NDLI: Dynamic spanning forest with worst-case update time: adaptive, Las Vegas, and O(n1/2 - ε)-time

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Dynamic spanning forest with worst-case update time: adaptive, Las Vegas, and O(n1/2 - ε)-time

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Dynamic spanning forest with worst-case update time: adaptive, Las Vegas, and O(n1/2 - ε)-time

Content Provider	ACM Digital Library
Author	Saranurak, Thatchaphol Nanongkai, Danupon
Abstract	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(â n) bound of [FredericksonÂ STOCâ 84, Eppstein, Galil, Italiano and NissenzweigÂ FOCSâ 92] for such type of algorithms. The previously best improvement was O(â n $(loglogn)^{2}/logn)$ [Kejlberg-Rasmussen, Kopelowitz, Pettie and ThorupÂ ESAâ 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(â loglogn/logn) factor. Our second algorithm is Las Vegas and guarantee an $O(n^{0.49306})$ worst-case update time with high probability. Algorithms with better update time either needed to assume that the adversary is oblivious (e.g. [Kapron, King and MountjoyÂ SODAâ 13]) or can only guarantee an amortized update time. Our second result answers an open problem by KapronÂ etÂ al. To the best of our knowledge, our algorithms are among a few non-trivial randomized dynamic algorithms that work against adaptive adversaries. The key to our results is a decomposition of graphs into subgraphs that either have high expansion or sparse. This decomposition serves as an interface between recent developments on (static) flow computation and many old ideas in dynamic graph algorithms: On the one hand, we can combine previous dynamic graph techniques to get faster dynamic spanning forest algorithms if such decomposition is given. On the other hand, we can adapt flow-related techniques (e.g. those from [Khandekar, Rao and VaziraniÂ STOCâ 06], [PengÂ SODAâ 16], and [Orecchia and ZhuÂ SODAâ 14]) to maintain such decomposition. To the best of our knowledge, this is the first time these flow techniques are used in fully dynamic graph algorithms. .
Starting Page	1122
Ending Page	1129
Page Count	8
File Format	PDF
ISBN	9781450345286
DOI	10.1145/3055399.3055447
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2017-06-19
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Graph decomposition Local graph partitioning Dynamic spanning forest Sparse recovery
Content Type	Text
Resource Type	Article

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