Fast Local Computation Algorithms

For input x, let F(x) denote the set of outputs that are the “legal” answers for a co mputational problem F . Suppose x and members of F(x) are so large that there is not time to read them in their entirety. We propose a model of local computation algorithmswhich for a given input x, support queries by a user to values of specified locations yi in a legal output y ∈ F(x). When more than one legal output y exists for a given x, the local computation algorithm should output in a way that is consistent with at least one such y. Local computation algorithms are intended to distill the c ommon features of several concepts that have appeared in various algorithmic subfield s, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on known constructions of small sample spaces ofk-wise independent random variables and Beck’s analysis in his algorithmic app roach to the Lovasz Local Lemma, which under certain conditions can be applied to construct local computation algorithms that run in polylogarithmic time and space. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying k-SAT formulas.

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

[2]  Fabian Kuhn,et al.  Distributed Approximation of Capacitated Dominating Sets , 2007, SPAA '07.

[3]  P. Erdos-L Lovász.  Problems and Results on 3-chromatic Hypergraphs and Some Related Questions , 2022 .

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

[5]  Robin A. Moser.  A constructive proof of the Lovász local lemma , 2008, STOC '09.

[6]  L. Lovász.  Combinatorial problems and exercises , 1979 .

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

[8]  Shang-Hua Teng,et al.  Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems , 2003, STOC '04.

[9]  M. Kaufmann.  What Can Be Computed Locally ? , 2003 .

[10]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[11]  Krzysztof Onak,et al.  Local Graph Partitions for Approximation and Testing , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[12]  Moni Naor,et al.  Local computations on static and dynamic graphs , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

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

[14]  Fabian Kuhn.  Local Multicoloring Algorithms: Computing a Nearly-Optimal TDMA Schedule in Constant Time , 2009, STACS.

[15]  Fan Chung Graham,et al.  Local Graph Partitioning using PageRank Vectors , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[16]  GoldreichOded,et al.  Property testing and its connection to learning and approximation , 1998 .

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

[18]  Roger Wattenhofer,et al.  The price of being near-sighted , 2006, SODA '06.

[19]  Roger Wattenhofer,et al.  Fast Deterministic Distributed Maximal Independent Set Computation on Growth-Bounded Graphs , 2005, DISC.

[20]  Richard M. Karp,et al.  A fast parallel algorithm for the maximal independent set problem , 1984, STOC '84.

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

[22]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[23]  Pavel Berkhin,et al.  Bookmark-Coloring Algorithm for Personalized PageRank Computing , 2006, Internet Math..

[24]  Bernard Chazelle,et al.  Online geometric reconstruction , 2006, SCG '06.

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

[26]  Krzysztof Onak,et al.  Constant-Time Approximation Algorithms via Local Improvements , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[27]  Yuichi Yoshida,et al.  An improved constant-time approximation algorithm for maximum~matchings , 2009, STOC '09.

[28]  Richard M. Karp,et al.  A fast parallel algorithm for the maximal independent set problem , 1985, JACM.

[29]  Noga Alon,et al.  Single Round Simulation on Radio Networks , 1992, J. Algorithms.

[30]  Jonathan Katz,et al.  On the efficiency of local decoding procedures for error-correcting codes , 2000, STOC '00.

[31]  Bernard Chazelle,et al.  Property-Preserving Data Reconstruction , 2004, Algorithmica.

[32]  Michael E. Saks,et al.  Local Monotonicity Reconstruction , 2010, SIAM J. Comput..

[33]  Dana Ron,et al.  Distance Approximation in Bounded-Degree and General Sparse Graphs , 2006, APPROX-RANDOM.

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

[35]  Andrew V. Goldberg,et al.  Parallel symmetry-breaking in sparse graphs , 1987, STOC.

[36]  Noga Alon,et al.  On the complexity of radio communication , 1989, STOC '89.

[37]  Vahab S. Mirrokni,et al.  Local Computation of PageRank Contributions , 2007, Internet Math..

[38]  Dana Ron,et al.  On Approximating the Minimum Vertex Cover in Sublinear Time and the Connection to Distributed Algorithms , 2007, Electron. Colloquium Comput. Complex..

[39]  Jennifer Widom,et al.  Scaling personalized web search , 2003, WWW '03.

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

[41]  Noga Alon,et al.  Space-efficient local computation algorithms , 2011, SODA.

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

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

[44]  András A. Benczúr,et al.  To randomize or not to randomize: space optimal summaries for hyperlink analysis , 2006, WWW '06.

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