Series Editors. Editorial Board

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4 Alexander Shen Laboratoire d Informatique Fondamentale de Marseille (LIF) CNRS, Université delaméditerranée, Université deprovence CMI 39 Rue Joliot-Curie Marseille Cedex 13 France alexander.shen@lif.univ-mrs.fr and Russian Academy of Sciences Institute for Information Transmission Problems Bolshoy Karetny per. 19 Moscow, GSP-4, Russia Series Editors Jonathan M. Borwein, FRSC Professor Laureate Director Centre for Computer Assisted Research Mathematics and its Applications, CARMA School of Mathematical & Physical Sciences University of Newcastle Callaghan NSW 2308 Australia Jonathan.Borwein@newcastle.edu.au Helge Holden Department of Mathematical Sciences Norwegian University of Science and Technology Alfred Getz vei 1 NO-7491 Trondheim Norway holden@math.ntnu.no ISSN e-issn ISBN e-isbn DOI / Springer New York Dordrecht Heidelberg London Library of Congress Control Number: Mathematics Subject Classification (2000): 65K05, 65Yxx, 90Cxx, 68-01, 68W40 1st edition: c Birkhäuser 1997 Reprint of 1st edition in series: Modern Birkhäuser Classics c Birkhäuser nd edition: c Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 To the memory of Anna Pogossiants

6 Contents Preface to the second edition xi 1 Variables, expressions, assignments Problems without arrays Arrays Inductive functions Generation of combinatorial objects Sequences Permutations Subsets Partitions Gray codes and similar problems Some remarks Counting Tree traversal (backtracking) Queens not attacking each other: position tree Tree traversal Queens: position tree implementation Backtracking in other problems Sorting Quadratic algorithms Sorting in n log n operations Applications of sorting Lower bound for the number of comparisons Problems related to sorting vii

7 viii Contents 5 Finite-state algorithms in text processing Compound symbols, comments, etc Numbers input Data types Stacks Queues Sets Priority queues Recursion Examples Trees: recursive processing The generation of combinatorial objects; search Other applications of recursion Recursive and non-recursive programs Table of values (dynamic programming) Stack of postponed tasks Difficult cases Graph algorithms Shortest paths Connected components, breadth- and depth-first search Pattern matching Simple example Repetitions in the pattern Auxiliary lemmas Knuth Morris Pratt algorithm Boyer Moore algorithm Rabin Karp algorithm Automata and more complicated patterns Suffix trees Games analysis Game examples Game cost Computing the game cost by backtracking Alpha-beta pruning A retrospective analysis

8 Contents ix 12 Optimal coding Codes The Kraft McMillan inequality Huffman code Shannon Fano code Set representation. Hashing Hashing with open addressing Hashing using lists Sets, trees, and balanced trees Set representation using trees Balanced trees Context-free grammars General parsing algorithm Recursive-descent parsing Parsing algorithm for LL(1)-grammars Left-to-right parsing (LR) LR-processes LR(0)-grammars SLR(1)-grammars LR(1)-grammars, LALR(1)-grammars General remarks about parsing algorithms Further reading Appendix: C and Pascal examples Index

9 Preface to the Second Edition Somebody once said that one may prove the correctness of an algorithm, but not of a program. One of the main goals of this book is to convince the reader that things are not so bad. A well-known programmer, C.A.R. Hoare, said that the beauty of a program is not an additional benefit but a criterion that separates success from failure. If, while solving problems in this book, you come to appreciate the beauty of a well-written program with each part in its correct place, the author s goal will have been reached. Theoretically this book can be used to study programming without a computer: one could write (correct) programs with pencil and paper. But in practice the ability to run the programs is a challenge and a reward that makes programming a fun. We have utilized the problem-solution format. Some chapters are collections of problems having a common topic, while others are devoted to one specific algorithm (e.g., chapter 16 covers LR(1)-parsing). The chapters are more or less independent, but the concluding chapters are more difficult. Chapters 1 7 cover material usually included in undergraduate courses while chapters are more appropriate for a graduate compiler course. In each chapter we have tried to give problems at different levels starting with easy exercises. Problems are usually provided with solutions, answers or hints. However, we strongly recommend to read the solution only after the reader makes a good faith attempt to solve it independently. The book is restricted to micro-programming leaving aside another very important topic: how to split the program into a manageable parts with nice interfaces between them. (Probably this can be learned only by reading and modifying rather large programs.) Pascal is used as a programming language; though being outdated, it is reasonably clear, so the readers familiar with any other procedural language (C, Modula, Oberon, etc.) will encounter no difficulties. For the reader s convenience, a short appendix is added that lists basic differences between Pascal and C. It is intended to help the reader who knows C to understand the program notation in the book (but cannot replace textbooks on C). xi

10 xii Preface to the second edition Most of the problems, of course, are well known. References are rare, but absence of references does not mean that the problem or algorithm is new. However, we hope that in some cases the algorithm or the proof is explained better than what is found in other sources. This book is addressed both to the ambitious student who wants to test and improve his/her skills and to the instructor looking for problems for his/her class. I thank all the people I met while teaching programming (first of all, my former students from 57th school and A.G. Kushnirenko, who was my programming teacher) and all readers that sent me corrections for the preliminary versions of this book (especially Yu.V. Matijasevich). I also thank American Mathematical Society (former Soviet Union aid fund), International Science Foundation, Open Society Foundation, MIT, University of Bordeaux, Bonn University, the Rosenbaum Foundation, INTAS, University of Provence, CNRS, Institute of Problems of Information Transmission and even the Russian government for support during writing this book. I thank Ann Kostant, Elizabeth Loew and the other nice people at Birkhäuser and Springer for their help. Tom Scavo did a great job correcting my English (as well as several other errors) but in no case should he be blamed for the remaining mistakes. Peter Panov helped to prepare the second edition by translating new material in chapters and language editing. The Russian version of this book is freely distributable as a TEX source and camera-ready copy; please look at ftp://ftp.mccme.ru/users/shen/progbook and/or contact the author ( addresses: shen@landau.ac.ru, shen@mccme.ru, sasha.shen@gmail.com, alexander.shen@lif.univ-mrs.fr) for details. I d be grateful if bug reports could be sent to the same addresses. Alexander Shen September, 2009

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