The Discrepancy Method: Randomness and Complexity

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Cambridge University Press, 2001 - Computers - 475 pages
The discrepancy method is the glue that binds randomness and complexity. It is the bridge between randomized computation and discrepancy theory, the area of mathematics concerned with irregularities in distributions. The discrepancy method has played a major role in complexity theory; in particular, it has caused a mini-revolution of sorts in computational geometry. This book tells the story of the discrepancy method in a few short independent vignettes. It is a varied tale which includes such topics as communication complexity, pseudo-randomness, rapidly mixing Markov chains, points on the sphere and modular forms, derandomization, convex hulls, Voronoi diagrams, linear programming and extensions, geometric sampling, VC-dimension theory, minimum spanning trees, linear circuit complexity, and multidimensional searching. The mathematical treatment is thorough and self-contained. In particular, background material in discrepancy theory is supplied as needed. Thus the book should appeal to students and researchers in computer science, operations research, pure and applied mathematics, and engineering.

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Combinatorial Discrepancy
Upper Bound Techniques
Lower Bound Techniques
Geometric Searching
Complexity Lower Bounds
Linear Programming and Extensions
Communication Complexity
Minimum Spanning Trees
A Probability Theory
B Harmonic Analysis
Convex Geometry

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Page 457 - J. Beck. Sums of distances between points on a sphere - an application of the theory of irregularities of distribution to discrete geometry. Mathematika, 31:33-41, 1984.
Page 456 - J. Erickson, Geometric range searching and its relatives, in: Advances in Discrete and Computational Geometry (B.
Page 456 - I., Furedi, Z., Kleitman, D. Point selections and weak e-nets for convex hulls, Combinatorics, Probability and Computing, 3 (1992), 189-200. 14. Alon, N., Goldreich, O., Hastad, J., Peralta, R. Simple constructions of almost k-wise independent random variables, Random Structures & Algorithms, 3 (1992), 289-304. 15. Alon, N., Spencer, JH The Probabilistic Method, John Wiley fc Sons, 1992.
Page 456 - Alon, N., Goldreich, O., Hastad, J., Peralta, R., "Simple Constructions of Almost /t-wise Independent Random Variables" , Proc. 31st FOCS, 1990. [3] Azar, Y., Motwani, R., Naor, J., "An efficient construction of a multiple value small bias probability space
Page 457 - Roth's estimate of the discrepancy of integer sequences is nearly sharp. Combinatorica 1, 319-325.

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