Sleeping is Superefficient: MIS in Exponentially Better Awake Complexity

Abstract

Maximal Independent Set (MIS) is one of the central and most well-studied problems in distributed computing. Even after four decades of intensive research, the best-known (randomized) MIS algorithms take O(log n) worst-case rounds on general graphs (where n is the number of nodes), while the best-known lower bound is Ω log n log log n rounds. Breaking past the O(log n) worst-case bound or showing stronger lower bounds have been longstanding open problems.Motivated by resource-considerations in energy-constrained distributed networks such as ad hoc wireless and sensor networks, we study MIS algorithms in the sleeping model [Chatterjee, Gmyr, and Pandurangan, PODC 2020], a model for design and analysis of resource-efficient distributed algorithms. The sleeping model is a generalization of the traditional model, that allows nodes to enter either "sleep" or "waking" states in any round. While the waking state corresponds to the default state in the traditional model, in the sleeping state a node does not send or receive messages (and messages sent to it are also lost) and does not incur any time, communication, or local computation cost. Thus, in the sleeping state, a node spends little or no energy. Hence, in this model, only rounds in which a node is awake are counted and we are interested in minimizing the worst-case number of rounds a node spends in the awake state, i.e., the awake complexity. While the primary goal is to minimize awake complexity, a secondary goal is to reduce the (traditional) worst-case round complexity (that counts both the awake and sleeping rounds).Our main contribution is that we show that MIS can be computed in (worst-case) awake complexity of O(log log n) rounds that is (essentially) exponentially better compared to the (traditional) round complexity lower bound of Ω log n log log n . Specifically, we present the following results.

Distributed MIS in O(log log n) Awake Complexity

“…The first lemma below captures the worst-case awake and round complexities of their first construction algorithm: Algorithm LDT-Construct-Awake. On the other hand, the second lemma captures the properties of a distributed LDT construction algorithm (Algorithm LDT-Construct-Round) with faster round complexity but larger awake complexity; Corollary 1 of [2] states such an algorithm exists, and we give a full construction procedure in the full version of this paper [17].…”

“…[2] provides a distributed algorithm for broadcasting over an LDT (see Observation 2 in [2]). A distributed algorithm for ranking over an LDT is presented in the full version of this paper [17]. The properties of these two algorithms are captured by the following lemma.…”