Fundamentals of the Average Case Analysis of Particular Algorithms

Transcription

1 Fundamentals of the Average Case Analysis of Particular Algorithms

2 Wiley-Teubner Series in Computer Science Editorial Board Günter Hotz Universität des Saarlandes. Fachbereich Angewandte Mathematik und Informatik. Saarbrücken. Germanv Michael J. Fischer Department of Computer Science. New Haven, Connecticut. USA Maurice Nivat U.E.R. de Mathematique et lnformatique. Universite Paris VII. Paris. France David W. Barron Department of Computer Science. Universitv of Southampton. UK Kemp: Fundamentals of the Average Case Analysis of Particular Algorithms Loeckx/Sieber/Stansifer: The Foundations of Program Verification

3 Fundamentals of the Average Case Analysis of Particular Algorithms Rainer Kemp Fachbereich Informatik Universität Frankfurt Germany Springer Fachmedien Wiesbaden GmbH

4 Copyright 1984 by Springer Fachmedien Wiesbaden Originally published by B.G. Teubner, Stuttgart in 1984 Ali rights reserved. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher. Library of Congress Cataloging in Publication Data: Kemp, Rainer. Fundamentals of the average case analysis of particular algorithms. (Wiley-Teubner series in computer science) lncludes bibliographical references and index. 1. Electronic digital computers-programming. 2. Algorithms. 1. Title. 11. Series. QA76.6.K ISBN ISBN (ebook) DOI / British Library Cataloguing in Publication Data: Kemp, Rainer. Fundamentals of the average case analysis of particular algorithms.-(wiley-teubner series in computer science) 1. Algorithms. 1. Title. 511 '.6 QA9.58 ISBN CIP-Kurztitelaufnahme der Deutschen Bibliothek: Kemp, Rainer. Fundamentals of the average case analysis of particular algorithms/ Rai ner Kemp. (Wiley-Teubner series in computer science) ISBN Typeset by Mid-County Press, London, SW15.

5 Preface This book provides an introduction to the analysis of particular algorithms. lt has its origin in lecture courses given at the Universität des Saarlandes, Saarbrücken in 1980 and atthejohann Wolfgang Goethe-Universität, Frankfurt a.m. in The material can be covered in a one-semester course. ln preparing the notes for publication as a book, I have added a considerable amount of material additional to the lecture notes, with the intention of making the book more useful. My prime consideration has been to produce a textbook whose scope is selective; some of the omitted material is outlined in various exercises and should be useful in indicating possible approaches to certain problems. Moreover, problems are provided to furnish examples, to expand on the material or to indicate related results, and occasionally to guide the reader through the steps of lengthy proofs and derivations. I have referred, in various places, to those books and original papers which have been of particular assistance to me. I wish to take this opportunity to thank all those who have had apart in this work, and who have made this book possible. I am particularly indebted to Professor Dr. Günter Hotz for his encouragement in the writing of this textbook. Special thanks are due to Ute Schürfeld for careful reading of the text. Dr. P. Spuhler from Teubner Verlag provided co-operative and competent support in all editorial problems. Finally, I wish to thank Teubner-Verlag and John Wiley & Sons for very good and timely editorial work. Frankfurt a.m., West Germany December 1982 R. Kemp V

6 Contents Preface 1. lntroduction Exercises 2. A Simple Class of Algorithms 2.1 Definitior1 ot a Random Algorithm 2.2 Analysis of Random Algorithms Exercises 3. Permutationsand Their Applications 3.1 Motivation 3.2 Average Values of Characteristic Quantities of Permutations AverageNumber of Cycles of a Permutation Average Length of a Cycle A General Approach Average Number of Falls and Rises of a Permutation Average Length of a Run Average Length of the k-th Run Average Number of Inversions of a Permutation Exercises 4. Random Walks, Trees, Lists 4.1 Random Walks 4.2 Enumeration of Random Walks Uniform Random Walks (d, e. u) -Random Walks Non-negative Closed Random Walks 4.3 One-to-one Correspondences Random Walks-Ordered Trees Random Walks-Dynamic Data Structures V vii

7 viii 4.4 Exact and Asymptotical Enumeration Results Useful Techniques for Determining Asymptotics From Generating Functions Ordered Trees Derivation Trees of Context-free Grammars Dynamic Data Structures 114 Exercises Applications Reduction of Binary Trees The Reduction of Binary Trees by a Stack The Reduction of Binary Trees by an Optimal Algorithm The Reduction of Binary Trees by an lnput-restricted Deque Two Algorithms for the Recognition of Dycklanguages The Recognition by a Stack The Recognition by a Two-Way-One-Counter Automaten Batcher's Algorithm Networks for Sorting The Number of Exchanges in the Warst Case The Number of Exchanges in the Average Case 190 Exercises 201 APPENDIX A Basic Definitions of Probability Theory Grammars-Formal Power Series Generating Functions Linear Recurrences Dirichlet Series Laurent Series-Residues-Cauchy Integral Formula Euler's Summation Formula 208 APPENDIX B Numbers in Combinatorial Analysis Special Functions 214 References 221 Index of Notation 227