Dynamic Minimum Spanning Forest with Subpolynomial Worst-Case Update Time

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 [37] with update time O(n 0.5−ǫ ) for some constant ǫ > 0 and, independently, by Nanongkai and Saranurak [24] 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−ǫ ) in [37] 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 [15] and Holm et al. [19], 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 [37,24] which is based on Frederickson's 2-dimensional topology tree [11] and illustrates a new application to this old technique.

Dynamic Matrix Inverse: Improved Algorithms and Matching Conditional Lower Bounds

“…Indeed, nding a clever way to repeat the same ideas several times is a key approach to signi cantly speed up many dynamic algorithms. (For a recent example, consider the spanning tree problem where [NSW17] sped up the n 1 2−ϵ update time of [NS17,Wul17] to n o(1) by appropriately repeating the approach of [NS17,Wul17] for about log(n) times. See, e.g., [HKN18,HKN14,HKN13] for other examples.)…”

“…Typically once the approach can be repeated to speed up a dynamic algorithm, it can be repeated several times (e.g. [NSW17,HKN18,HKN13]). Given this, it might be tempting to get further speed-ups by writing A = A ′ T 1 T 2 T 3 T 4 .…”

Dynamic Algorithms for Graph Coloring

“…We believe that this is an important question, since it may help in understanding how to develop deterministic dynamic algorithms in general. It is very challenging and interesting to design deterministic dynamic algorithms with performances similar to the randomized ones for many dynamic graph problems such as maximal matching [5,10,9,11,12,13], connectivity [24,28,32,27], and shortest paths [6,8,7,21,22].…”

Coarse-Grained Complexity for Dynamic Algorithms

“…The main question is whether it is in coNP dy . (Techniques from [NSW17,Wul17,NS17] almost give this, with verification time n o(1) instead of polylogarithmic.) Having connectivity in NP dy ∩ coNP dy would be a strong evidence that it is in P dy , meaning that it admits a deterministic algorithm with polylogarithmic update time.…”

“…In this section, we study this problem from non-deterministic perspective. Nanongkai, Saranurak and Wulff-Nilsen [NSW17] show a Las Vegas algorithm for dynamic minimum spanning forest (which implies dynamic connectivity). On an n-node graph, their algorithm has n o(log log log n/ log log n) worst-case update time, which is later slightly improved to n O(log log n/ √ log n ) [SW19].…”