Data Structures and Algorithms: A First Course

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1 Data Structures and Algorithms: A First Course

2 Springer London Berlin Heidelberg New York Barcelona Budapest Hong Kong Milan Paris Santa Clara Singapore Tokyo

3 lain T. Adamson Data Structures and Algorithms: A First Course i Springer

4 lain T. Adamson, BSc, MSc, AM, PhD Department of Mathematics University of Dundee 23 Perth Road, Dundee DDI 41IN, UK ISBN 13: British Library Cataloguing in Publication Data Adamson, lain T. Data structures and algorithms : a first course I.Data structures (Computer science) 2.A1gorithms I.Title 005.7'3 ISBN 13: Library of Congress Cataloging in.publication Data Adamson, lain T. Data structures and algorithms : a first course 1 lain T. Adamson. p. em. Includes index. ISBN 13: e-isbn 13: DOl: Data structures (Computer science) 2. Computer algorithms. I. Title. QA76.9.D33A '3 dc20 CIP Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permis~ion in writing of the publishers, or in the case of reprographic reproduction in accordance With the terms oflicences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. Springer Verlag London Limited 1996 The use of registered names, trademarks etc. in this publication docs not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: camera ready by author 34/ Printed on acid free paper

5 PREFACE In 1976 Niklaus Wirth, the inventor of Pascal, published a book entitled Algorithms + Data Structures = Progmms. If the assertion of Wirth's title is correct-and it would be hard to dispute it-all young computer scientists who aspire to write programs must learn something about algorithms and data structures. This book is intended to help them do that. It is based on lecture courses developed over the past few years and I hope that at least some of the informality of the classroom and the spoken word has been transferred to the printed page. The lectures were given to first and second year students in The University of Dundee who had been well-grounded in Pascal and who had therefore already met some elementary data structures and sorting and searching algorithms; but only the syntax of Pascal was taken for granted, as it is in the book. My students had rather varied mathematical backgrounds and some were not very well-disposed to old-fashioned algebraic manipulation. A little of this (and a brief mention of limits) does appear in the book; but readers of a book are more fortunate than students in a classroomthey can skip all the details and concentrate on the final results. The book is divided into four parts. Part Ion Data Structures introduces a variety of structures and the fundamental operations associated with them, together with descriptions of how they are implemented in Pascal. Chapter 1, on arrays, records and linked lists, and Chapter 3, on binary trees, are largely (but not entirely) recapitulation of ideas from any introductory programming course. Chapter 2, on stacks and queues, introduces these structures and describes various ways of implementing them; it also discusses some of the many applications of stacks in computer science. Heaps (priority queues) are introduced in Chapter v

6 vi Preface 4 as a way of modelling constantly changing collections in which the only item of interest at any moment is the one with highest priority. In Chapter 5 we introduce graphs, which are much less regular structures than any of the earlier ones, and we describe how, in spite of their irregularity, they can be handled in the computer science environment. Part II on Algorithms begins with the short Chapter 6 in which the concept of algorithm is discussed informally and the notion of the complexity of an algorithm is introduced. Although some of the mathematical details can be skipped over, it is important even at the beginning of one's education in computer science to have at least an inkling of the ideas introduced here. In Chapter 7 we cover a variety of sorting algorithms, both internal and external, paying particular attention to their complexity (though of course the mathematical details can be skipped). It may be that a smaller proportion of the world's computer resources are used for sorting nowadays than the 25 to 50 per cent mentioned by Knuth in his 1973 volume Sorting and Searching; but sorting is still an everyday problem and it is right for students to acquire an armoury of sorting methods. Chapter 8 is devoted to two types of graph algorithms, dealing with shortest path and spanning tree problems. In both sections of this chapter there are several pages which readers may find difficult; they may, I suppose, be described as 'mathematical', but only in the sense that they involve close logical reasoning and proof by contradiction. I would like to encourage readers to persevere with these hard proofs, which provide justification that the algorithms described do in fact solve the problems to which they apply. The shorter Chapter 9, as its title indicates, includes an assortment of algorithms which were included in my lectures 'as time allowed'. Most students enjoyed particularly the stable marriage algorithm which is described in Section 9.3. Part II refers in passing to various approaches to algorithm development-the divide-and-conquer method, greedy methods and dynamic programming. Part III is concerned with the description of successively more elaborate structures for the storage of records and algorithms for retrieving a record from such a structure by means of its key (this is what is

7 Preface vii meant by the 'searching' of the title). The collections of records we are concerned with are supposed to be constantly changing; so we must examine how to deal not only with searching but also with insertion and deletion of records. Part IV consists of very full solutions to nearly all the exercises in the book. It would have been more conventional (and easier for the author) to give more abbreviated solutions, perhaps only final results, to only a selection of the exercises; but my students have always said how helpful they have found my solutions to tutorial problems and claim to have learned from them. Of course readers ought to try the exercises for themselves before consulting the solutions. Acknowledgements My first word of thanks must go to the several generations of Dundee students whose favourable comments on my lectures encouraged me to turn them into a book. Next I have to thank Keith Edwards, who read and commented on some of my manuscript and gave me much helpful advice about mathematical typesetting. My chief thanks, as always, go to my wife for her constant love and support. Dundee, April lain T. Adamson

8 CONTENTS PREFACE. 1 I DATA STRUCTURES 1 1 ARRAYS, RECORDS AND LINKED LISTS 1.1 Arrays Storage of arrays 1.3 Records Linked lists 1.5 Exercises 1. 2 STACKS AND QUEUES 2.1 Stacks Applications of stacks. 2.3 Queues Exercises BINARY TREES 3.1 Binary trees Binary search trees 3.3 Exercises 3. 4 HEAPS 4.1 Priority queues and heaps 4.2 Exercises ix

9 x 5 GRAPHS 5.1 Graphs and their implementation 5.2 Graph traversals 5.3 Exercises Contents II ALGORITHMS ALGORITHMS AND COMPLEXITY 6.1 Algorithms Complexity of algorithms. 6.3 Exercises SORTING ALGORITHMS 7.1 Internal sorting by comparisons 7.2 Other internal sorting algorithms 7.3 External sorting algorithms 7.4 Exercises GRAPH ALGORITHMS 8.1 Shortest path algorithms 8.2 Spanning tree algorithms. 8.3 Exercises SOME MISCELLANEOUS ALGORITHMS 9.1 Numerical multiplication algorithms. 9.2 Matrix multiplication algorithms 9.3 A stable marriage algorithm 9.4 Exercises III STORING AND SEARCHING STORING IN ARRAYS AND LISTS 10.1 Sequential and binary searching 10.2 Hashing Exercises

10 Contents xi 11 STORING IN BINARY TREES Storing in binary search trees Storing in AVL-trees Exercises STORING IN MULTIWAY TREES Multiway search trees B-trees Tries Exercises IV SOLUTIONS SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES SOLUTIONS TO EXERCISES INDEX 417

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