Geometric discrepancy revisited

Discrepancy theory addresses the general issue of approximating one measure by another one. Originally an offshoot of diophantine approximation theory, the area has expanded into applied mathematics, and now, computer science. Besides providing the theoretical foundation for sampling, it holds some of the keys to understanding the computational power of randomization. A few applications of discrepancy theory are listed. We give elementary algorithms for estimating the discrepancy between various measures arising in practice. We also present a general technique for proving discrepancy lower bounds.<<ETX>>

[1]  József Beck,et al.  Irregularities of distribution: Index of theorems and corollaries , 1987 .

[2]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[3]  Bernard Chazelle,et al.  Cutting hyperplanes for divide-and-conquer , 1993, Discret. Comput. Geom..

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

[5]  Jirí Matousek,et al.  Discrepancy and approximations for bounded VC-dimension , 1993, Comb..

[6]  K. Stolarsky.  Sums of distances between points on a sphere. II , 1972 .

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

[8]  David Eppstein,et al.  Computing the discrepancy , 1993, SCG '93.

[9]  Prabhakar Raghavan,et al.  Probabilistic construction of deterministic algorithms: Approximating packing integer programs , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[10]  Peter Shirley,et al.  Discrepancy as a Quality Measure for Sample Distributions , 1991, Eurographics.

[11]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[12]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[13]  Bernard Chazelle,et al.  An optimal convex hull algorithm in any fixed dimension , 1993, Discret. Comput. Geom..

[14]  W. Schmidt.  Irregularities of distribution , 1968 .

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

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

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

[18]  L. Santaló.  Integral geometry and geometric probability , 1976 .

[19]  József Beck.  Quasi-Random 2-Colorings of Point Sets , 1991, Random Struct. Algorithms.

[20]  H. Weyl.  Über die Gleichverteilung von Zahlen mod. Eins , 1916 .

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

[22]  Michael E. Saks,et al.  Efficient construction of a small hitting set for combinatorial rectangles in high dimension , 1993, Comb..

[23]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[24]  Don P. Mitchell,et al.  Spectrally optimal sampling for distribution ray tracing , 1991, SIGGRAPH.

[25]  Bonnie Berger,et al.  Simulating (log/sup c/n)-wise independence in NC , 1989, 30th Annual Symposium on Foundations of Computer Science.

[26]  Glyn Harman,et al.  Sums of distances between points of a sphere , 1982 .

[27]  J. Beck,et al.  Irregularities of distribution , 1987 .

[28]  Bernard Chazelle,et al.  Quasi-optimal range searching in spaces of finite VC-dimension , 1989, Discret. Comput. Geom..

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

[30]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[31]  Ralph Alexander,et al.  Geometric methods in the study of irregularities of distribution , 1990, Comb..

[32]  William W. L. Chen.  On irregularities of distribution. , 1980 .

[33]  K. F. Roth.  On irregularities of distribution , 1954 .

[34]  W. Schmidt.  IRREGULARITIES OF DISTRIBUTION (Cambridge Tracts in Mathematics 89) , 1988 .