DICTIONARY OF DISTANCES This page intentionally left blank DICTIONARY OF DISTANCES ELENA DEZA Moscow Pedagogical State University and Donghua University MICHEL-MARIE DEZA Ecole Normale Superieure, Paris, and Shanghai University Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo ELSEVIER Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2006 Copyright © 2006 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; Email: [email protected]. 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Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publica tion Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-444-52087-6 ISBN-10: 0-444-52087-2 For information on all Elsevier publications visit our website at books.elsevier.com Printed and bound in The Netherlands 0607080910 10987654321 On 2 April 2006 it was 100 years from the submission by Maurice FRÉCHET of his outstanding doctoral dissertation Sur quelques points du calcul functionnel that introduced (within a systematic study of functional operations) the notion of metric space. It was also 92 years from 1914, the date of publication by Felix HAUSDORFF of his famous Grundzüge der Mengenlehre where the theory of topological and metric spaces was created. Let this Dictionary be our homage to the memory of these great mathematicians and their lives of dignity through the hard ways of the first half of the XX century. Maurice FRÉCHET (1878–1973) coined Felix HAUSDORFF (1868–1942) coined in 1906 the concept of écart (semi-metric) in 1914 the term metric space v This page intentionally left blank Preface The concept of distance is one of the basic ones in the whole of human experience. In everyday life it usually means some degree of closeness between two physical objects or ideas, i.e., length, time interval, gap, rank difference, coolness or remoteness, while the term metric is often used as a standard for a measurement. But here we consider, except in the last two chapters, the mathematical meaning of these terms. The mathematical notions of distance metric (i.e., a function d(x,y) from X × X to the set of real numbers satisfying d(x,y) 0 with equality only for x = y, d(x,y) = d(y,x), and d(x,y) d(x,z) + d(z,y)) and of metric space (X, d) were originated a century ago by M. Fréchet (1906) and F. Hausdorff (1914) as a special case of an infinite topological space. The triangle inequality above appears already in Euclid. The infinite metric spaces are seen usually as a generalization of the metric |x − y| on the real numbers. Their main classes are the measurable spaces (add measure) and Banach spaces (add norm and completeness). However, starting from K. Menger (1928) and, especially, L.M. Blumenthal (1953), an explosion of interest in finite metric spaces occurred. Another trend: many mathematical theories, in the process of their generalization, settled on the level of metric space. Now finite distance metrics have become an essential tool in many areas of Mathemat- ics and its applications include Geometry, Probability, Statistics, Coding/Graph Theory, Clustering, Data Analysis, Pattern Recognition, Networks, Engineering, Computer Graph- ics/Vision, Astronomy, Cosmology, Molecular Biology, and many other areas of science. Devising the most suitable distance metrics has become a standard task for many re- searchers. Especially intense ongoing searches for such distances occur, for example, in Genetics, Image Analysis, Speech Recognition, Information Retrieval. Often the same dis- tance metric appears independently in several different areas; for example, the edit distance between words, the evolutionary distance in Biology, the Levenstein distance in Coding Theory, and the Hamming+Gap or shuffle-Hamming distance. This body of knowledge has become too large and disparate to operate within. The number of worldwide web entries offered by Google on the topics “distance”, “metric space” and “distance metric” approach 300 million (i.e., about 4% of all), 12 million and 6 million, respectively, not to mention all the printed information outside the Web, or the vast “invisible Web” of searchable databases. However, this huge amount of information on distances is too scattered: the works evaluating distance from some list usually treat very specific areas and are hardly accessible for non-experts. Therefore, many researchers, including us, keep and cherish a collection of distances for use in their own areas of science. In view of the growing general need for an accessible interdisciplinary source for a vast multitude of researchers, we have expanded our private collection into this Dictionary. Some additional material was reworked from various en- vii viii Preface cyclopedia, especially, Encyclopedia of Mathematics ([EM98]), MathWorld ([Weis99]), PlanetMath ([PM]), and Wikipedia ([WFE]). However, the majority of distances should be extracted directly from specialist literature. The vast reservoir of concepts defined in this Dictionary, aims to be a thought-provoking archive and a valuable resource. Besides distances themselves, we have collected here many distance-related notions (especially in Chapter 1) and paradigms, enabling people from applications to get those, arcane for non-specialists, research tools, in ready-to-use fashion. This and the appearance of some distances in different contexts can be a source of new research. At a time when over-specialization and terminology barriers isolate researchers, this Dictionary tries to be “centripetal” and “ecumenical”, providing some access and altitude of vision but without taking the route of scientific vulgarization. This attempted balance defined the structure and style of the Dictionary. The Dictionary is divided into 28 chapters grouped into 7 Parts of about the same length. The titles of parts are purposely approximative: they just allow a reader to figure out her/his area of interest and competence. For example, Parts II, III and IV, V require some culture in, respectively, pure and applied Mathematics. Part VII can be read by a layman. The chapters are thematic lists, by areas of Mathematics or applications which can be read independently. When necessary, a chapter or a section starts with a short introduction: a field trip with the main concepts. Besides those introductions, the main properties and uses of distances are given, within items, only exceptionally. We also tried, when it was easy, to trace distances to their originator(s), but the proposed extensive bibliography has a less general ambition: it is just to provide convenient sources for a quick search. Each chapter consists of items ordered in a way that hints of connections between them. All item titles and selected key terms can be traced in the large Subject Index (about 1400 entries); they are boldfaced unless the meaning is clear from the context. So, the definitions are easy to locate, by subject, in chapters and/or, by alphabetic order, in the index. The introductions and definitions are reader-friendly and as far as possible independent of each other; still they are interconnected, in the 3-dimensional HTML manner, by hyperlink-like boldfaced references to similar definitions. Many nice curiosities appear in this “Who is Who” of distances. Examples of such sundry terms are: ubiquitous Euclidean distance (“as-the-crow-flies”), flower-shop met- ric (shortest way between two points, visiting a “flower-shop” point first), knight-move metric on a chessboard, Gordian distance of knots, Earth Mover distance, biotope distance, Procrustes distance, lift metric, post-office metric, Internet hop metric, WWW hyperlink quasi-metric, Moscow metric, dogkeeper distance. Besides abstract distances, the distances having physical meaning appear also (especially in Part VI); they range from 1.6×10−35 m (Planck length) to 7.4×1026 m (the estimated size of observable Universe, about 46×1060 Planck lengths). The number of distance metrics is infinite and therefore, our Dictionary cannot enu- merate all of them. But we were inspired by several successful thematic dictionaries on other infinite lists; for example, on Integer Sequences, Inequalities, Numbers, Random Processes, and by atlases of Functions, Groups, Fullerenes, etc. On the other hand, the largeness of the scope forced us often to switch into a laconic tutorial style. Preface ix The target audience consists of all researchers working on some measuring schemes and, to a certain degree, of students and a part of the general public interested in science. We have tried to address, even if incompletely, all scientific uses of the notion of dis- tance. However some distances did not made it into this Dictionary due to space limitations (being too specific and/or complex) or our oversight. In general, the size/interdisciplinarity cut-off, i.e., decision where to stop, was our main headache. We would be grateful to the readers who send us their favorite distances missed here.