Concentration of Measure for the Analysis of Randomised Algorithms

Concentration of Measure for the Analysis of Randomised Algorithms

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Unformatted text preview: D R A F T Concentration of Measure for the Analysis of Randomised Algorithms Devdatt P. Dubhashi Alessandro Panconesi October 21, 2005 D R A F T 2 D R A F T Contents 1 Chernoff–Hoeffding Bounds 17 1.1 What is “Concentration of Measure”? . . . . . . . . . . . . . . . . 17 1.2 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . 19 1.3 The Chernoff Bound . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4 Heterogeneous Variables . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 The Hoeffding Extension . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 Useful Forms of the Bound . . . . . . . . . . . . . . . . . . . . . . 22 1.7 A Variance Bound . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.8 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 Interlude: Probabilistic Recurrences 33 3 Applying the CH-bounds 39 3.1 Probabilistic Amplification . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Load Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Data Structures: Skip Lists . . . . . . . . . . . . . . . . . . . . . 41 3.3.1 Skip Lists: The Data Structure . . . . . . . . . . . . . . . 41 3.3.2 Skip Lists: Randomization makes it easy . . . . . . . . . . 42 3 D R A F T 4 CONTENTS 3.3.3 Quicksort . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4 Packet Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 Randomized Rounding . . . . . . . . . . . . . . . . . . . . . . . . 49 3.6 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4 Chernoff-Hoeffding Bounds in Dependent Settings 55 4.1 Negative Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Local Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Janson’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Limited Independence . . . . . . . . . . . . . . . . . . . . . . . . 64 4.5 Markov Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.5.2 Statement of the Bound . . . . . . . . . . . . . . . . . . . 68 4.5.3 Application: Probability Amplification . . . . . . . . . . . 68 4.6 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5 Martingales and Azuma’s Inequality 73 5.1 Review of Conditional Probabilities and Expectations . . . . . . . 74 5.2 Martingales and Azuma’s Inequality. . . . . . . . . . . . . . . . . 76 5.3 Generalizing Martingales and Azuma’s Inequality. . . . . . . . . . 81 5.4 The Method of Bounded Differences . . . . . . . . . . . . . . . . . 83 5.5 Bibliographic Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 D...
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.

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Concentration of Measure for the Analysis of Randomised Algorithms

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