We present two algorithms for dynamically maintaining a spanning forest of a graph undergoing edge insertions and deletions. Our algorithms guarantee worst-case update time and work against an adaptive adversary, meaning that an edge update can depend on previous outputs of the algorithms. We provide the first polynomial improvement over the long-standing O( √ n) bound of [Frederickson STOC'83, Eppstein, Galil, Italiano and Nissenzweig FOCS'92] for such type of algorithms. The previously best improvement was O( n(log log n) 2 /log n) [Kejlberg-Rasmussen, Kopelowitz, Pettie and Thorup ESA'16]. We note however that these bounds were obtained by deterministic algorithms while our algorithms are randomized. Our first algorithm is Monte Carlo and guarantees an O(n 0.4+o(1) ) worst-case update time, where the o(1) term hides the O( log log n/log n) factor. Our second algorithm is Las Vegas and * The full version of this paper is available at https://arxiv.org/abs/1611.03745
Recently, Nanongkai and Saranurak REF presented a randomized Las-Vegas algorithm for dynamic MSF with an expected worst-case update time of O(n 1 2 −ϵ ) for some constant ϵ > 0.
Dynamic spanning forest with worst-case update time: adaptive, Las Vegas, and O(n1/2 - ε)-time
STOC 2017
Computer Science