The Locality of Distributed Symmetry Breaking

We present new bounds on the locality of several classical symmetry breaking tasks in distributed networks. A sampling of the results include 1) A randomized algorithm for computing a maximal matching (MM) in O(log Δ + (log log n)<sup>4</sup>) rounds, where Δ is the maximum degree. This improves a 25-year old randomized algorithm of Israeli and Itai that takes O(log n) rounds and is provably optimal for all log Δ in the range [(log log n)<sup>4</sup>, √log n]. 2) A randomized maximal independent set (MIS) algorithm requiring O(log Δ√log n) rounds, for all Δ, and only 2<sup>O</sup>(√log log n) rounds when Δ = poly(log n). These improve on the 25-year old O(log n)-round randomized MIS algorithms of Luby and Alon, Babai, and Itai when log Δ ≫ √log n. 3) A randomized (Δ + 1)-coloring algorithm requiring O(log Δ + 2<sup>O</sup>(<sup>(√log log n)</sup>) rounds, improving on an algorithm of Schneider and Wattenhofer that takes O(log Δ + √log n) rounds. This result implies that an O(Δ)-coloring can be computed in 2<sup>O(√log log n)</sup> rounds for all Δ, improving on Kothapalli et al.'s O(√log n)-round algorithm. We also introduce a new technique for reducing symmetry breaking problems on low arboricity graphs to low degree graphs. Corollaries of this reduction include MM and MIS algorithms for low arboricity graphs (e.g., planar graphs and graphs that exclude any fixed minor) requiring O(√log n) and O(log<sup>2/3</sup> n) rounds w.h.p., respectively.

[1]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[2]  C. Nash-Williams.  Decomposition of Finite Graphs Into Forests , 1964 .

[3]  Christoph Lenzen,et al.  MIS on trees , 2011, PODC '11.

[4]  Öjvind Johansson.  Simple Distributed Delta+1-coloring of Graphs , 1999, Inf. Process. Lett..

[5]  Sriram V. Pemmaraju,et al.  Distributed graph coloring in a few rounds , 2011, PODC '11.

[6]  C. Scheideler,et al.  Distributed coloring in O~(⎷(log n)) bit rounds , 2006, IPDPS.

[7]  Roger Wattenhofer,et al.  Distributed Weighted Matching , 2004, DISC.

[8]  Alessandro Panconesi,et al.  Concentration of Measure for the Analysis of Randomized Algorithms , 2009 .

[9]  Rocco A. Servedio,et al.  Martingale Boosting , 2005, COLT.

[10]  Beat Gfeller,et al.  A randomized distributed algorithm for the maximal independent set problem in growth-bounded graphs , 2007, PODC '07.

[11]  Roger Wattenhofer,et al.  Symmetry breaking depending on the chromatic number or the neighborhood growth , 2013, Theor. Comput. Sci..

[12]  Richard Cole,et al.  Deterministic Coin Tossing with Applications to Optimal Parallel List Ranking , 2018, Inf. Control..

[13]  Aravind Srinivasan,et al.  On the Complexity of Distributed Network Decomposition , 1996, J. Algorithms.

[14]  Sriram V. Pemmaraju,et al.  Super-Fast 3-Ruling Sets , 2012, FSTTCS.

[15]  Alessandro Panconesi,et al.  A faster distributed algorithm for computing maximal matchings deterministically , 1999, PODC '99.

[16]  Noga Alon,et al.  A Parallel Algorithmic Version of the Local Lemma , 1991, Random Struct. Algorithms.

[17]  Leonid Barenboim,et al.  Deterministic Distributed Vertex Coloring in Polylogarithmic Time , 2010, JACM.

[18]  Christian Scheideler,et al.  Distributed coloring in O/spl tilde/(/spl radic/(log n)) bit rounds , 2006, Proceedings 20th IEEE International Parallel & Distributed Processing Symposium.

[19]  Leonid Barenboim,et al.  Distributed (Delta+1)-Coloring in Linear (in Delta) Time , 2014, SIAM J. Comput..

[20]  Nathan Linial,et al.  Distributive graph algorithms Global solutions from local data , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[21]  Devdatt P. Dubhashi,et al.  Balls and bins: A study in negative dependence , 1996, Random Struct. Algorithms.

[22]  Roger Wattenhofer,et al.  An optimal maximal independent set algorithm for bounded-independence graphs , 2010, Distributed Computing.

[23]  Alessandro Panconesi,et al.  Some simple distributed algorithms for sparse networks , 2001, Distributed Computing.

[24]  Fabian Kuhn.  Weak graph colorings: distributed algorithms and applications , 2009, SPAA '09.

[25]  Svante Janson,et al.  Large deviations for sums of partly dependent random variables , 2004, Random Struct. Algorithms.

[26]  N Linial,et al.  Low diameter graph decompositions , 1993, Comb..

[27]  József Beck,et al.  An Algorithmic Approach to the Lovász Local Lemma. I , 1991, Random Struct. Algorithms.

[28]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[29]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[30]  Sundar Vishwanathan,et al.  Locality based graph coloring , 1993, STOC.

[31]  Roger Wattenhofer,et al.  A new technique for distributed symmetry breaking , 2010, PODC '10.

[32]  Alon Itai,et al.  A Fast and Simple Randomized Parallel Algorithm for Maximal Matching , 1986, Inf. Process. Lett..

[33]  Roger Wattenhofer,et al.  What cannot be computed locally! , 2004, PODC '04.

[34]  Dan Suciu,et al.  Journal of the ACM , 2006 .

[35]  Roger Wattenhofer,et al.  A log-star distributed maximal independent set algorithm for growth-bounded graphs , 2008, PODC '08.

[36]  Hsin-Hao Su,et al.  Distributed algorithms for the Lovász local lemma and graph coloring , 2014, PODC '14.

[37]  Alessandro Panconesi,et al.  On the distributed complexity of computing maximal matchings , 1997, SODA '98.

[38]  Gábor Tardos,et al.  A constructive proof of the general lovász local lemma , 2009, JACM.

[39]  Leonid Barenboim,et al.  Distributed Graph Coloring: Fundamentals and Recent Developments , 2013, Distributed Graph Coloring: Fundamentals and Recent Developments.

[40]  Andrew V. Goldberg,et al.  Network decomposition and locality in distributed computation , 1989, 30th Annual Symposium on Foundations of Computer Science.

[41]  Sriram V. Pemmaraju,et al.  Brief announcement: Super-fast t-ruling sets , 2014, PODC '14.

[42]  Jean-Sébastien Sereni,et al.  Toward more localized local algorithms: removing assumptions concerning global knowledge , 2011, PODC '11.

[43]  Ronitt Rubinfeld,et al.  Fast Local Computation Algorithms , 2011, ICS.

[44]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

[45]  Nico Eigenmann.  ( Δ + 1 )-COLORING IN LINEAR ( IN Δ ) TIME , 2009 .

[46]  Leonid Barenboim,et al.  Distributed (δ+1)-coloring in linear (in δ) time , 2009, STOC '09.

[47]  Yves Métivier,et al.  An optimal bit complexity randomized distributed MIS algorithm , 2011, Distributed Computing.

[48]  Leonid Barenboim,et al.  The Locality of Distributed Symmetry Breaking , 2012, FOCS.

[49]  Roger Wattenhofer,et al.  On the complexity of distributed graph coloring , 2006, PODC '06.