Converting Online Algorithms to Local Computation Algorithms

We propose a general method for converting online algorithms to local computation algorithms, by selecting a random permutation of the input, and simulating running the online algorithm. We bound the number of steps of the algorithm using a query tree, which models the dependencies between queries. We improve previous analyses of query trees on graphs of bounded degree, and extend this improved analysis to the cases where the degrees are distributed binomially, and to a special case of bipartite graphs. Using this method, we give a local computation algorithm for maximal matching in graphs of bounded degree, which runs in time and space O(log3n). We also show how to convert a large family of load balancing algorithms (related to balls and bins problems) to local computation algorithms. This gives several local load balancing algorithms which achieve the same approximation ratios as the online algorithms, but run in O(logn) time and space. Finally, we modify existing local computation algorithms for hypergraph 2-coloring and k-CNF and use our improved analysis to obtain better time and space bounds, of O(log4n), removing the dependency on the maximal degree of the graph from the exponent.

[1]  R. Otter.  The Multiplicative Process , 1949 .

[2]  T. E. Harris,et al.  The Theory of Branching Processes. , 1963 .

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

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

[5]  Allan Borodin,et al.  Online computation and competitive analysis , 1998 .

[6]  Desh Ranjan,et al.  Balls and bins: A study in negative dependence , 1996, Random Struct. Algorithms.

[7]  Eli Upfal,et al.  Balanced Allocations , 1999, SIAM J. Comput..

[8]  Berthold Vöcking,et al.  Balanced allocations: the heavily loaded case , 2000, STOC '00.

[9]  Ramesh K. Sitaraman,et al.  The power of two random choices: a survey of tech-niques and results , 2001 .

[10]  Jeffrey Considine,et al.  Simple Load Balancing for Distributed Hash Tables , 2003, IPTPS.

[11]  Berthold Vöcking,et al.  How asymmetry helps load balancing , 1999, JACM.

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

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

[14]  RonDana,et al.  Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms , 2007 .

[15]  Kunal Talwar,et al.  Balanced allocations: the weighted case , 2007, STOC '07.

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

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

[18]  André Brinkmann,et al.  Balls into non-uniform bins , 2010, 2010 IEEE International Symposium on Parallel & Distributed Processing (IPDPS).

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

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