Description

This book presents a coherent and unified account of classical and more advanced techniques for analyzing the performance of randomized algorithms. Randomized algorithms have become a central part of the algorithms curriculum, based on their increasingly widespread use in modern applications. This book presents a coherent and unified account of classical and more advanced techniques for analyzing the performance of such algorithms. The presentation emphasizes discrete settings and elementary notions of probability, making it accessible to computer scientists and applied discrete mathematicians. Randomized algorithms have become a central part of the algorithms curriculum, based on their increasingly widespread use in modern applications. This book presents a coherent and unified account of classical and more advanced techniques for analyzing the performance of such algorithms. The presentation emphasizes discrete settings and elementary notions of probability, making it accessible to computer scientists and applied discrete mathematicians. Randomized algorithms have become a central part of the algorithms curriculum, based on their increasingly widespread use in modern applications. This book presents a coherent and unified treatment of probabilistic techniques for obtaining high probability estimates on the performance of randomized algorithms. It covers the basic toolkit from the Chernoff–Hoeffding bounds to more sophisticated techniques like martingales and isoperimetric inequalities, as well as some recent developments like Talagrand's inequality, transportation cost inequalities and log-Sobolev inequalities. Along the way, variations on the basic theme are examined, such as Chernoff–Hoeffding bounds in dependent settings. The authors emphasise comparative study of the different methods, highlighting respective strengths and weaknesses in concrete example applications. The exposition is tailored to discrete settings sufficient for the analysis of algorithms, avoiding unnecessary measure-theoretic details, thus making the book accessible to computer scientists as well as probabilists and discrete mathematicians. 1. Chernoff–Hoeffding bounds; 2. Applying the CH-bounds; 3. CH-bounds with dependencies; 4. Interlude: probabilistic recurrences; 5. Martingales and the MOBD; 6. The MOBD in action; 7. Averaged bounded difference; 8. The method of bounded variances; 9. Interlude: the infamous upper tail; 10. Isoperimetric inequalities and concentration; 11. Talagrand inequality; 12. Transportation cost and concentration; 13. Transportation cost and Talagrand's inequality; 14. Log–Sobolev inequalities; Appendix A. Summary of the most useful bounds. Review of the hardback: 'It is beautifully written, contains all the major concentration results, and is a must to have on your desk.' Richard Lipton Review of the hardback: 'Concentration bounds are at the core of probabilistic analysis of algorithms. This excellent text provides a comprehensive treatment of this important subject, ranging from the very basic to the more advanced tools, including some recent developments in this area. The presentation is clear and includes numerous examples, demonstrating applications of the bounds in analysis of algorithms. This book is a valuable resource for both researchers and students in the field.' Eli Upfal, Brown University Review of the hardback: 'Concentration inequalities are an essential tool for the analysis of algorithms in any probabilistic setting.