Fully-Dynamic Minimum Spanning Forest with Improved Worst-Case Update Time

TL;DR

This paper introduces a novel Las Vegas data structure for fully-dynamic minimum spanning forests that improves worst-case update times below the long-standing O(n^{1/2}) barrier, using advanced reduction and graph techniques.

Contribution

It presents the first data structure surpassing the 25-year-old worst-case update time barrier for dynamic minimum spanning forests, with a reduction approach and new techniques for low-conductance cuts.

Findings

Achieves expected worst-case update time of O(n^{1/2 - c}) for some c>0

First to improve worst-case bounds for unweighted spanning forest maintenance

Provides a polynomially faster worst-case update time for dynamic connectivity

Abstract

We give a Las Vegas data structure which maintains a minimum spanning forest in an n-vertex edge-weighted dynamic graph undergoing updates consisting of any mixture of edge insertions and deletions. Each update is supported in O(n^{1/2 - c}) expected worst-case time for some constant c > 0 and this worst-case bound holds with probability at least 1 - n^{-d} where d is a constant that can be made arbitrarily large. This is the first data structure achieving an improvement over the O(n^{1/2}) deterministic worst-case update time of Eppstein et al., a bound that has been standing for nearly 25 years. In fact, it was previously not even known how to maintain a spanning forest of an unweighted graph in worst-case time polynomially faster than Theta(n^{1/2}). Our result is achieved by first giving a reduction from fully-dynamic to decremental minimum spanning forest preserving worst-case update time up to logarithmic factors. Then decremental minimum spanning forest is solved using several novel techniques, one of which involves keeping track of low-conductance cuts in a dynamic graph. An immediate corollary of our result is the first Las Vegas data structure for fully-dynamic connectivity where each update is handled in worst-case time polynomially faster than Theta(n^{1/2}) w.h.p.; this data structure has O(1) worst-case query time.