An Improved Distributed Algorithm for Maximal Independent Set

TL;DR

This paper introduces a simple randomized distributed algorithm for the Maximal Independent Set problem that achieves near-optimal local and global complexities, improving previous bounds and applying to various graph classes and models.

Contribution

The paper presents a new simple randomized algorithm for MIS with degree-dependent local complexity and improved global complexity bounds, advancing the state of the art.

Findings

Achieves local termination in O(log deg(v)+log 1/ε) rounds with high probability.

Global complexity improved to O(log Δ) + 2^{O(√log log n)} rounds.

Applicable to graphs with bounded arboricity, girth, and other models like LCA and Lovász Local Lemma.

Abstract

The Maximal Independent Set (MIS) problem is one of the basics in the study of locality in distributed graph algorithms. This paper presents an extremely simple randomized algorithm providing a near-optimal local complexity for this problem, which incidentally, when combined with some recent techniques, also leads to a near-optimal global complexity. Classical algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86] provide the global complexity guarantee that, with high probability, all nodes terminate after $O(\log n)$ rounds. In contrast, our initial focus is on the local complexity, and our main contribution is to provide a very simple algorithm guaranteeing that each particular node $v$ terminates after $O(\log \mathsf{deg}(v)+\log 1/\epsilon)$ rounds, with probability at least $1-\epsilon$. The guarantee holds even if the randomness outside $2$-hops neighborhood of $v$ is determined adversarially. This degree-dependency is optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Interestingly, this local complexity smoothly transitions to a global complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider [FOCS'12, arXiv: 1202.1983v3], we get a randomized MIS algorithm with a high probability global complexity of $O(\log \Delta) + 2^{O(\sqrt{\log \log n})}$, where $\Delta$ denotes the maximum degree. This improves over the $O(\log^2 \Delta) + 2^{O(\sqrt{\log \log n})}$ result of Barenboim et al., and gets close to the $\Omega(\min\{\log \Delta, \sqrt{\log n}\})$ lower bound of Kuhn et al. Corollaries include improved algorithms for MIS in graphs of upper-bounded arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local Computation Algorithms (LCA) model, and a faster distributed algorithm for the Lov\'asz Local Lemma.