## The Discrepancy Method: Randomness and ComplexityThe 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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### Contents

Combinatorial Discrepancy | 1 |

Upper Bound Techniques | 41 |

Lower Bound Techniques | 133 |

Sampling | 169 |

Geometric Searching | 206 |

Complexity Lower Bounds | 228 |

Linear Programming and Extensions | 307 |

Pseudorandomness | 318 |

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### Common terms and phrases

algorithm apply associated assume bits called cell Chapter choose coloring complexity compute condition consider constant construction contains contraction convex corresponding cost curve cutting defined denote derive dimension discrepancy discussion distinct distribution easily edges eigenvalues elements elliptic equal example exists expected fact factor finite fixed follows Fourier function geometric given gives graph heap hence hyperplanes incidence independent integer intersection interval invariant least leaves Lemma linear lower bound matrix means method modular node Note obtained operator optimal original plane points probability problem proof prove query random range space rank Recall resp respect result root sample satisfies searching set system simple simplex Step subset takes Theorem theory transform tree triangle upper variables vector vertex vertices weight

### Popular passages

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.