Improved Network Decompositions using Small Messages with Applications on MIS, Neighborhood Covers, and Beyond

Abstract

Network decompositions, as introduced by Awerbuch, Luby, Goldberg, and Plotkin [FOCS'89], are one of the key algorithmic tools in distributed graph algorithms. We present an improved deterministic distributed algorithm for constructing network decompositions of power graphs using small messages, which improves upon the algorithm of Ghaffari and Kuhn [DISC'18]. In addition, we provide a randomized distributed network decomposition algorithm, based on our deterministic algorithm, with failure probability exponentially small in the input size that works with small messages as well. Compared to the previous algorithm of Elkin and Neiman [PODC'16], our algorithm achieves a better success probability at the expense of its round complexity, while giving a network decomposition of the same quality. As a consequence of the randomized algorithm for network decomposition, we get a faster randomized algorithm for computing a Maximal Independent Set, improving on a result of Ghaffari [SODA'19]. Other implications of our improved deterministic network decomposition algorithm are: a faster deterministic distributed algorithms for constructing spanners and approximations of distributed set cover, improving results of Ghaffari, and Kuhn [DISC'18] and Deurer, Kuhn, and Maus [PODC'19]; and faster a deterministic distributed algorithm for constructing neighborhood covers, resolving an open question of Elkin [SODA'04].CONGEST model, Ghaffari and Kuhn [GK18] showed that k-hop separated network decompositions can be used for computing spanners and approximating minimum dominating set. State of the Art-DeterministicConstructions: There are four known deterministic distributed constructions of network decompositions, successively improving either quantitatively or qualitatively [ALGP89, PS92, GK18, Gha19]. Awerbuch et al. [ALGP89] provided an algorithm for computing (2 O( √ log n log log n) , 2 O( √ log n log log n) ) network decompositions of an n node graph G in 2 O( √ log n log log n) rounds, which works in the CONGEST model. Subsequently, this was improved by Panconesi and Srinivasan [PS92] showing that all 2 O( √ log n log log n) terms could be replaced by 2 O( √ log n). However, their algorithm requires large messages. For network decompositions with higher levels of separation, Ghaffari and Kuhnnetwork decomposition of G k , which works with small messages. Note that extending network decomposition algorithms to compute a decomposition of G k is trivial in the LOCAL model: As nodes can send messages of arbitrary size, communication on G k can be simulate in k rounds of communication on G. Thus, with a k factor overhead in the round complexity (and a k factor increase in the diameter with respect to distances in G), we can use any LOCAL-model network decomposition algorithm to also compute k-hop separated decompositions.Recently, Ghaffarinetwork decomposition can also be computed in 2 O( √ log n) rounds in the CONGEST model. However, his construction cannot extend to G k , which is one of the issues we address in this paper. In con...