DOCSLIB.ORG

Smallest Enclosing Balls of Balls

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Smallest Enclosing Balls of Balls DISS. ETH NO. 16168, 2005 Smallest enclosing balls of balls Combinatorial structure & algorithms A dissertation submitted to the Swiss Federal Institute of Technology, ETH Z¨urich for the degree of Doctor of Technical Sciences presented by Kaspar Fischer Dipl. Ing. Inf. ETH born 23rd May 1975 citizen of Switzerland accepted on the recommendation of Prof. Dr. Emo Welzl, ETH Z¨urich, examiner Dr. Bernd G¨artner, ETH Z¨urich, co-examiner Prof. Dr. Jiˇr´ıMatouˇsek, co-examiner Acknowledgments I would like to thank all the people without whom this thesis would not have been written. In the first place, my thanks go to Emo Welzl for giving me the opportunity to work in his research group and the graduate program Combinatorics, Geometry, and Computation. I would like to express my deep gratitude to my advisor Bernd G¨artner. His support, character, and working style were a warm com- panion during my stay at ETH. And without the interesting time I spent doing my diploma thesis under his guidance I would not have started my PhD in the first place. I also want to warmly thank Jirka Matouˇsek for being my third su- pervisor and for taking the time to read and provide feedback. Oscar Chinelato, Menelaos Karavelas, Sylvain Pion and others on the Cgal mailing list helped me a lot with ideas, discussions, and program- ming knowledge. I also owe a big thank you to Martin Kutz; working with him was inspiring. Good memories come from and go to the members of the Gremo group: Andreas Razen, Bettina Speckmann, Dieter Mitsche, Eva Schu- bert, Falk Tschirschnitz, Floris Tschurr, Frans Wessendorp, Franziska Hefti, Joachim Giesen, Leo R¨ust, Matthias John, Michael Hoffmann, MiloˇsStojakovi´c,Philipp Zumstein, Robert Berke, Shankar Ram Lak- shminarayanan, Tibor Szab´o, Uli Wagner, and Yoshio Okamoto. In par- ticular, I am very thankful to Ingo Schurr (for tons of good moments) and P´eter Csorba (for the best humour on the floor). Behind the scenes there were, of course, other people who helped and influenced me substantially during my PhD time. I thank my girlfriend Nicole Schmidlin, my best friends Klaus Labhardt and Florian D¨ublin, and my sister Alexandra Ryser—no, not just for feeding me. Katharina Adiecha, B´eatrice Goetz, Joachim Dietiker and Miriam K¨undig, together with a dozen of anonymous Tango dancers all over the world changed a lot inside me. Last but not least, I would like to thank my parents, Elisabeth Zumthor and Franz Fischer, for their lively and lovely support. Abstract This subject of this thesis is the miniball problem which asks for the smallest ball that contains a set of balls in d-dimensional Euclidean space. We try to answer three main questions concerning this problem. What structural properties does the ‘miniball’ exhibit? What practical algorithms can be used to compute it? And how efficiently can instances in high dimensions be tackled? In all these questions, it is of interest how the problem and algorithms for it compare to the more specific variant of the problem in which all input balls are points (all radii zero). In connection with the first question, we show that many of the al- ready known properties of the miniball of points translate also to the miniball of balls. However, some important properties (that allow for subexponential algorithms in the point case, for instance) do not gener- alize, and we provide counterexamples and appropriate new characteri- zations for the balls case. The change in structure between the point and ball case also reflects itself in the algorithmic picture, and we demonstrate that Welzl’s algo- rithm does not work for balls in general, and likewise a known reduction to unique sink orientations does not apply either. Our main result here is that under a simple general-position assumption, both the correctness of Welzl’s algorithm and a reduction to unique sink orientations can be established. The result has an appealing geometric interpretation and practical significance in that it allows for pivoting algorithms to solve the problem; the latter have the potential of being very fast in practice. As a byproduct, we develop a deeper (but not yet complete) understanding of the general applicability of Welzl’s algorithm to certain optimization problems on the combinatorial cube. Our contributions concerning the last question are twofold. One the one side, we provide a combinatorial, simplex-like algorithm for the point case that turns out to be very efficient and robust in practice and for which we can show that in theory it does not cycle. On the other hand, we formulate the problem with balls as input as a mathematical pro- gram and show that the latter can be solved in subexponential time by using G¨artner’s algorithm for abstract optimization problems. In fact, our method works for other convex mathematical programs as well, and as a second application of it we present a subexponential algorithm for finding the distance between two convex hulls of balls. Zusammenfassung Diese Arbeit besch¨aftigt sich mit dem Miniballproblem, welches nach der kleinsten Kugel verlangt, die eine gegebene Menge von Kugeln im d-di- mensionalen Euklidschen Raum einschließt. Drei Fragen versuchen wir zu beantworten: Welche strukturellen Eigenschaften weist der ‘Miniball’ auf? Wie kann man ihn in der Praxis schnell berechnen? Und wie verh¨alt sich die Komplexit¨at des Problems in hohen Dimensionen? Von speziel- lem Interesse ist dabei, wie sich das Problem und die Algorithmen von der Variante unterscheiden, in welcher alle Eingabekugeln Punkte sind. Im Zusammenhang mit der ersten Frage zeigen wir, daß sich viele der schon bekannten Eigenschaften des Miniball von Punkten auf den Ballfall ¨ubertragen lassen. Einige wichtige Merkmale jedoch—sie erm¨oglichen es zum Beispiel, das Problem im Punktfall in subexponentieller Zeit zu l¨osen—verallgemeinern sich nicht, was wir anhand von Gegenbeispielen und angepaßten Charakterisierungen aufzeigen. Der strukturelle Unterschied zwischen dem Punkt- und Ballfall dr¨uckt sich auch in den Algorithmen aus. Wir stellen fest, daß Welzl’s Punkt- Algorithmus f¨urB¨alle nicht mehr funktioniert und gleichfalls eine Reduk- tion zu Unique Sink Orientations (USO) scheitert. Unser Beitrag dazu zeigt auf, daß eine einfache Annahme ¨uber die Lage der Eingabekugeln sowohl die Korrektheit von Welzl’s Algorithmus garantiert als auch eine Reduktion zum USO-Problem erm¨oglicht. Das Resultat kommt mit einer anschaulichen geometrischen Erkl¨arung und ist von praktischer Bedeu- tung, da damit Pivotieralgorithmen angewandt werden k¨onnen, die in der Praxis oft sehr schnell sind. Als Nebenprodukt entwickeln wir ein tieferes (aber noch unvollst¨andiges) Verst¨andnis f¨urjene Optimierungsprobleme, auf welche Welzl’s Algorithmus angewandt werden kann. Unser Beitrag zur dritten Frage ist zweigeteilt. Zum einen liefern wir einen kombinatorischen, Simplex-artigen Algorithmus f¨urden Punktfall. Dieser ist in der Praxis sehr effizient und robust, und wir beweisen, daß er nicht ‘zykeln’ kann. Zum anderen formulieren wir dass urspr¨ungliche Problem als mathematisches Programm und zeigen, daß sich dieses mit- tels G¨artner’s Algorithmus f¨ur Abstract Optimization Problems in subex- ponentieller Zeit l¨osen l¨aßt. Die resultierende Methode erfaßt auch an- dere mathematische Programme, insbesondere erhalten wir einen subex- ponentiellen Algorithmus zur Berechnung der Distanz zwischen den kon- vexen H¨ullen zweier Kugelmengen. Contents 1 Introduction 1 1.1 Background.......................... 1 1.2 Contributions and outline of the thesis . 7 2 Combinatorial frameworks 11 2.1 Complexitymodel ...................... 11 2.2 TheLP-typeframework . 14 2.2.1 LP-type problems . 15 2.2.2 Basis-regularity . 19 2.2.3 The MSW-algorithm . 21 2.3 TheAOPframework..................... 25 2.4 TheUSOframework ..................... 28 2.5 Weak LP-type problems . 30 2.5.1 Reducibility . 36 2.6 Strong LP-type problems . 43 3 Properties of the smallest enclosing ball 49 3.1 The problem . 49 3.2 Properties........................... 52 3.3 Properties of mb(U, V ).................... 62 3.4 LP-type formulations . 70 3.5 Smallest superorthogonal ball . 73 4 Smallest enclosing balls of points 79 4.1 Sketch of the algorithm . 79 4.2 The algorithm in detail . 81 4.3 Remarks............................ 87 v vi Contents 5 Smallest enclosing balls of balls 89 5.1 Welzl’s algorithm . 90 5.2 Algorithm msw ........................ 94 5.3 Signed balls and shrinking . 97 5.4 Inversion............................101 5.4.1 A dual formulation for sebb0 ............101 5.4.2 The distance to the convex hull . 104 5.5 Smallcases ..........................108 5.5.1 The unique sink orientation . 118 5.5.2 Symbolic perturbation . 123 6 More programs in subexponential time 127 6.1 A mathematical program for sebb0 .............128 6.2 Overview of the method . 131 6.3 Theoracle...........................135 6.3.1 The primitive for sebb0 ...............139 6.4 Thedetails ..........................142 6.5 Themainresult........................149 6.6 Further applications . 150 6.6.1 Smallest enclosing ellipsoid . 150 6.6.2 Distance between convex hulls of balls . 153 6.7 Remarks............................163 Bibliography 165 Notation N ....................................the natural numbers including 0 Rd .................................the d-dimensional Euclidean space d d d R+ ........................the positive orthant x R x 0 of R im(f) ..................the image f(x) x dom({ ∈f) of| a≥ function} f sgn(α) ..................the sign of{ a real| number∈ α (with} sgn(0) = 0) d Uδ(x) . the open δ-neighborhood x′ R x′ x < δ of the point x {Rd ∈for real| k radius− kδ > 0} ∈ d U˙ δ(x) ..................the dotted neighborhood Uδ(x) x of x R 2T ....................... V V T , i.e., the power set\ { of} the set∈ T U V ..................the{ symmetric| ⊆ } difference of the sets U and V U ⊕ V .......................U is any subset of V (U = V is possible) U ⊆ V .............
Load more

Recommended publications
  • Fast Distributed Algorithms for LP-Type Problems of Low Dimension
    Fast Distributed Algorithms for LP-Type Problems of Low Dimension Kristian Hinnenthal Paderborn University, Germany [email protected] Christian Scheideler Paderborn University, Germany [email protected] Martijn Struijs TU Eindhoven, The Netherlands [email protected] Abstract In this paper we present various distributed algorithms for LP-type problems in the well-known gossip model. LP-type problems include many important classes of problems such as (integer) linear programming, geometric problems like smallest enclosing ball and polytope distance, and set problems like hitting set and set cover. In the gossip model, a node can only push information to or pull information from nodes chosen uniformly at random. Protocols for the gossip model are usually very practical due to their fast convergence, their simplicity, and their stability under stress and disruptions. Our algorithms are very efficient (logarithmic rounds or better with just polylogarithmic communication work per node per round) whenever the combinatorial dimension of the given LP-type problem is constant, even if the size of the given LP-type problem is polynomially large in the number of nodes. 2012 ACM Subject Classification Theory of computation → Distributed algorithms; Theory of computation → Mathematical optimization Keywords and phrases LP-type problems, linear optimization, distributed algorithms, gossip al- gorithms Digital Object Identifier 10.4230/LIPIcs.DISC.2019.23 Funding K. Hinnenthal and C. Scheideler are supported by the DFG SFB 901 (On-The-Fly Com- puting). 1 Introduction 1.1 LP-type Problems LP-type problems were defined by Sharir and Welzl [18] as problems characterized by a tuple (H, f) where H is a finite set of constraints and f : 2H → T is a function that maps subsets from H to values in a totally ordered set (T, ≤) containing −∞.
    [Show full text]
  • Call for Papers RIMS Workshop on Computational Geometry and Discrete Mathematics October 16-18, 2008, Kyoto, Japan
    Call For Papers RIMS Workshop on Computational Geometry and Discrete Mathematics October 16-18, 2008, Kyoto, Japan RIMS Workshop on Computational Geometry and Workshop Organizers: Discrete Mathematics will be held at RIMS (Research Institute for Mathematical Sciences), Tetsuo Asano (JAIST, Japan) Kyoto, October 16-18, 2008. It is an event of RIMS Satoru Fujishige (Kyoto U., Japan) International Project Research 2008, Discrete Satoru Iwata (Kyoto U., Japan) Structures and Algorithms, and intended to provide a Naoki Katoh (Kyoto U., Japan) forum for researchers working in the field of Yoshio Okamoto (Tokyo Inst. Tech, Japan) computational geometry and related topics. Takeshi Tokuyama (Tohoku U., Japan, Chair) Scope: Invited Speakers: The workshop consists of invited talks and presentations on contributed papers. We call for Oswin Aichholzer (T. U. Graz, Austria) submissions of contributed papers on original Boris Aronov (Polytechnic U., USA) research on computational geometry and related Sergey Bereg (U. Texas, USA) topics. Also survey papers on recent results of the Siu Wing Cheng (HKUST, Hong Kong) authors are welcome. Otfried Cheong (KAIST, Korea) Danny Chen (U. Notre Dame, USA) Submissions: Jiri Matousek (Charles U., Czech Republic) Authors should submit short extended abstract in Kurt Mehlhorn (MPI, Germany) PDF format not exceeding 4 pages on A4 (or US Kokichi Sugihara (U. Tokyo, Japan) letter) paper, including author’s names, affiliations, Emo Welzl (ETH, Switzerland) references, and figures. One may use double-column format (we recommend the style of EuroCG). Workshop Web Page : Authors may replace the abstracts with longer ones (up to 10 pages) for the camera-ready version. http://www.dais.is.tohoku.ac.jp/~rimscg08 The submissions should be sent via email to [email protected] Symposium Site: together with your contact information.
    [Show full text]
  • Dagst%Uh|-Seminar-Report; 59
    Helmut Alt, Bernard Chazelle, Emo Welzl (editors): Computational Geometry Dagst%uh|-Seminar-Report;59 22.03.-26.03.93 (9312) ISSN 0940-1 121 Copyright © 1993 by IBF l GmbH,Schloß Dagstuhl, 66687Wadern, Germany TeI.: +49-6871 - 2458 Fax: +49-6871 - 5942 Das Intemationale Begegnungs- und Forschungszentrum für Informatik (IBFI) ist eine gemein- nützige GmbH. Sie veranstaltet regelmäßig wissenschaftliche Seminare, welche nach Antrag der Tagungsleiter undBegutachtung durchdas wissenschaftliche Direktorium mit persönlich eingeladenen Gästen durchgeführt werden. &#39; Verantwortlich für das Programm: Prof. Dr.-Ing. José Encamagao, Prof. Dr. Winfried Görke, Prof. Dr. Theo Härder Dr. Michael Laska, Prof. Dr. Thomas Lengauer, Prof. Walter Fchy Ph. D. Prof. Dr. Reinhard Wilhelm (wissenschaftlicher Direktor) Gesellschafter: Universität des Saarlandes, Universität Kaiserslautern, Universität Karlsruhe, Gesellschaft für Informatik e.V. Bonn Träger: Die Bundesländer Saarland und Rheinland-Pfalz Bezugsadresse: Geschäftsstelle Schloß Dagstuhl Informatik, Bau 36 Universität des Saarlandes Postfach 1150 66041 Saarbrücken Gennany TeI.: +49 681 - 302 - 4396 Fax: +49 -681 302 - 4397 Report of the Third Dagstuhl Seminar on Computational Geometry March 22-26, 1993 The Third DagstuhlSeminar on ComputationalGeometry was organized by Helmut.Alt. (FU Berlin), BernardCha.zelle (Princeton University), and Emo Welzl (F U Berlin). The 32 participants ca.mefrom 10 countries, among them 13 who came from North America a.nd Israel. This report contains abstracts of the 30 talks (in chronologicalorder) given at the meeting as well as Micha.Sharirs report of the open problem sessionwhich took place on Tuesday afternoon and was chaired by Kurt Mehlhorn. Berichterstatter: Frank Hoffmann Participants Pankaj Agarwal, Duke University Helmut Alt, Freie Universität Berlin Boris Aronov, PolytechnicUniversity New York Franz Aurenharmner, TU Graz Jean-Daniel Boissounat, INRIA Sophia.
    [Show full text]
  • Pseudo Unique Sink Orientations Arxiv:1704.08481V1 [Math.CO] 27
    Pseudo Unique Sink Orientations Vitor Bosshard Bernd G¨artner July 17, 2018 Abstract A unique sink orientation (USO) is an orientation of the n-dimensional cube graph (n-cube) such that every face (subcube) has a unique sink. The number of unique sink orientations is nΘ(2n) [13]. If a cube orientation is not a USO, it contains a pseudo unique sink orientation (PUSO): an orientation of some subcube such that every proper face of it has a unique sink, but the subcube itself hasn't. In this paper, we characterize and count PUSOs of the n-cube. We show that PUSOs have a much more rigid structure than USOs and that their number is between 2Ω(2n−log n) and 2O(2n) which is negligible compared to the number of USOs. As tools, we introduce and characterize two new classes of USOs: border USOs (USOs that appear as facets of PUSOs), and odd USOs which are dual to border USOs but easier to understand. 1 Introduction Unique sink orientations. Since more than 15 years, unique sink orientations (USOs) have been studied as particularly rich and appealing combinatorial ab- stractions of linear programming (LP) [6] and other related problems [3]. Orig- inally introduced by Stickney and Watson in the context of the P-matrix linear complementarity problem (PLCP) in 1978 [19], USOs have been revived by Szab´o and Welzl in 2001, with a more theoretical perspective on their structural and algorithmic properties [20]. The major motivation behind the study of USOs is the open question whether arXiv:1704.08481v1 [math.CO] 27 Apr 2017 efficient combinatorial algorithms exist to solve PLCP and LP.
    [Show full text]
  • The Epsilon-T-Net Problem, Proc. Symposium on Computational
    The -t-Net Problem Noga Alon Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv 69978, Israel [email protected] Bruno Jartoux Department of Computer Science, Ben-Gurion University of the Negev, Be’er-Sheva, Israel [email protected] Chaya Keller Department of Computer Science, Ariel University, Ariel, Israel [email protected] Shakhar Smorodinsky Department of Mathematics, Ben-Gurion University of the Negev, Be’er-Sheva, Israel [email protected] Yelena Yuditsky Department of Mathematics, Ben-Gurion University of the Negev, Be’er-Sheva, Israel [email protected] Abstract We study a natural generalization of the classical -net problem (Haussler–Welzl 1987), which we t call the -t-net problem: Given a hypergraph on n vertices and parameters t and ≥ n , find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least n contains a set in S. When t = 1, this corresponds to the -net problem. We prove that any sufficiently large hypergraph with VC-dimension d admits an -t-net of (1+log t)d 1 size O( log ). For some families of geometrically-defined hypergraphs (such as the dual 1 hypergraph of regions with linear union complexity), we prove the existence of O( )-sized -t-nets. We also present an explicit construction of -t-nets (including -nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of -nets (i.e., for t = 1), it does not rely on advanced derandomization techniques.
    [Show full text]
  • Crossing-Free Configurations on Planar Point Sets
    Diss. ETH No. 18607 Crossing-Free Configurations on Planar Point Sets A dissertation submitted to the Swiss Federal Institute of Technology Zurich for the degree of Doctor of Sciences presented by Andreas Razen Dipl. Math. ETH born May 7, 1982 citizen of Austria accepted on the recommendation of Prof. Dr. Emo Welzl, examiner Prof. Dr. Jack Snoeyink, co-examiner Dr. Uli Wagner, co-examiner 2009 to my parents Acknowledgments I am deeply grateful to my supervisor Emo Welzl for his motivating and encouraging support during my time as Ph.D. student. He introduced me to the beautiful field of geometric graph theory, and with his inspiring advice he made it an unforgettable experience to learn from him. It was an honor being part of his research group Gremo, which provided both a stimulating research environment and a wonderful academic home where I met people who I will greatly miss. My sincerest thanks go to Jack Snoeyink for working with me and giving me the opportunity to learn about applications of geometric computations in molecular biology, for reviewing my thesis and providing me with many valuable suggestions and comments which greatly helped to improve the presentation of this dissertation. I am also indebted to Uli Wagner for working together with me on the transformation graphs of compatible spanning trees, for broadening my horizon by teaching vivid courses on Fourier-analytic, topological, and algebraic methods in combinatorics, and for co-refereeing my thesis and commenting on its write-up. I would like to thank all the current and previous members of Gremo for making my time as Ph.D.
    [Show full text]
  • Federated Computing Research Conference, FCRC’96, Which Is David Wise, Steering Being Held May 20 - 28, 1996 at the Philadelphia Downtown Marriott
    CRA Workshop on Academic Careers Federated for Women in Computing Science 23rd Annual ACM/IEEE International Symposium on Computing Computer Architecture FCRC ‘96 ACM International Conference on Research Supercomputing ACM SIGMETRICS International Conference Conference on Measurement and Modeling of Computer Systems 28th Annual ACM Symposium on Theory of Computing 11th Annual IEEE Conference on Computational Complexity 15th Annual ACM Symposium on Principles of Distributed Computing 12th Annual ACM Symposium on Computational Geometry First ACM Workshop on Applied Computational Geometry ACM/UMIACS Workshop on Parallel Algorithms ACM SIGPLAN ‘96 Conference on Programming Language Design and Implementation ACM Workshop of Functional Languages in Introductory Computing Philadelphia Skyline SIGPLAN International Conference on Functional Programming 10th ACM Workshop on Parallel and Distributed Simulation Invited Speakers Wm. A. Wulf ACM SIGMETRICS Symposium on Burton Smith Parallel and Distributed Tools Cynthia Dwork 4th Annual ACM/IEEE Workshop on Robin Milner I/O in Parallel and Distributed Systems Randy Katz SIAM Symposium on Networks and Information Management Sponsors ACM CRA IEEE NSF May 20-28, 1996 SIAM Philadelphia, PA FCRC WELCOME Organizing Committee Mary Jane Irwin, Chair Penn State University Steve Mahaney, Vice Chair Rutgers University Alan Berenbaum, Treasurer AT&T Bell Laboratories Frank Friedman, Exhibits Temple University Sampath Kannan, Student Coordinator Univ. of Pennsylvania Welcome to the second Federated Computing Research Conference, FCRC’96, which is David Wise, Steering being held May 20 - 28, 1996 at the Philadelphia downtown Marriott. This second Indiana University FCRC follows the same model of the highly successful first conference, FCRC’93, in Janice Cuny, Careers which nine constituent conferences participated.
    [Show full text]
  • LINEAR PROGRAMMING Martin Dyer, Bernd G¨Artner, Nimrod Megiddo, and Emo Welzl
    49 LINEAR PROGRAMMING Martin Dyer, Bernd G¨artner, Nimrod Megiddo, and Emo Welzl INTRODUCTION Linear programming has many important practical applications, and has also given rise to a wide body of theory. See Section 49.9 for recommended sources. Here we consider the linear programming problem in the form of maximizing a linear function of d variables subject to n linear inequalities. We focus on the relationship of the problem to computational geometry, i.e., we consider the problem in small dimension. More precisely, we concentrate on the case where d n, i.e., d = d(n) is a function that grows very slowly with n. By linear programming≪ duality, this also includes the case n d. This has been called fixed-dimensional linear programming, though our viewpoint≪ here will not treat d as constant. In this case there are strongly polynomial algorithms, provided the rate of growth of d with n is small enough. The plan of the chapter is as follows. In Section 49.2 we consider the sim- plex method, in Section 49.3 we review deterministic linear time algorithms, in Section 49.4 randomized algorithms, and in Section 49.5 we consider the deran- domization of the latter. Section 49.6 discusses the combinatorial framework of LP-type problems, which underlie most current combinatorial algorithms and al- lows their application to a host of optimization problems. We briefly describe the more recent combinatorial framework of unique sink orientations, in the context of striving for algorithms with a milder dependence on d. In Section 49.7 we examine parallel algorithms for this problem, and finally in Section 49.8 we briefly discuss related issues.
    [Show full text]
  • Computational Geometry – 99102
    Computational Geometry – 99102 Michael Goodrich (Johns Hopkins) Rolf Klein (Hagen) Raimund Seidel (Saarbr¨ucken) 07.03.1999 to 12.03.1999 Abstract Geometric computing is creeping into virtually every corner of science and engineering, from design and manufacturing to astrophysics and cartography. This report describes presentations made at a workshop focused on recent advances in this computational geometry field. Previous Dagstuhl workshops on computational geometry dealt mostly with theoretical issues: the development of provably efficient algorithms for various geometric problems on point sets, arrangements of curves and sur- faces, triangulations and other sets of objects; proving various combinatorial results on sets of geomtric objects, which usually have implications on the performance of performance guarantees of geometric algorithms; describing the intrinsic computational complexity of various geometric problems. This workshop continued some of this tradition, but as one more point there was a strong emphasis on the exchange of ideas regarding carrying the many theoretical findings of the last years into computational practice. There were presentations about the recent development of software libraries such as CGAL, LEDA, JDSL, VEGA, and TPIE, and also some experimentation with them. These libraries should help to simplify the realization of abstractly conceived geometric algorithms as actually executable software. The participants and organizers of this workshop would like to thank the Dagstuhl office and the local personnel for the engaged and competent support in all matters. The cordial atmosphere provided for an unusually successful and enjoyable workshop. 1 Computational Geometry Program - Monday, 8 March 1999 9.00-9.30: Marc Van Kreveld, Utrecht University Six CG Problems from Cartographic Generalization 9.30-10.00: Rolf Klein, FernUniversität Hagen Optimal Strategy for Walking an Unknown Street 10.00-10.45 COFFEE BREAK 10.45-11.15: Subhash Suri, Washington University Analysis of a Bounding Box Heuristic 11.15-11.45: Ferran Hurtado, Univ.
    [Show full text]
  • Fast Distributed Algorithms for LP-Type Problems of Bounded Dimension
    Fast Distributed Algorithms for LP-Type Problems of Bounded Dimension Kristian Hinnenthal Paderborn University, Germany [email protected] Christian Scheideler Paderborn University, Germany [email protected] Martijn Struijs TU Eindhoven, The Netherlands [email protected] Abstract In this paper we present various distributed algorithms for LP-type problems in the well-known gossip model. LP-type problems include many important classes of problems such as (integer) linear programming, geometric problems like smallest enclosing ball and polytope distance, and set problems like hitting set and set cover. In the gossip model, a node can only push information to or pull information from nodes chosen uniformly at random. Protocols for the gossip model are usually very practical due to their fast convergence, their simplicity, and their stability under stress and disruptions. Our algorithms are very efficient (logarithmic rounds or better with just polylogarithmic communication work per node per round) whenever the combinatorial dimension of the given LP-type problem is constant, even if the size of the given LP-type problem is polynomially large in the number of nodes. 2012 ACM Subject Classification Theory of computation → Distributed algorithms; Theory of computation → Mathematical optimization Keywords and phrases LP-type problems, abstract optimization problems, abstract linear programs, distributed algorithms, gossip algorithms 1 Introduction 1.1 LP-type problems LP-type problems were defined by Sharir and Welzl [30] as problems characterized by a tuple (H, f) where H is a finite set and f : 2H → T is a function that maps subsets from H to values in a totally ordered set (T, ≤) containing ∞.
    [Show full text]
  • A Subexponential Bound for Linear Programming∗†
    appeared in Algorithmica 16 (1996) 498-516. A Subexponential Bound for Linear Programming∗† Jirˇ´i Matouˇsek‡ Micha Sharir Department of Applied Mathematics School of Mathematical Sciences Charles University, Malostransk´e n´am. 25 Tel Aviv University, Tel Aviv 69978, Israel 118 00 Praha 1, Czechoslovakia e-mail: [email protected] e-mail: [email protected] and Courant Institute of Mathematical Sciences New York University, New York, NY 10012, USA Emo Welzl Institut f¨ur Informatik, Freie Universit¨at Berlin Takustraße 9, 14195 Berlin, Germany e-mail: [email protected] Abstract We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected min O(d22dn),e2√d ln(n/√d )+O(√d+ln n) { } time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorithm computes the lexicographically smallest nonnegative point satisfying n given linear inequalities in d variables. The expectation is over the internal randomizations performed by the algorithm, and holds for any input. In conjunction with Clarkson’s linear programming algorithm, this gives an expected bound of O(d2n + eO(√d ln d)) . The algorithm is presented in an abstract framework, which facilitates its appli- cation to several other related problems like computing the smallest enclosing ball (smallest volume enclosing ellipsoid) of n points in d-space, computing the distance of two n-vertex (or n-facet) polytopes in d-space, and others. The subexponential running time can also be established for some of these problems (this relies on some recent results due to G¨artner).
    [Show full text]
  • Fast and Robust Smallest Enclosing Balls *
    Fast and Robust Smallest Enclosing Balls ? Bernd GÄartner Institut furÄ Theoretische Informatik, ETH ZuricÄ h, ETH-Zentrum, CH-8092 ZuricÄ h, Switzerland ( [email protected]) Abstract. I describe a C++ program for computing the smallest en- closing ball of a point set in d-dimensional space, using oating-point arithmetic only. The program is very fast for d · 20, robust and simple (about 300 lines of code, excluding prototype de¯nitions). Its new fea- tures are a pivoting approach resembling the simplex method for linear programming, and a robust update scheme for intermediate solutions. The program with complete documentation following the literate pro- gramming paradigm [3] is available on the Web.1 1 Introduction The smallest enclosing ball (or Euclidean 1-center) problem is a classical problem of computational geometry. It has a long history which dates back to 1857 when Sylvester formulated it for points in the plane [8]. The ¯rst optimal linear-time algorithm for ¯xed dimension was given by Megiddo in 1982 [4]. In 1991, Emo Welzl developed a simple randomized method to solve the problem in expected linear time [9]. In contrast to Megiddo's method, his algorithm is easy to imple- ment and very fast in practice, for dimensions 2 and 3. In higher dimensions, a heuristic move-to-front variant considerably improves the performance. The roots of the program I will describe go back to 1991 when I ¯rst imple- mented Emo Welzl's new method. Using the move-to-front variant I was able to solve problems on 5000 points up to dimension d = 10 (see [9] for all results).
    [Show full text]