Awake-Efficient Distributed Algorithms for Maximal Independent Set

Abstract

We present a simple algorithmic framework for designing efficient distributed algorithms for the fundamental symmetry breaking problem of Maximal Independent Set (MIS) in the sleeping model [Chatterjee et al, PODC 2020]. In the sleeping model, only the rounds in which a node is awake are counted for the awake complexity, while sleeping rounds are ignored. This is motivated by the fact that a node spends resources only in its awake rounds and hence the goal is to minimize the awake complexity.Our framework allows us to design distributed MIS algorithms that have ${\mathcal{O}}({\text{polyloglog }}n)$ (worst-case) awake complexity in certain important graph classes which satisfy the so-called adjacency property. Informally, the adjacency property guarantees that the graph can be partitioned into an appropriate number of classes so that each node has at least one neighbor belonging to every class. Graphs that can satisfy the adjacency property are random graphs with large clustering coefficient such as random geometric graphs as well as line graphs of regular (or near regular) graphs.We first apply our framework to design two randomized distributed MIS algorithms for random geometric graphs of arbitrary dimension d (even non-constant). The first algorithm has ${\mathcal{O}}({\text{polyloglog }}n)$ (worst-case) awake complexity with high probability, where n is the number of nodes in the graph. 1 This means that any node in the network spends only ${\mathcal{O}}({\text{polyloglog }}n)$ awake rounds; this is almost exponentially better than the (traditional) time complexity of ${\mathcal{O}}({\text{log }}n)$ rounds (where there is no distinction between awake and sleeping rounds) known for distributed MIS algorithms on general graphs or even the faster ${\mathcal{O}}\left({\sqrt {\frac{{{\text{log }}n}}{{{\text{loglog }}n}}} }\right)$ rounds known for Erdos-Renyi random graphs. However, the (traditional) time complexity of our first algorithm is quite large—essentially proportional to the degree of the graph. Our second algorithm has a slightly worse awake complexity of ${\mathcal{O}}(d\,{\text{polyloglog }}n)$, but achieves a significantly better time complexity of ${\mathcal{O}}(d\,\log n\,{\text{polyloglog }}n)$ rounds whp.We also show that our framework can be used to design ${\mathcal{O}}({\text{polyloglog }}n)$ awake complexity MIS algorithms in other types of random graphs, namely an augmented Erdos-Renyi random graph that has a large clustering coefficient.