## 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 |

Sampling | 11 |

Upper Bound Techniques | 41 |

Lower Bound Techniques | 133 |

Geometric Searching | 206 |

Complexity Lower Bounds | 228 |

Convex Hulls and Voronoi Diagrams | 306 |

Pseudorandomness | 318 |

Communication Complexity | 346 |

Minimum Spanning Trees | 376 |

A Probability Theory | 430 |

B Harmonic Analysis | 442 |

Convex Geometry | 449 |

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### Common terms and phrases

algebraic algorithm arithmetic progressions assume boxes cell Chazelle coloring complexity Comput constant convex hull coordinates corresponding cross-ratio cusp cusp form defined denote derive dimension discrepancy theory discussion distribution e-nets edges eigenvalues elements elliptic curve entropy equation Euclidean Euler product factor finite fixed follows Fourier coefficients Fourier transform geometric given graph halfplane halfspace Hecke operators hyperbolic hyperplanes incidence matrix inequality integer intersection key-set Laplacian least Lemma linear low-discrepancy lower bound Matoušek modular forms modular functions modular group modulo multiplicative node norm Note operator optimal plane point set polynomial probability problem proof prove query queue random range searching range space Recall recursively resp Riemann surface root sampling sequence set system shatter function simplex soft heap subgroup subset Theorem theory tree triangle upper bound variables VC-dimension vector vertex vertices