The Locality of Distributed Symmetry Breaking (1202.1983v3)

Abstract: Symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this paper we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes - An MIS algorithm running in $O(\log^2\Delta + 2^{O(\sqrt{\log\log n})})$ time, where $\Delta$ is the maximum degree. This is the first MIS algorithm to improve on the 1986 algorithms of Luby and Alon, Babai, and Itai, when $\log n \ll \Delta \ll 2^{\sqrt{\log n}}$, and comes close to the $\Omega(\log \Delta)$ lower bound of Kuhn, Moscibroda, and Wattenhofer. - A maximal matching algorithm running in $O(\log\Delta + \log^4\log n)$ time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on $\Delta$ is provably optimal. - A method for reducing symmetry breaking problems in low arboricity/degeneracy graphs to low degree graphs. (Roughly speaking, the arboricity or degeneracy of a graph bounds the density of any subgraph.) Corollaries of this reduction include an $O(\sqrt{\log n})$-time maximal matching algorithm for graphs with arboricity up to $2^{\sqrt{\log n}}$ and an $O(\log^{2/3} n)$-time MIS algorithm for graphs with arboricity up to $2^{(\log n)^{1/3}}$. Each of our algorithms is based on a simple, but powerful technique for reducing a randomized symmetry breaking task to a corresponding deterministic one on a poly$(\log n)$-size graph.

• The paper introduces novel randomized algorithms for MIS, maximal matching, and (Δ+1)-coloring, offering significant runtime improvements over classic approaches.
• The paper achieves optimal dependency on maximum degree Δ in the LOCAL model, with an MIS algorithm that nearly matches the Ω(log Δ) lower bound.
• The paper employs graph reduction techniques for low arboricity graphs, enhancing the practical efficiency of distributed symmetry-breaking in networked systems.

An Academic Insight into "The Locality of Distributed Symmetry Breaking"

The paper "The Locality of Distributed Symmetry Breaking" by Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider presents significant advancements in the domain of distributed computing by focusing on the LOCAL model. It revisits fundamental symmetry-breaking problems, specifically maximal independent sets (MIS), maximal matchings, vertex colorings, and ruling sets, and improves upon classic results in the randomized complexity of these problems.

Key Contributions and Results

The authors developed new algorithms that effectively leverage the capabilities of the LOCAL model to achieve better run-time performance:

1. Maximal Independent Set: They propose a randomized algorithm that executes in $O(\log \Delta + 2^{O(\sqrt{\log \log n})})$ time, for a graph with maximum degree $\Delta$. This marks the first significant improvement over classical algorithms such as those by Luby (1986) when the degree $\Delta$ is within a particular range. This is quite close to the established $\Omega(\log \Delta)$ lower bound.
2. Maximal Matching: A novel algorithm running in $O(\log \Delta + \log \log n)$ time emerges as another cornerstone of this research, being the first to advance existing algorithms from the late 80s. Its dependency on $\Delta$ is provably optimal in the LOCAL model environment.
3. Vertex Coloring: Their (Δ+1)-coloring algorithm requires $O(\log Δ + 2^{O(\sqrt{\log \log n})})$ time, further optimizing existing techniques and showcasing substantial improvements in computing complexity-related tasks.
4. Graph Reduction Techniques: The authors introduced a method for translating symmetry-breaking problems on graphs with low arboricity (a measure capturing the extent to which a graph can be partitioned into dense components) to problems on graphs with low degree. This allows achieving impressive time bounds on specific problem instances, such as an $O(\log^{2/3} n)$-time MIS algorithm on graphs with bounded arboricity.

Theoretical Implications

This work underlines the substantial gap between the efficiency of deterministic and randomized algorithms in distributed systems. Typically, randomized approaches achieve exponential speed-ups over deterministic equivalents due to their inherent ability to break symmetry more effectively. By improving these bounds, the paper not only challenges but also sets a higher benchmark in the distributed computation landscape.

Practical Implications

The implications of these algorithms are considerable in the field of networked systems where rapid, distributed decision-making is paramount. Specifically, improved algorithms for tasks like MIS and maximal matching can directly influence collaborative processes in network structures, load balancing, and resource allocation in computational clusters and distributed sensor networks.

Future Research Directions

Several areas for future exploration arise from this research. Assessing whether these methods could be further improved, especially regarding the dependence on $n$, remains a pertinent question. A deeper exploration of the possible limitations of deterministic algorithms, especially in the context of achieving similar time complexity to their randomized counterparts, also presents a fertile ground for future inquiry.

Conclusion

The paper is a robust contribution to distributed computing, enhancing our understanding of symmetry-breaking problems in the LOCAL model. The methodologies presented offer improved performance bounds and contribute significantly to both theoretical and practical facets of distributed systems. It stands as a guiding reference for ongoing research seeking to further optimize distributed algorithms with a focus on efficiency and scalability.