Sriram V. Pemmaraju
ABSTRACT
Until recently, the fastest distributed MIS algorithm, even for simple graphs, e.g., unoriented trees, has been the simple randomized algorithm discovered in the 80s. This algorithm (commonly called Luby's algorithm) computes an MIS in O(log n) rounds (with high probability). This situation changed when Lenzen and Wattenhofer (PODC 2011) presented a randomized O(√log n} ⋅ log\log n)-round MIS algorithm for unoriented trees. This algorithm was improved by Barenboim et al. (FOCS 2012), resulting in an MIS algorithm running in O(√log n ⋅ log\log n) rounds.
The analyses of these tree MIS algorithms depends on "near independence" of probabilistic events, a feature of the tree structure of the network. In their paper, Lenzen and Wattenhofer hope that their algorithm and analysis could be extended to graphs with bounded arboricity. We show how to do this in this note. By using a new tail inequality for read-k families of random variables due to Gavinsky et al. Random Struct Algorithms, 2015), we show how to deal with dependencies induced by the recent tree MIS algorithms when they are executed on bounded arboricity graphs. Specifically, we analyze a version of the tree MIS algorithm of Barenboim et al. and show that it runs in O(poly(α) ⋅ √log n ⋅ log log n) rounds in the CONGEST model for arboricity-α graphs.
- N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms, 7(4):567--583, 1986. Google Scholar
Digital Library
- L. Barenboim, M. Elkin, S. Pettie, and J. Schneider. The locality of distributed symmetry breaking. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20--23, 2012, pages 321--330, 2012. Google ScholarDigital Library
- L. Barenboim, M. Elkin, S. Pettie, and J. Schneider. The locality of distributed symmetry breaking. CoRR, abs/1202.1983, 2015. Google ScholarDigital Library
- D. Gavinsky, S. Lovett, M. Saks, and S. Srinivasan. A tail bound for read-k families of functions. Random Structures & Algorithms, 47:99--108, 2015. Google ScholarDigital Library
- A. Israeli and A. Itai. A fast and simple randomized parallel algorithm for maximal matching. Inf. Process. Lett., 22(2):77--80, 1986. Google ScholarDigital Library
- C. Lenzen and R. Wattenhofer. MIS on trees. In Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing, PODC '11, pages 41--48, New York, NY, USA, 2011. ACM. Google ScholarDigital Library
- M. Luby. A simple parallel algorithm for the maximal independent set. SIAM Journal on Computing, 15:1036--1053, 1986. Google ScholarDigital Library
- S. V. Pemmaraju and T. Riaz. Using Read-k Inequalities to Analyze a Distributed MIS Algorithm. ArXiv e-prints, May 2016.Google Scholar
Index Terms
- Brief Announcement: Using Read-k Inequalities to Analyze a Distributed MIS Algorithm
Recommendations
Brief Announcement: Symmetry Breaking in the CONGEST Model: Time- and Message-Efficient Algorithms for Ruling Sets
PODC '17: Proceedings of the ACM Symposium on Principles of Distributed ComputingWe study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. Our work is motivated by the following central question: can we break the long-...
Brief announcement: almost-tight approximation distributed algorithm for minimum cut
PODC '14: Proceedings of the 2014 ACM symposium on Principles of distributed computingIn this short paper, we present an improved algorithm for approximating the minimum cut on distributed (CONGEST) networks. Let λ be the minimum cut. Our algorithm can compute λ exactly in O((√n+D) poly(λ)) time, where n is the number of nodes (...
Brief Announcement: An Exponential Separation Between Randomized and Deterministic Complexity in the LOCAL Model
PODC '16: Proceedings of the 2016 ACM Symposium on Principles of Distributed ComputingOver the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially ...
Comments