Probabilistic Methods for Algorithmic Discrete Mathematics

1. Front Matter

   Pages I-XVII

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2. The Probabilistic Method

   • Michael Molloy

   Pages 1-35

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3. Probabilistic Analysis of Algorithms

   • Alan M. Frieze, Bruce Reed

   Pages 36-92

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4. An Overview of Randomized Algorithms

   • Rajeev Motwani, Prabhakar Raghavan

   Pages 93-115

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5. Mathematical Foundations of the Markov Chain Monte Carlo Method

   • Mark Jerrum

   Pages 116-165

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6. Percolation and the Random Cluster Model: Combinatorial and Algorithmic Problems

   • Dominic Welsh

   Pages 166-194

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7. Concentration

   • Colin McDiarmid

   Pages 195-248

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8. Branching Processes and Their Applications in the Analysis of Tree Structures and Tree Algorithms

   • Luc Devroye

   Pages 249-314

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9. Back Matter

   Pages 315-325

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Leave nothing to chance. This cliche embodies the common belief that ran­ domness has no place in carefully planned methodologies, every step should be spelled out, each i dotted and each t crossed. In discrete mathematics at least, nothing could be further from the truth. Introducing random choices into algorithms can improve their performance. The application of proba­ bilistic tools has led to the resolution of combinatorial problems which had resisted attack for decades. The chapters in this volume explore and celebrate this fact. Our intention was to bring together, for the first time, accessible discus­ sions of the disparate ways in which probabilistic ideas are enriching discrete mathematics. These discussions are aimed at mathematicians with a good combinatorial background but require only a passing acquaintance with the basic definitions in probability (e.g. expected value, conditional probability). A reader who already has a firm grasp on the area will be interested in the original research, novel syntheses, and discussions of ongoing developments scattered throughout the book. Some of the most convincing demonstrations of the power of these tech­ niques are randomized algorithms for estimating quantities which are hard to compute exactly. One example is the randomized algorithm of Dyer, Frieze and Kannan for estimating the volume of a polyhedron. To illustrate these techniques, we consider a simple related problem. Suppose S is some region of the unit square defined by a system of polynomial inequalities: Pi (x. y) ~ o.

• Graph
• Monte Carlo Method
• Probabilistic method
• Sim
• algorithm
• algorithms
• calculus
• probabilistic combinatorics
• randomized algorithms

• LIRMM, Montpellier Cedex 5, France

  Michel Habib

• Department of Statistics, University of Oxford, Oxford, UK

  Colin McDiarmid

• Equipe Combinatoire, Université Pierre et Marie Curie, Paris 6 Case 189, Paris Cedex 5, France

  Jorge Ramirez-Alfonsin, Bruce Reed

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