Mathematics for the Analysis of Algorithms

1. Front Matter

   Pages i-viii

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2. Binomial Identities

   Pages 1-10

3. Recurrence Relations

   Pages 11-30

4. Operator Methods

   Pages 31-41

5. Asymptotic analysis

   Pages 42-76

6. Back Matter

   Pages 77-132

   PDF

This monograph, derived from an advanced computer science course at Stanford University, builds on the fundamentals of combinatorial analysis and complex variable theory to present many of the major paradigms used in the precise analysis of algorithms, emphasizing the more difficult notions. The authors cover recurrence relations, operator methods, and asymptotic analysis in a format that is terse enough for easy reference yet detailed enough for those with little background. Approximately half the book is devoted to original problems and solutions from examinations given at Stanford.

• algorithmic analysis
• algorithms
• asymptotic analysis
• binomial identities
• combinatorial analysis
• complex variable theory
• operator methods
• recurrence relations
• saddle point method
• calculus
• computer science
• Hashing
• Mathematica
• mathematics
• Variable
• algorithm analysis and problem complexity

“This is a short cookbook of methods for analyzing the run time of computer algorithms, aimed at computer scientists … . a very erudite book, full of interesting things for both mathematicians and computer scientists … .” (Allen Stenger, MAA Reviews, September, 2015)

“Mathematics for the Analysis of Algorithms covers a variety of topics in a relatively small amount of pages. Despite its briefness, most of the topics are clearly and fully explained using detailed examples for better understanding. As such, the book is suitable for use as study material, as well as a good reference guide…The reviewer recommends this book to anyone interested in advanced theory of algorithms and the mathematics behind it, either as an exposition to the topic or as reference material in future work.”   —SIGACT NEWS

"This book collects some fundamental mathematical techniques which are required for the analysis of algorithms... This book arose from handouts for an advanced course on the analysis of algorithms at Standard University, and the appendices list lectures, homework assignments and problems for the midterm and the final exams with their solutions. In summary, this book is a very valuable collection of mathematical techniques for the analysis of algorithms and accompanies, as well as complements, the second author's series The Art of Computer Programming

."   —Mathematical Reviews

"The book covers the important mathematical tools used in computer science, especially in the exact analysis of algorithms. A wide range of topics are covered, from the binomial theorem to the saddle point method and Laplace's techniques for asymptotic analysis...The book is very well written. The style and the mathematical exposition make the book pleasant to read...It covers many of the major paradigms used in the analysis of algorithms in its one hundred plus pages."   —SIAM Review

"The book presents a welcome selection and careful exposition of material that can be (and is) covered in a single course...In this reviewer's opinion, this would be an interesting text to use with a group of advanced students well-grounded in undergraduate mathematics and computer science, and would produce a valuable course for the participating students."   — Computing Reviews

"The reader has probably heard of the expression 'good things come in small packages.' The validity of that maxim is no more in evidence than in the work under review, which is nothing less than a mathematical wellspring among the otherwise parched world of theoretical algorithm analysis. In only 76 pages (not counting the bibliography and amazing appendices), the authors cover four important topics in algorithm analysis, all from a rudimentary, but highly original,

point of view: Binomial Identities, Recurrence Relations, Operator Methods, and Asymptotic Analysis. Each of these topics is critical to understanding the modern analysis of algorithms, primarily from the speed of execution perspective... In summary, the book under review should not be underestimated in its powerful use of mathematics for the analysis of algorithms arising from computer science considerations."   —Timothy Hall, Process Quality Improvement Consulting

"The analysis of algorthms is possible on mathematical and on computer scientific ways. This [book] is a mathematical look at this topic. It is based on an advanced course in computer science at Stanford University... The Appendices contain further difficult problems for applying the methods of this outstanding, full-of-thoughts book."   —P.L. Erdos (Periodica Mathematica Hungarica)

• Computer Science Laboratory, Xerox Palo Alto Research Center, Stanford, USA

  Daniel H. Greene

• Department of Computer Science, Stanford University, Stanford, USA

  Donald E. Knuth

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