An Improved Distributed Algorithm for Maximal Independent Set (1506.05093v2)

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.

• The paper presents a novel randomized distributed algorithm for the Maximal Independent Set (MIS) problem, achieving near-optimal local complexity.
• The algorithm ensures any node decides its MIS participation locally within O(log deg(v) + log 1/ε) rounds with high probability, using information from its 2-hop neighborhood.
• Leveraging prior techniques, the algorithm attains a global complexity of O(log Δ) + 2^{O(√log log n)}, improving upon known results and having implications for Ruling Sets, LCA, and LLL computations.

Overview of "An Improved Distributed Algorithm for Maximal Independent Set"

The paper addresses the Maximal Independent Set (MIS) problem, a fundamental problem within distributed graph algorithms. The research introduces a novel and randomized distributed algorithm that achieves near-optimal local complexity for solving the MIS problem. The proposed algorithm distinguishes itself by its simplicity and its efficient handling of both local and global computational complexities.

Key Contributions

1. Local Complexity Optimization:
   • The paper presents an algorithm that guarantees any specific node $v$ will decide its participation in the MIS within $O(\log \mathsf{deg}(v) + \log 1/\varepsilon)$ rounds, with high probability. The computation is localized within a $2$-hop neighborhood around $v$, allowing the algorithm to potentially operate under adversarial conditions because it includes randomness only within this neighborhood.
2. Transition to Global Complexity:
   • By leveraging techniques from previous works [Barenboim et al., FOCS'12], the algorithm achieves a global complexity of $O(\log \Delta) + 2^{O(\sqrt{\log \log n})}$, where $\Delta$ denotes the maximum degree in the graph. This represents an enhancement over known results, approaching the theoretical lower bound of Kuhn et al. without exceeding computational feasibility for large graphs.
3. Algorithm Implications:
   • The paper also explores various implications. It accommodates graphs of bounded arboricity and girth, resulting in improved algorithms for particular graph classes. Furthermore, applications extend to Ruling Sets, the Local Computation Algorithm (LCA) model, and improvements for distributed Lovász Local Lemma computations.

Implications and Future Work

The results of this research underscore the practical and theoretical significance suggested by the capability of efficiently solving MIS problems with respect to local information in distributed networks. The improvements in complexity potentially reduce resource requirements for network operations and facilitate scalability in big data applications or internet settings characterized by high node degrees.

In terms of future directions, the work sets a critical foundation for exploring further optimization under varied constraints and assumptions of network structures. Challenges remain in closing the gap between existing upper and lower bounds when $\log \Delta = \omega(\sqrt{\log n})$, thus pushing forward the understanding of locality and randomness roles in distributed computation. Moreover, testing these theoretical advancements in real-world distributed network scenarios could offer valuable insights and validation, encouraging a push towards practical distributed computing solutions.

In summary, the contributions presented not only advance the algorithmic techniques available for solving the MIS problem but also expand the scope of application and understanding of distributed algorithms in graph theory. The work opens avenues for future theoretical exploration and immediate applicability in distributed systems requiring efficient resource coordination.