## Randomized AlgorithmsFor many applications a randomized algorithm is either the simplest algorithm available, or the fastest, or both. This tutorial presents the basic concepts in the design and analysis of randomized algorithms. The first part of the book presents tools from probability theory and probabilistic analysis that are recurrent in algorithmic applications. Algorithmic examples are given to illustrate the use of each tool in a concrete setting. In the second part of the book, each of the seven chapters focuses on one important area of application of randomized algorithms: data structures; geometric algorithms; graph algorithms; number theory; enumeration; parallel algorithms; and on-line algorithms. A comprehensive and representative selection of the algorithms in these areas is also given. This book should prove invaluable as a reference for researchers and professional programmers, as well as for students. |

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### Contents

I | ix |

II | 1 |

III | 3 |

IV | 28 |

V | 43 |

VI | 67 |

VII | 101 |

VIII | 127 |

X | 195 |

XI | 197 |

XII | 234 |

XIII | 278 |

XIV | 306 |

XV | 335 |

XVI | 368 |

XVII | 392 |

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### Common terms and phrases

adversary analysis apply assume binary bipartite graph Boolean cache Chapter Chernoff bound choice choose chosen clauses competitiveness coefficient compute Consider constraints contains corresponding cost define Definition deleted described determine deterministic algorithm distribution elements evaluation Exercise expanding graphs expected number expected running given graph G hash function independent inequality input integer intersection iteration least Lemma linear program lower bound Markov chain martingale matrix multiplication min-cut modulo node number of edges number of steps O(logn obtain offline online algorithm output pairwise independent partition path perfect matching permutation pointers polynomial prime processors proof prove quadratic residue random bits random variable random walk randomized algorithm recursive result RNC algorithm sampling satisfying Section segments Show skip list space square roots sub-tree subset Suppose technique Theorem treap truth assignment uniformly at random upper bound vector verify vertex weight

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Page 463 - JP Schmidt, A. Siegel, and A. Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence. In Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 331-340, 1993.

Page 451 - U. Manber and G. Myers. Suffix arrays: A new method for on-line string searches. In Proceedings of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms, pages 319-327, 1990.

Page 452 - KL Clarkson. A Las Vegas algorithm for linear programming when the dimension is small. In Proc. 29th Annu. IEEE Sympos. Found. Comput. Sei., pages 452-456, 1988. [21] KL Clarkson and PW Shor. Applications of random sampling in computational geometry, II.