DOCSLIB.ORG

Geometric Algorithms for Object Placement and Planarity in a Terrain

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometric Algorithms for Object Placement and Planarity in a Terrain Geometric Algorithms for Object Placement and Planarity in a Terrain Rahul Ray Max-Planck Institut fur¨ Informatik Geometric Algorithms for Object Placement and Planarity in a Terrain Geometric Algorithms for Object Placement and Planarity in a Terrain Rahul Ray Dissertation zur Erlangung des Grades Doktor der Ingenieurwissenschaften (Dr.-Ing.) der Naturwissenschaftlich-Technischen Fakultat¨ I der Universitat¨ des Saarlandes Saarbruck¨ en July 27th, 2004 Max-Planck Institut fur¨ Informatik Tag des Kolloquiums: 27 July 2004 Dekan: Prof. Dr. Philipp Slusallek Gutachter: Prof. Dr. Kurt Mehlhorn Prof. Dr. Stefan Schirra Dedicated to my parents Acknowledgements Many people have taught, inspired, encouraged, supported, helped and advised me during the time in which I worked on this thesis. I wish to express my deepest gratitude to all of them, without which this thesis would never happen. First of all, I would like to thank my advisor, Prof. Dr. Kurt Mehlhorn, for providing me a perfect balance of scientific guidance and scientific freedom. He has always been a great source of motivation whenever I needed. I would like to thank him for creating a wonderful research atmosphere in Max-Planck Institut fur¨ Informatik, Saarbruck¨ en. I also wish to thank Dr. Stefan Funke and Dr. Theocharis Malamatos for co-guiding my research work that led to this thesis. They are great persons to work with and I consider myself privileged to have collaborated with them in my research. I share some of the most enjoyable moments in MPI with them which came along during many of our enlighting discussions or exchanges of innu- merable emails. They also helped me in proof-reading this thesis manuscript and improving the presentation a lot. When I go back to my uninteresting days in industry, I shall never forget Prof. Michiel Smid who gave me an opportunity to think about a PhD option. I am grateful to him for infinite number of reasons. He introduced me to the planarity problem in terrain and many other interesting aspects of it while I was in Magdeburg University (Germany). I am thankful to ever smiling Prof. Ulrich Wendt for his wonderful collaboration from material science department in our project. I learned a lot from my co-authors, whose ideas and insightful suggestions were invaluable in discovering many of the algorithms presented in this thesis. My sincere thanks to them. I thank Prof. Stefan Schirra for kindly agreeing to review the thesis and to act as an examiner. I would like to thank and express my sincere gratitude to Suman Mukherjee, who pur- suaded me to continue this research at a time I thought to abandon it. My friend Arnab Paul has always kept on inspiring me even before I started my research. All my friends in Tajmahal, Chhannachhara have been a constant source of inspiration for me to finish this thesis one day. My special thanks to them. My MPII experience would not be the same without my friends, colleagues and office- mates (categories not mutually exclusive). I thank Naveen Sivadasan, Debapriyo, Hisao, Venkatesh, Tobias, Ingmar, Christian, Holgar, Kerstin, Petra, Arno, Nicola, Kavitha, Sunil and many others who are/were my colleagues in various times in MPII for their friendly help and making my life fun and enjoyable. I express my gratitude and thanks to Deb, Sudip, Chandrima, Pintu, Raja, Sai, Reddy, Venkat, Fawad, Tarang, Hareesh and many others in Saarcricket Club for providing wonderful social interaction which has tunred my stay in Saarbruck¨ en an unforgettable event in my life. x Finally, I would like to thank those whom I owe the most. My parents have always en- couraged me to pursue higher studies even when I was in a kindergarten !! They have greatly influenced my life with love and guidance and most importantly freedom of my soul which allowed me to reach where I am now. My then class mate and now wife, Piyali, was a great emotional support for me, whether it was a time for hard work or for taking it easy. To mark the submission of this thesis, we were blessed with a baby son, Rishav, on 22nd April. I gratefully acknowledge the fellowship from International Max-Planck Research School (IMPRS) , that supported my research at Max-Planck Institut fur¨ Informatik, Saarbruck¨ en. Abstract We consider the following placement problem: Let C be a compact set in R2 and let S be a set of n points in R2. The objective is to compute a translate of C that contains the maximum number, κ∗, of points of S. Motivated by the applications in clustering and pattern recognition, this optimal placement problem has received much attention over the last two decades. It is known that this problem can be solved in a time that is roughly quadratic in n. We show for a given " > 0 how random-sampling and bucketing techniques can be used to develop a near-linear-time Monte Carlo algorithm that computes a placement of C containing at least (1 ")κ points of S with high probability. − ∗ When C is a unit disk, we give an approximation algorithm for the optimal placement problem by approximating the constraining radius of the disk. Here, our algorithm computes a placement of the disk of radius (1 + ") which contains at least κ∗ points, where κ∗ denotes the maximum number of points covered by any unit disk. The running time of this algorithm is O(n="2). Then, we turn to the problem of computing the largest connected region in a triangulated terrain of size n for which the normals of the triangles deviate by at most some small fixed angle. We devise an exact algorithm by adapting dynamic graph connectivity algorithm to compute the largest planar region in O(n2 log n(log log n)3) time. We also give a heuristic that can be used to compute sufficiently large planar regions in a terrain much faster. We discuss an implementation of this heuristic, and show some experimental results for terrains representing three-dimensional (topographical) images of fracture surfaces of metals obtained by confocal laser scanning microscopy, which directly motivated our research into this direction. Since the output of this heuristic comes with no quality assurance, we present a new ap- proximation scheme for the same problem which apart from giving a guarantee on the quality of the produced solution also has been implemented in practice and shows good performance on real-world data representing fracture surfaces. This approximate deterministic algorithm computes in O(n="2) time the largest approximately connected planar region in the terrain. We also give a variant of the above algorithm with a better dependence on " at the cost of an extra poly-logarithmic factor on n. For a sufficiently large n, both the algorithms consume optimal O(n) space. Kurzzusammenfassung Wir betrachten das folgende Platzierungsproblem: Sei C eine kompakte Menge im R2 and S eine Menge von n Punkten im R2. Das Ziel ist es, ein Verschiebung von C zu berechnen, welche eine maximale Anzahl κ∗ an Punkten aus S enthalt.¨ Aufgrund zahlreichen Anwendun- gen im Clustering und in der Mustererkennung wurde dieses Problem in den letzten Jahrzehn- ten oft betrachtet. Es gibt bekannte Algorithmen, die dieses Problem in fast-quadratischer Zeit losen.¨ Wir prasentieren¨ einen Monte-Carlo Algorithmus, der fur¨ ein gegebenes " > 0 in fast- linearer Zeit eine Platzierung von C berechnet, welche mindestens (1 ")κ Punkte aus S − ∗ mit hoher Wahrscheinlichkeit enthalt.¨ Unser Algorithmus basiert auf der Entnahme zufalliger¨ Stichproben und Bucketingtechniken. Im Falle, dass C eine Einheitskreisscheibe ist, stellen wir einen Approximationsalgorith- mus fur¨ das Plazierungsproblem vor, wobei wir den Radius der Kreisscheibe relaxieren. So erhalten wir eine Platzierung einer Kreisscheibe mit Radius (1 + ") welche mindestens κ∗ Punkte enthalt.¨ Hierbei bezeichnet κ∗ die maximale Anzahl von Punkten, die von einer Ein- heitskreisscheibe uberdeckt¨ werden konnen.¨ Die Laufzeit dieses Algorithmus ist O(n="2). Danach betrachten wir das Problem, die grosste¨ zusammenhangende¨ Region in einem tri- angulierten Terrain der Grosse¨ n zu berechnen, fur¨ welche die Dreiecksnormalen der Re- gion um nicht mehr als einen vorgegebenen Winkel abweichen. Wir prasentieren¨ einen ex- akten Algorithmus, der auf einer Datenstruktur zur dynamischen Verwaltung von Zusammen- hangskomponenten eines Graphen basiert und das Problem in O(n2 log n(log log n)3) Zeit lost.¨ Wir stellen auch eine heuristische Losung¨ vor, die schnell relativ grosse planare Regio- nen berechnet. Diese Heuristik wurde implementiert und experimentell evaluiert. Dazu lagen uns dreidimensionale topografische Daten von Bruchoberflachen¨ verschiedenster Metallsorten vor. Diese Anwendung war auch die ursprungliche¨ Motivation fur¨ unsere Forschung. Da diese Heuristik jedoch keinerlei Qualitatsgarantie¨ fur¨ die berechnete Region liefert, stellen wir schliesslich ein neues Approximationsschema vor, welches sowohl eine garantierte Gute¨ liefert, als auch praktisch implementiert wurde und auf unseren Testdaten gute Resultate erzielte. Die Laufzeit dieses Approximationsalgorithmus ist O(n="2). Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 Placement Problem . 2 1.1.1 Problem Definition . 2 1.1.2 Motivation and State of the Art . 2 1.1.3 Main Results . 3 1.2 Planarity in a Terrain . 4 1.2.1 Problem Definition . 4 1.2.2 Motivation . 4 1.2.3 Main Results . 5 1.3 Finding Large Planar Regions in a Terrain with a Guarantee . 6 1.3.1 Problem Definition . 6 1.3.2 Main Results . 6 2. Placement Algorithms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.1 Introduction . 9 2.2 Model of Computation . 10 2.3 Overview of Results . 11 2.4 Preliminaries and Exact Algorithms . 12 2.4.1 Bucketing and Estimating κ∗(C; S). 15 2.4.2 A Bucketing Algorithm . 16 2.5 Monte-Carlo Algorithms . 17 2.5.1 A Random Sampling Approach . 17 2.5.2 Bucketing and Sampling Combined .
Load more

Recommended publications
  • Covered Objects∗
    The Complexity of the Union of (α, β)-Covered Objects∗ Alon Efrat† January 28, 2000 Abstract An (α, β)-covered object is a simply connected planar region c with the property that for each point p ∈ ∂c there exists a triangle contained in c and having p as a vertex, such that all its angles are at least α and all its edges are at least β · diam(c)- long. This notion extends that of fat convex objects. We show that the combinatorial complexity of the union of n (α, β)-covered objects of ‘constant description complexity’ 2 is O(λs+2(n)log n log log n), where s is the maximum number of intersections between the boundaries of any pair of the given objects. 1 Introduction A planar object c is (α, β)-covered if the following conditions are satisfied. 1. c is simply-connected; 2. For each point p ∈ ∂c we can place a triangle ∆ fully inside c, such that p is a vertex of ∆, each angle of ∆ is at least α, and the length of each edge of ∆ is at least β ·diam(c). We call such a triangle ∆ a good triangle for c. The notion of (α, β)-covered objects generalizes the notion of convex fat objects. A planar convex object c is α-fat if the ratio between the radii of the balls s+ and s− is at most α, where s+ is the smallest ball containing c and s− is a largest ball that is contained in c. It is easy to show that an α-fat convex object is an (α′, β′)-covered object, for appropriate constants α′, β′ that depend on α.
    [Show full text]
  • On the Theoretical Complexity of the Silhouette of a Polyhedron
    On the theoretical complexity of the silhouette of a polyhedron M´emoire soutenu le vendredi 5 septembre 2003 par Marc Glisse 1 pour l'obtention du dipl^ome d'´etudes approfondies sous la direction de Sylvain Lazard 2 1Ecole´ Normale Sup´erieure, 45, rue d'Ulm, 75005 Paris, France. Email: [email protected] 2Inria Lorraine, Nancy. Email: [email protected] Abstract We study conditions under which the silhouette of a polyhedron is guaranted to be sublinear, either on average or in the worst case, and give some counter-examples when not. Contents 1 Introduction 2 2 Definitions and general remarks 2 2.1 View, silhouette . 2 2.2 Projective invariance . 2 2.3 kD-fatness . 2 2.4 Duality . 3 3 Polytopes 3 3.1 First examples . 3 3.2 Apparent length of a polytope . 4 3.3 Worst-case complexity for polytopes . 5 3.3.1 The cylinder example . 5 3.3.2 Local theorem . 6 3.3.3 Global theorem . 6 3.3.4 Existence of such polytopes . 7 3.4 Average complexity for polytopes . 7 3.5 Lower bound for polytopes . 8 4 Approximation of a surface 9 4.1 Kettner and Welzl . 9 4.2 Average case . 9 4.2.1 Smooth surface . 9 4.2.2 Generalization . 10 4.2.3 Shadow . 10 4.3 Worst case . 10 4.3.1 The sphere . 10 4.3.2 Other surfaces . 10 5 Conclusion 11 1 1 Introduction Given a viewpoint, the apparent boundary of a polyhedron (in 3D), or silhouette, is the set of edges incident to a visible face and an invisible one, and in the neighbourhood of which one can see infinity; a face whose supporting plane contains the viewpoint is considered invisible.
    [Show full text]
  • Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs∗
    Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs∗ Sariel Har-Peledy Kent Quanrudz September 23, 2015 Abstract We investigate the family of intersection graphs of low density objects in low dimensional Eu- clidean space. This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion. We present efficient (1 + ")-approximation algorithms for polynomial expansion graphs, for Independent Set, Set Cover, and Dominating Set problems, among others, and these results seem to be new. Naturally, PTAS's for these problems are known for subclasses of this graph family. These results have immediate interesting applications in the geometric domain. For example, the new algorithms yield the only PTAS known for covering points by fat triangles (that are shallow). We also prove corresponding hardness of approximation for some of these optimization problems, characterizing their intractability with respect to density. For example, we show that there is no PTAS for covering points by fat triangles if they are not shallow, thus matching our PTAS for this problem with respect to depth. 1. Introduction Many classical optimization problems are intractable to approximate, let alone solve. Motivated by the discrepancy between the worst-case analysis and real-world success of algorithms, more realistic models of input have been developed, alongside algorithms that take advantage of their properties. In this paper, we investigate approximability of some classical optimization problems (e.g., set cover and independent set, among others) for two closely-related families of graphs: Graphs with polynomially- bounded expansion, and intersection graphs of geometric objects with low-density.
    [Show full text]
  • TANGRAM TREEMAPS Method with Various Shapes
    An enclosure geometrical partitioning TANGRAM TREEMAPS method with various shapes Tangram Treemaps An enclosure geometrical partitioning method with various shapes By Jie Liang Supervisor: A/Prof. Mao Lin Huang Co-supervisor: Dr. Quang Vinh Nguyen A thesis submitted in fulfilment for the Degree of Doctor of Philosophy In the Faculty of Engineering and Information Technology 1 | Page DEC 2012 Tangram Treemaps CERTIFICATE OF AUTHORSHIP/ORIGINALITY UNIVERSITY OF TECHNOLOGY SYDNEY I certify that the work in this thesis has not previously been submitted for a degree nor has it been submitted as part of requirements for a degree except as fully acknowledged within the text. I also certify that the thesis has been written by me. Any help that I have received in my research work and the preparation of the thesis itself has been acknowledged. In addition, I certify that all information sources and literature used are indicated in the thesis. SIGNATURE OF STUDENT Jie Liang i | Page ACKNOWLEDGEMENTS I would like to gratefully acknowledge the enthusiastic supervision of A/Prof Mao Lin Huang, during this research. He brought me closer to the reality I had initially perceived, eventually enabling me to grasp its rich complexity. This thesis grew out of a series of dialogues with him. The guidance, motivation and friendship of my co-supervisor, Dr. Quang Vinh Nguyen, has been invaluable on both an academic and a personal level, for which I am extremely grateful. Furthermore, I owe sincere thankfulness to the members of Visualization Team and fellow researchers and the professors of iNext Research centre. This research also benefited tremendously from many researchers and staffs in The University of Technology, Sydney.
    [Show full text]
  • Exploration, Interpolation and Extrapolation-Each Approached in a Different Manner
    I dedicate this thesis to my husband, Amiel, for his constant support and unconditional love. Acknowledgements I would like to express my special appreciation and thanks to my PhD advisors, Professors Shlomi Dolev and Zvi Lotker, for supporting me during these past five years. You have been a tremendous mentors for me. I would like to thank you for encouraging my research and for allowing me to grow as a research scientist. Your scientific advice and knowledge and many insightful discussions and suggestions have been priceless. I would also like to thank my committee members, professor Eitan Bachmat, professor Chen Avin, professor Amnon Ta-Shma for their helpful comments and suggestions in general. A heartfelt thanks to the really supportive and active BGU community here in Beer-Sheva and all my friends who made the research experience something special, in particular, Ariel, Dan, Nisha, Shantanu, Martin, Nova, Eyal, Guy and Marina. Special thanks to Daniel and Elisa for proof reading my final draft. A special thanks to my family. Words cannot express how grateful I am to my mother-in law, father-in-law, my mother, and father for all of the sacrifices that youve made onmy behalf. Finally, I would like to acknowledge the most important person in my life my husband Amiel. He has been a constant source of strength and inspiration. There were times during the past five years when everything seemed hopeless and I didnt have any hope. I can honestly say that it was only his determination and constant encouragement (and sometimes a kick on my backside when I needed one) that ultimately made it possible for me to see this project through to the end.
    [Show full text]
  • New Data Structures and Algorithms for Mobile Data
    New data structures and algorithms for mobile data Citation for published version (APA): Abam, M. A. (2007). New data structures and algorithms for mobile data. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR630204 DOI: 10.6100/IR630204 Document status and date: Published: 01/01/2007 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
    [Show full text]
  • Decompositions and Applications
    Algorithms for fat objects : decompositions and applications Citation for published version (APA): Gray, C. M. (2008). Algorithms for fat objects : decompositions and applications. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR636648 DOI: 10.6100/IR636648 Document status and date: Published: 01/01/2008 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
    [Show full text]
  • Domination in Geometric Intersection Graphs
    Domination in Geometric Intersection Graphs Thomas Erlebach1 and Erik Jan van Leeuwen2,? 1 Department of Computer Science, University of Leicester, University Road, Leicester LE1 7RH, UK, [email protected] 2 CWI, Kruislaan 413, 1098 SJ Amsterdam, the Netherlands, [email protected] Abstract. For intersection graphs of disks and other fat objects, polynomial-time approximation schemes are known for the independent set and vertex cover prob- lems, but the existing techniques were not able to deal with the dominating set problem except in the special case of unit-size objects. We present approximation algorithms and inapproximability results that shed new light on the approxima- bility of the dominating set problem in geometric intersection graphs. On the one hand, we show that for intersection graphs of arbitrary fat objects, the dominat- ing set problem is as hard to approximate as for general graphs. For intersec- tion graphs of arbitrary rectangles, we prove APX-hardness. On the other hand, we present a new general technique for deriving approximation algorithms for various geometric intersection graphs, yielding constant-factor approximation al- gorithms for r-regular polygons, where r is an arbitrary constant, for pairwise homothetic triangles, and for rectangles with bounded aspect ratio. For arbitrary fat objects with bounded ply, we get a (3 + )-approximation algorithm. 1 Introduction We study the approximability of the minimum dominating set problem in geometric in- tersection graphs. Given an undirected graph G = (V, E), a set D ⊆ V is a dominating set if every v ∈ V is in D or has a neighbor in D.
    [Show full text]
  • Christer Ericson — «Real Time Collision Detection
    Accurate and efficient collision detection in complex environments is one of the foundations of today’s cutting-edge computer games. Yet collision detection is notoriously difficult to implement robustly and takes up an increasingly large fraction of compute cycles in current game engines as increasingly detailed environments are becoming the norm. Real-time Collision Detection is a comprehensive reference on this topic,covering it with both breadth and depth. Not only are the fundamental algorithms explained clearly and in detail, but Ericson’s book covers crucial implementation issues, including geometric and numeric robustness and cache-efficient implementations of the algorithms. Together, these make this book a“must have”practical reference for anyone interested in developing interactive applications with complex environments. –Matt Pharr, Senior Software Developer, NVIDIA Christer Ericson’s Real-time Collision Detection is an excellent resource that covers the fundamentals as well as a broad array of techniques applicable to game development. –Jay Stelly, Senior Engineer,Valve Christer Ericson provides a practical and very accessible treatment of real-time collision detection.This includes a comprehensive set of C++ implementations of a very large number of routines necessary to build such applications in a context which is much broader than just game programming. The programs are well-thought out and the accompanying discussion reveals a deep understanding of the graphics, algorithms, and ease of implementation issues. It will find a welcome home on any graphics programmer’s bookshelf although it will most likely not stay there long as others will be constantly borrowing it. –Hanan Samet, Professor of Computer Science, University of Maryland Real-Time Collision Detection is an excellent resource that every serious engine programmer should have on his bookshelf.
    [Show full text]
  • Improved Bounds for the Union of Locally Fat Objects in the Plane∗
    SIAM J. COMPUT. c 2014 Society for Industrial and Applied Mathematics Vol. 43, No. 2, pp. 543–572 IMPROVED BOUNDS FOR THE UNION OF LOCALLY FAT OBJECTS IN THE PLANE∗ † ‡ § ¶ BORIS ARONOV ,MARKDEBERG,ESTHEREZRA, AND MICHA SHARIR Abstract. We show that, for any γ>0, the combinatorial complexity of the union of n locally γ- ∗ n O(log n) γ fat objects of constant complexity in the plane is γ4 2 . For the special case of -fat triangles, O n ∗ n n 2 1 the bound improves to ( log + γ log γ ). Key words. combinatorial geometry, union complexity, fat objects AMS subject classifications. 05D99, 52C45, 68U05, 68R05 DOI. 10.1137/120891241 1. Introduction. In this paper we obtain sharper upper bounds on the complex- ity of the union of n locally γ-fat objects of constant complexity in the plane, and of nγ-fat triangles in the plane; see below for the definitions of these classes of objects and for the precise statements of our bounds. Background. Consider a family F of well-behaved and simply shaped geometric objects in the plane; we will formally refer to them as objects of constant complexity, and give a precise definition of this notion below. For now assume F to consist of Jordan regions with interiors so that every pair of boundaries intersect in at most some fixed constant number of points. We denote the union of F by U(F). The (combinatorial) complexity of U(F), which we denote by |U(F)|, is defined as the total number of vertices of the union boundary, which can either be vertices of original objects or intersections between pairs of object boundaries.
    [Show full text]
  • Guarding Fat Polygons and Triangulating Guarded Polygons
    Guarding Fat Polygons and Triangulating Guarded Polygons G. Aloupis ∗ P. Bose † V. Dujmovic ‡ C. Gray § S. Langerman ¶ B. Speckmann § Abstract In this paper we study three related problems: (i) Given a fat polygon P , we show how to find a set G of points in P such that every point on the boundary of P sees at least one point of G. The set G is said to guard the boundary of P and its cardinality depends on the shape parameters of P . Fat polygons are often used to model more realistic inputs. (ii) Given k points that guard the boundary of P , we show how to compute O(k2) points that guard the interior of P . (iii) Given k points that guard P , we provide two simple O(kn) time algorithms to triangulate P : one that triangulates the polygon when the guards are given as input and the other slightly more complicated algorithm that triangulates the polygon when the guards are not given. 1 Introduction Many algorithms and data structures developed in Computational Geometry often display their worst-case performance on intricate configurations that seem artificial. Indeed, in practical situ- ations, these algorithms tend to perform much better than predicted by the theoretical bounds. An attempt to understand this disparity has led to the study of geometric problems with so-called realistic input models [5]. Here one places certain restrictions on the shape and/or distribution of the input objects, so that hypothetical worst-case examples are excluded. One of the most widely studied realistic input models assumes that the input objects are fat, that is, they are not arbitrar- ily long and skinny.
    [Show full text]
  • Motion Planning Amidst Fat Obstacles
    Motion Planning amidst Fat Obstacles Motion Planning tussen Vette Obstakels met een samenvatting in het Nederlands PROEFSCHRIFT ter verkrijging van de graad van do ctor aan de Universiteit Utrecht op gezag van de Rector Magnicus Prof dr JA van Ginkel ingevolge het b esluit van het College van Decanen in het op enbaar te verdedigen op vrijdag oktob er des middags te uur do or Arnoldus Franciscus van der Stapp en geb oren op oktob er te Eindhoven Promotor Prof dr MH Overmars Faculteit Wiskunde en Informatica CIPGEGEVENS KONINKLIJKE BIBLIOTHEEK DEN HAAG Stapp en Arnoldus Franciscus van der Motion planning amidst fat obstacles Arnoldus Franciscus van der Stapp en Utrecht Universiteit Utrecht Faculteit Wiskunde Informatica Pro efschrift Universiteit Utrecht Met index lit opg Met samenvatting in het Nederlands ISBN Trefw geometrie rob otica algoritmen The research in this thesis was supp orted by the Netherlands Organization for Sci entic Research NWO and partially supp orted by the ESPRIT I I BRA Pro ject ALCOM and the ESPRIT I I I BRA Pro ject PROMotion Contents Introduction The general motion planning problem Exact motion planning algorithms Fatness in geometry and thesis outline Fatness in computational geometry Fatness Computing the fatness of an ob ject Prop erties of scenes of fat ob jects Fatness implies low density Arrangements of fat ob ject wrappings Assembling and disassembling fat ob jects Fatness dened with resp ect to other shap es Range searching and p oint lo cation among fat ob jects
    [Show full text]