Improved algorithms via approximations of probability distributions (extended abstract)

We present two techniques for approximating probability distributions. The first is a simple method for constructing the small-bias probability spaces introduced in [21]. This construction can be efficiently combined with the method of conditional probabilities to yield improved NC algorithms for many problems such as set cover, set discrepancy, finding large cuts in graphs etc. The second is a construction of small probability spaces approximating general independent distributions, which is of smaller size than the constructions of [13].

[1]  Moni Naor,et al.  The Probabilistic Method Yields Deterministic Parallel Algorithms , 1994, J. Comput. Syst. Sci..

[2]  Joel H. Spencer,et al.  Balancing vectors in the max norm , 1986, Comb..

[3]  Aravind Srinivasan,et al.  Chernoff-Hoeffding bounds for applications with limited independence , 1995, SODA '93.

[4]  Miklos Santha,et al.  Generating Quasi-random Sequences from Semi-random Sources , 1986, J. Comput. Syst. Sci..

[5]  Rudolf Lide,et al.  Finite fields , 1983 .

[6]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[8]  Moni Naor,et al.  Small-Bias Probability Spaces: Efficient Constructions and Applications , 1993, SIAM J. Comput..

[9]  Pierre Kelsen,et al.  On the parallel complexity of computing a maximal independent set in a hypergraph , 1992, STOC '92.

[10]  J. Spencer.  Ten lectures on the probabilistic method , 1987 .

[11]  Bonnie Berger,et al.  Approximation alogorithms for the maximum acyclic subgraph problem , 1990, SODA '90.

[12]  Noam Nisan,et al.  A parallel approximation algorithm for positive linear programming , 1993, STOC.

[13]  Bonnie Berger,et al.  Efficient NC Algorithms for Set Cover with Applications to Learning and Geometry , 1994, J. Comput. Syst. Sci..

[14]  Endre Szemerédi,et al.  Undirected Connectivity in O(l~gl*~ n) Space* , 1992 .

[15]  Noam Nisan,et al.  Approximations of general independent distributions , 1992, STOC '92.

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

[17]  Noga Alon,et al.  Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs , 1992, IEEE Trans. Inf. Theory.

[18]  Bonnie Berger,et al.  Simulating (logcn)-wise independence in NC , 1991, JACM.

[19]  Vijay V. Vazirani,et al.  Primal-dual RNC approximation algorithms for (multi)-set (multi)-cover and covering integer programs , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[20]  P. Erdös,et al.  On coloring graphs to maximize the proportion of multicolored k-edges , 1968 .

[21]  Derick Wood,et al.  An Optimal Worst Case Algorithm for Reporting Intersections of Rectangles , 1980, IEEE Transactions on Computers.

[22]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[23]  Noga Alon,et al.  Simple Construction of Almost k-wise Independent Random Variables , 1992, Random Struct. Algorithms.

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

[25]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[26]  Aravind Srinivasan,et al.  Improved distributed algorithms for coloring and network decomposition problems , 1992, STOC '92.

[27]  Michael Luby.  Removing Randomness in Parallel Computation without a Processor Penalty , 1993, J. Comput. Syst. Sci..

[28]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1990, Comb..

[29]  Noga Alon,et al.  Simple construction of almost k-wise independent random variables , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[30]  Bernard Chazelle,et al.  On linear-time deterministic algorithms for optimization problems in fixed dimension , 1996, SODA '93.