CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS A series of lectures on topics of current research interest in applied mathematics under the direction of the Conference Board of the Mathematical Sciences, supported by the National Science Foundation and published by SIAM G A K R H T B i R K i i o n , The Numerical Solution of Elliptic Equations D V L I N D I Y , Bayesian Statistics, A Review R S V A R < ; A Functional Analysis and Approximation Theory in Numerical Analysis R R H : \ I I \ D I : R , Some Limit Theorems in Statistics P X I K K K Bin I.VISLI -y Weak Convergence of Measures: Applications in Probability I LIONS Some Aspects of the Optimal Control of Distributed Parameter Systems R ( H ; I : R PI-NROSI-: Tecltniques of Differentia/ Topology in Relativity H i K M \N C'ui KNOI r Sequential Analysis and Optimal Design D I ' K H I N Distribution Theory for Tests Based on the Sample Distribution Function Soi I Ri BINO\\, Mathematical Problems in the Biological Sciences P D L \ x Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves I .1 Soioi.NUiiRci Cardinal Spline Interpolation \\.\\ SiMii.R The Theory of Best Approximation and Functional Analysis WI-.KNI R C RHHINBOLDT, Methods of Solving Systems of Nonlinear Equations HANS I-' WHINBKRQKR, Variational Methods for Eigenvalue Approximation R TYRRM.I ROCKAI-KLI.AK, Conjugate Dtialitv and Optimization SIR JAMKS LIGHTHILL, Mathematical Biofhtiddynamics GI-.RAKD SAI.ION, Theory of Indexing C \ rnLi-:i;.N S MORAWKTX, Notes on Time Decay and Scattering for Some Hyperbolic Problems F Hoi'i'hNSTKAm, Mathematical Theories of Populations: Demographics, Genetics and Epidemics RK HARD ASKF;Y Orthogonal Polynomials and Special Functions L H PAYNI: Improperly Posed Problems in Partial Differential Equations S ROSI:N, lectures on the Measurement and Evaluation of the Performance of Computing Systems HHRBHRT B KI;I.I.I:R Numerical Solution of Two Point Boundary Value Problems } P L.ASxLi.i., The Stability of Dynamical Systems - Z ARTSTKIN, Appendix A: Limiting Equations and Stability of Nonautonomous Ordinary Differential Equations I), ( i o n in B AND S A ORS/AC,, Numerical Analysis of Spectral Methods: Theon and Applications Pi ii R H I B I - R Robust Statistical Procedures Hi RBI K r SOLOMON, Geometric Probability FRI:D S ROBF.RIS, Graph Theory and Its Applications to Problems of Society Ii RIS H A R I M - \ N I S Feasible Computations and Provable Complexity Properties ZOIIAR M A N N A , Lectures on the Logic of Computer Programming F.I I is L JOHNSON, Integer Programming: Facets, Subadditivitv, and Duality for Group and SemiGroup Problems S H N H I - I WINOGRAD, Arithmetic Complexity of Computations J F C KiNCiMAN Mathematics of Genetic Diversity M O R I O N F GiuiTiN Topics in Finite Elasticity T I I O M X S G K t i R f X , Approximation of Population Processes (continued on inside back coven Robert Endre Tarjan Bell Laboratories Murray Hill, New Jersey Data Structures and Network Algorithms Siam SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS PHILADELPHIA Copyright ©1983 by the Society for Industrial and Applied Mathematics 109 All rights reserved Printed in the United States of America No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688 Library of Congress Catalog Card Number: 83-61374 ISBN 0-89871-187-8 • Siamumm is ais a is a registered trademark To Gail Maria Zawacki This page intentionally left blank Contents Preface vii Chapter FOUNDATIONS 1.1 Introduction 1.2 Computational complexity 1.3 Primitive data structures 1.4 Algorithmic notation 1.5 Trees and graphs 12 14 Chapter DISJOINT SETS 2.1 Disjoint sets and compressed trees 2.2 An amortized upper bound for path compression 2.3 Remarks 23 24 29 Chapter HEAPS 3.1 Heaps and heap-ordered trees 3.2 Cheaps 3.3 Leftist heaps 3.4 Remarks 33 34 38 42 Chapter SEARCH TREES 4.1 Sorted sets and binary search trees 4.2 Balanced binary trees 4.3 Self-adjusting binary trees 45 48 53 Chapter L I N K I N G AND CUTTING TREES 5.1 The problem of linking and cutting trees 5.2 Representing trees as sets of paths 5.3 Representing paths as binary trees 5.4 Remarks 59 60 64 70 V VI CONTENTS Chapter MINIMUM SPANNING TREES 6.1 The greedy method 6.2 Three classical algorithms 6.3 The round robin algorithm 6.4 Remarks 71 72 77 81 Chapter SHORTEST PATHS 7.1 Shortest-path trees and labeling and scanning 7.2 Efficient scanning orders 7.3 All pairs 85 89 94 Chapter NETWORK FLOWS 8.1 Flows, cuts, and augmenting paths 8.2 Augmenting by blocking flows 8.3 Finding blocking flows 8.4 Minimum cost flows 97 102 104 108 Chapter MATCHINGS 9.1 Bipartite matchings and network flows 9.2 Alternating paths 9.3 Blossoms 9.4 Algorithms for nonbipartite matching 113 114 115 119 References 125 Preface In the last fifteen years there has been an explosive growth in the field of combinatorial algorithms Although much of the recent work is theoretical in nature, many newly discovered algorithms are quite practical These algorithms depend not only on new results in combinatorics and especially in graph theory, but also on the development of new data structures and new techniques for analyzing algorithms My purpose in this book is to reveal the interplay of these areas by explaining the most efficient known algorithms for a selection of combinatorial problems The book covers four classical problems in network optimization, including a development of the data structures they use and an analysis of their running times This material will be included in a more comprehensive two-volume work I am planning on data structures and graph algorithms My goal has been depth, precision and simplicity I have tried to present the most advanced techniques now known in a way that makes them understandable and available for possible practical use I hope to convey to the reader some appreciation of the depth and beauty of the field of graph algorithms, some knowledge of the best algorithms to solve the particular problems covered, and an understanding of how to implement these algorithms The book is based on lectures delivered at a CBMS Regional Conference at the Worcester Polytechnic Institute (WPI) in June, 1981 It also includes very recent unpublished work done jointly with Dan Sleator of Bell Laboratories I would like to thank Paul Davis and the rest of the staff at WPI for their hard work in organizing and running the conference, all the participants for their interest and stimulation, and the National Science Foundation for financial support My thanks also to Cindy Romeo and Marie Wenslau for the diligent and excellent job they did in preparing the manuscript, to Michael Garey for his penetrating criticism, and especially to Dan Sleator, with whom it has been a rare pleasure to work vii This page intentionally left blank 118 CHAPTER FIG 9.4 Execution of the blossom-shrinking algorithm Plus denotes an even vertex, minus an odd vertex Arrows denote parents, (a) After examining [a, c] (Case 2) (b) After examining [b, a] (Case 4), [c, d], [