edge deletions. Second, by revisiting the "contraction technique" by Henzinger and King [1997] and Holm et al. [STOC'98], we show a new approach for maintaining a minimum spanning forest in connected graphs with very few (at most $(1+o(1))n$) edges. This significantly improves the previous approach in [Wulff-Nilsen STOC'17] and [Nanongkai and Saranurak STOC'17] which is based on Frederickson's 2-dimensional topology tree and illustrates a new application to this old technique.