Polygon Partitions
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Optimal Geometric Partitions, Covers and K-Centers
Optimal Geometric Partitions, Covers and K-Centers MUGUREL IONU Ţ ANDREICA, ELIANA-DINA TÎR ŞA Politehnica University of Bucharest Splaiul Independentei 313, sector 6, Bucharest ROMANIA {mugurel.andreica,eliana.tirsa}@cs.pub.ro https://mail.cs.pub.ro/~mugurel.andreica/ CRISTINA TEODORA ANDREICA, ROMULUS ANDREICA Commercial Academy Satu Mare Mihai Eminescu 5, Satu Mare ROMANIA [email protected] http://www.academiacomerciala.ro MIHAI ARISTOTEL UNGUREANU Romanian-American University Bd. Expozitiei 1B, sector 1, Bucharest ROMANIA [email protected] http://www.rau.ro Abstract: - In this paper we present some new, practical, geometric optimization techniques for computing polygon partitions, 1D and 2D point, interval, square and rectangle covers, as well as 1D and 2D interval and rectangle K-centers. All the techniques we present have immediate applications to several cost optimization and facility location problems which are quite common in practice. The main technique employed is dynamic programming, but we also make use of efficient data structures and fast greedy algorithms. Key-Words: - polygon partition, point cover, interval cover, rectangle cover, interval K-center, dynamic programming. 1 Introduction In this paper we present some geometric optimization techniques for computing polygon partitions, interval and rectangle K-centers and 1D and 2D point, interval, square and rectangle covers. The techniques have immediate applications to several cost optimization and facility location problems which appear quite often in practical settings. The main technique we use is dynamic programming, together with efficient data structures. For some problems, we also make use of binary search and greedy algorithms. For most of the problems we will only show how to compute the best value of the optimization function; the actual solution will be easy to compute using some auxiliary information. -
Conflict-Free Chromatic Art Gallery Coverage Andreas Bärtschi, Subhash Suri
Conflict-free Chromatic Art Gallery Coverage Andreas Bärtschi, Subhash Suri To cite this version: Andreas Bärtschi, Subhash Suri. Conflict-free Chromatic Art Gallery Coverage. STACS’12 (29th Symposium on Theoretical Aspects of Computer Science), Feb 2012, Paris, France. pp.160-171. hal- 00678165 HAL Id: hal-00678165 https://hal.archives-ouvertes.fr/hal-00678165 Submitted on 3 Feb 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Conflict-free Chromatic Art Gallery Coverage∗ Andreas Bärtschi1 and Subhash Suri2 1 Institute of Theoretical Computer Science, ETH Zürich 8092 Zürich, Switzerland ([email protected]) 2 Department of Computer Science, University of California Santa Barbara, CA 93106, USA ([email protected]) Abstract We consider a chromatic variant of the art gallery problem, where each guard is assigned one of k distinct colors. A placement of such colored guards is conflict-free if each point of the polygon is seen by some guard whose color appears exactly once among the guards visible to that point. What is the smallest number k(n) of colors that ensure a conflict-free covering of all n-vertex polygons? We call this the conflict-free chromatic art gallery problem. -
Exploring Topics of the Art Gallery Problem
The College of Wooster Open Works Senior Independent Study Theses 2019 Exploring Topics of the Art Gallery Problem Megan Vuich The College of Wooster, [email protected] Follow this and additional works at: https://openworks.wooster.edu/independentstudy Recommended Citation Vuich, Megan, "Exploring Topics of the Art Gallery Problem" (2019). Senior Independent Study Theses. Paper 8534. This Senior Independent Study Thesis Exemplar is brought to you by Open Works, a service of The College of Wooster Libraries. It has been accepted for inclusion in Senior Independent Study Theses by an authorized administrator of Open Works. For more information, please contact [email protected]. © Copyright 2019 Megan Vuich Exploring Topics of the Art Gallery Problem Independent Study Thesis Presented in Partial Fulfillment of the Requirements for the Degree Bachelor of Arts in the Department of Mathematics and Computer Science at The College of Wooster by Megan Vuich The College of Wooster 2019 Advised by: Dr. Robert Kelvey Abstract Created in the 1970’s, the Art Gallery Problem seeks to answer the question of how many security guards are necessary to fully survey the floor plan of any building. These floor plans are modeled by polygons, with guards represented by points inside these shapes. Shortly after the creation of the problem, it was theorized that for guards whose positions were limited to the polygon’s j n k vertices, 3 guards are sufficient to watch any type of polygon, where n is the number of the polygon’s vertices. Two proofs accompanied this theorem, drawing from concepts of computational geometry and graph theory. -
Locating Guards for Visibility Coverage of Polygons∗
Locating Guards for Visibility Coverage of Polygons∗ Yoav Amity Joseph S. B. Mitchellz Eli Packerx n Abstract trivial b 3 c-approximation that comes from the classical We propose heuristics for visibility coverage of a polygon combinatorial bound. with the fewest point guards. This optimal coverage prob- Our Contribution. We propose heuristics for lem, often called the \art gallery problem", is known to be computing a small set of point guards to cover a given NP-hard, so most recent research has focused on heuristics polygon. While we are not able to prove good worst- and approximation methods. We evaluate our heuristics case approximation bounds for our methods (indeed, through experimentation, comparing the upper bounds on each can be made \bad" in certain cases), we conduct the optimal guard number given by our methods with com- an experimental analysis of their performance. We give puted lower bounds based on heuristics for placing a large methods also to compute lower bounds on the optimal number of visibility-independent \witness points". We give number of guards for each instance in our experiments, experimental evidence that our heuristics perform well in using another set of implemented heuristics for deter- practice, on a large suite of input data; often the heuristics mining a set of visibility-independent witness points. give a provably optimal result, while in other cases there (The cardinality of such a set is a lower bound on the op- is only a small gap between the computed upper and lower timal number of guards.) We show that on a wide range bounds on the optimal guard number. -
AN EXPOSITION of MONSKY's THEOREM Contents 1. Introduction
AN EXPOSITION OF MONSKY'S THEOREM WILLIAM SABLAN Abstract. Since the 1970s, the problem of dividing a polygon into triangles of equal area has been a surprisingly difficult yet rich field of study. This paper gives an exposition of some of the combinatorial and number theoretic ideas used in this field. Specifically, this paper will examine how these methods are used to prove Monsky's theorem which states only an even number of triangles of equal area can divide a square. Contents 1. Introduction 1 2. The p-adic Valuation and Absolute Value 2 3. Sperner's Lemma 3 4. Proof of Monsky's Theorem 4 Acknowledgments 7 References 7 1. Introduction If one tried to divide a square into triangles of equal area, one would see in Figure 1 that an even number of such triangles would work, but could an odd number work? Fred Richman [3], a professor at New Mexico State, considered using this question in an exam for graduate students in 1965. After being unable to find any reference or answer to this question, he decided to ask it in the American Mathematical Monthly. n Figure 1. Dissection of squares into an even number of triangles After 5 years, Paul Monsky [4] finally found an answer when he proved that only an even number of triangles of equal area could divide a square, which later came to be known as Monsky's theorem. Formally, the theorem states: Theorem 1.1 (Monsky [4]). Let S be a square in the plane. If a triangulation of S into m triangles of equal area is given, then m is even. -
CSE 5319-001 (Computational Geometry) SYLLABUS
CSE 5319-001 (Computational Geometry) SYLLABUS Spring 2012: TR 11:00-12:20, ERB 129 Instructor: Bob Weems, Associate Professor, http://ranger.uta.edu/~weems Office: 627 ERB, 817/272-2337 ([email protected]) Hours: TR 12:30-1:50 PM and by appointment (please email by 8:30 AM) Prerequisite: Advanced Algorithms (CSE 5311) Objective: Ability to apply and expand geometric techniques in computing. Outcomes: 1. Exposure to algorithms and data structures for geometric problems. 2. Exposure to techniques for addressing degenerate cases. 3. Exposure to randomization as a tool for developing geometric algorithms. 4. Experience using CGAL with C++/STL. Textbooks: M. de Berg et.al., Computational Geometry: Algorithms and Applications, 3rd ed., Springer-Verlag, 2000. https://libproxy.uta.edu/login?url=http://www.springerlink.com/content/k18243 S.L. Devadoss and J. O’Rourke, Discrete and Computational Geometry, Princeton University Press, 2011. References: Adobe Systems Inc., PostScript Language Tutorial and Cookbook, Addison-Wesley, 1985. (http://Www-cdf.fnal.gov/offline/PostScript/BLUEBOOK.PDF) B. Casselman, Mathematical Illustrations: A Manual of Geometry and PostScript, Springer-Verlag, 2005. (http://www.math.ubc.ca/~cass/graphics/manual) CGAL User and Reference Manual (http://www.cgal.org/Manual) T. Cormen, et.al., Introduction to Algorithms, 3rd ed., MIT Press, 2009. E.D. Demaine and J. O’Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, 2007. (occasionally) J. O’Rourke, Art Gallery Theorems and Algorithms, Oxford Univ. Press, 1987. (http://maven.smith.edu/~orourke/books/ArtGalleryTheorems/art.html, occasionally) J. O’Rourke, Computational Geometry in C, 2nd ed., Cambridge Univ. -
Optimal Higher Order Delaunay Triangulations of Polygons*
Optimal Higher Order Delaunay Triangulations of Polygons Rodrigo I. Silveira and Marc van Kreveld Department of Information and Computing Sciences Utrecht University, 3508 TB Utrecht, The Netherlands {rodrigo,marc}@cs.uu.nl Abstract. This paper presents an algorithm to triangulate polygons optimally using order-k Delaunay triangulations, for a number of qual- ity measures. The algorithm uses properties of higher order Delaunay triangulations to improve the O(n3) running time required for normal triangulations to O(k2n log k + kn log n) expected time, where n is the number of vertices of the polygon. An extension to polygons with points inside is also presented, allowing to compute an optimal triangulation of a polygon with h ≥ 1 components inside in O(kn log n)+O(k)h+2n expected time. Furthermore, through experimental results we show that, in practice, it can be used to triangulate point sets optimally for small values of k. This represents the first practical result on optimization of higher order Delaunay triangulations for k>1. 1 Introduction One of the best studied topics in computational geometry is the triangulation. When the input is a point set P , it is defined as a subdivision of the plane whose bounded faces are triangles and whose vertices are the points of P .When the input is a polygon, the goal is to decompose it into triangles by drawing diagonals. Triangulations have applications in a large number of fields, including com- puter graphics, multivariate analysis, mesh generation, and terrain modeling. Since for a given point set or polygon, many triangulations exist, it is possible to try to find one that is the best according to some criterion that measures some property of the triangulation. -
Polygon Decomposition Motivation: Art Gallery Problem Art Gallery Problem
CG Lecture 3 Motivation: Art gallery problem Definition: two points q and r in a Polygon decomposition simple polygon P can see each other if the open segment qr R lies entirely within P. 1. Polygon triangulation p q • Triangulation theory A point p guards a region • Monotone polygon triangulation R ⊆ P if p sees all q∈R 2. Polygon decomposition into monotone r pieces Problem: Given a polygon P, what 3. Trapezoidal decomposition is the minimum number of guards 4. Convex decomposition required to guard P, and what are their locations? 5. Other results 1 2 Simple observations Art gallery problem: upper bound • Convex polygon: all points • Theorem: Every simple planar are visible from all other polygon with n vertices has a points only one guard in triangulation of size n-2 (proof any location is necessary! later). convex • Star-shaped polygon: all • n-2 guards suffice for an n-gon: points are visible from any • Subdivide the polygon into n–2 point in the kernel only triangles (triangulation). one guard located in its • Place one guard in each triangle. kernel is necessary. star-shaped 3 4 Art gallery problem: lower bound Simple polygon triangulation • There exists a Input: a polygon P polygon with n described by an ordered vertices, for which sequence of vertices ⎣n/3⎦ guards are <v0, …vn–1>. necessary. Output: a partition of P Can we improve the upper into n–2 non-overlapping • Therefore, ⎣n/3⎦ bound? triangles and the guards are needed in adjacencies between Yes! In fact, at most ⎣n/3⎦ the worst case. -
Two Algorithms for Constructing a Delaunay Triangulation 1
International Journal of Computer and Information Sciences, Vol. 9, No. 3, 1980 Two Algorithms for Constructing a Delaunay Triangulation 1 D. T. Lee 2 and B. J. Schachter 3 Received July 1978; revised February 1980 This paper provides a unified discussion of the Delaunay triangulation. Its geometric properties are reviewed and several applications are discussed. Two algorithms are presented for constructing the triangulation over a planar set of Npoints. The first algorithm uses a divide-and-conquer approach. It runs in O(Nlog N) time, which is asymptotically optimal. The second algorithm is iterative and requires O(N 2) time in the worst case. However, its average case performance is comparable to that of the first algorithm. KEY WORDS: Delaunay triangulation; triangulation; divide-and-con- quer; Voronoi tessellation; computational geometry; analysis of algorithms. 1. INTRODUCTION In this paper we consider the problem of triangulating a set of points in the plane. Let V be a set of N ~> 3 distinct points in the Euclidean plane. We assume that these points are not all colinear. Let E be the set of (n) straight- line segments (edges) between vertices in V. Two edges el, e~ ~ E, el ~ e~, will be said to properly intersect if they intersect at a point other than their endpoints. A triangulation of V is a planar straight-line graph G(V, E') for which E' is a maximal subset of E such that no two edges of E' properly intersect.~16~ 1 This work was supported in part by the National Science Foundation under grant MCS-76-17321 and the Joint Services Electronics Program under contract DAAB-07- 72-0259. -
An Introductory Study on Art Gallery Theorems and Problems
AN INTRODUCTORY STUDY ON ART GALLERY THEOREMS AND PROBLEMS JEFFRY CHHIBBER M Dept of Mathematics, Noorul Islam Centre of Higher Education, Kanyakumari, Tamil Nadu, India. E-mail: [email protected] Abstract - In computational geometry and robot motion planning, a visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane. Visibility graphs may also be used to calculate the placement of radio antennas, or as a tool used within architecture and urban planningthrough visibility graph analysis. This is a brief survey on the visibility graphs application in Art Gallery Problems and Theorems. Keywords- Art galley theorems, orthogonal polygon, triangulation, Visibility graphs I. INTRODUCTION point y outside of P if the segment xy is nowhere interior to P; xy may intersect ∂P, the boundary of P. In a visibility graph, each node in the graph Star polygon: A polygon visible from a single interior represents a point location, and each edge represents point. Diagonal: A segment inside a polygon whose a visible connectionbetween them. That is, if the line endpoints are vertices, and which otherwise does not segment connecting two locations does not pass touch ∂P. Floodlight: A light that illuminates from the through any obstacle, an edge is drawn between them apex of a cone with aperture α. Vertex floodlight: in the graph. Lozano-Perez & Wesley (1979) attribute One whose apex is at a vertex (at most one per the visibility graph method for Euclidean shortest vertex). paths to research in 1969 by Nils Nilsson on motion planning for Shakey the robot, and also cite a 1973 The problem: description of this method by Russian mathematicians What is the art gallery problem? M. -
Cluster Algebras in Kinematic Space of Scattering Amplitudes
Prepared for submission to JHEP Cluster Algebras in Kinematic Space of Scattering Amplitudes Marcus A. C. Torres IMPA, Est. Dona Castorina 110, 22460-320 Rio de Janeiro-RJ, Brazil E-mail: [email protected] Abstract: We clarify the natural cluster algebra of type A that exists in a residual and tropical form in the kinematical space as suggested in 1711.09102 by the use of triangu- lations, mutations and associahedron on the definition of scattering forms. We also show that this residual cluster algebra is preserved in a hypercube (diamond) necklace inside the n associahedron where cluster sub-algebras (A1) exist. This result goes in line with results with cluster poligarithms in 1401.6446 written in terms of A2 and A3 functions only and other works showing the primacy of A cluster sub-algebras as data input for scattering amplitudes. arXiv:1712.06161v1 [hep-th] 17 Dec 2017 Contents 1 Introduction1 2 Cluster Algebras3 2.1 An Cluster Algebras4 2.2 Snake triangulations5 3 Kinematic space and its An cluster structure6 3 4 Kn vs. Confn(P ) 9 5 Conclusion 10 1 Introduction In [1] cluster algebras made quite an appearance in the studies of Scattering Amplitudes. There, the authors showed that a judicious choice of kinematic variables was one of the main ingredients in a large simplification of the previously calculated two-loops six particle (2) MHV remainder function Rn [2–4] of N = 4 supersymmetric Yang-Mills (SYM). This choice is related to the cluster structure that is intrinsic to the kinematic configuration 3 space Confn(P ) of n external particles. -
Visibility-Monotonic Polygon Deflation
Volume 10, Number 1, Pages 1{21 ISSN 1715-0868 VISIBILITY-MONOTONIC POLYGON DEFLATION PROSENJIT BOSE, VIDA DUJMOVIC,´ NIMA HODA, AND PAT MORIN Abstract. A deflated polygon is a polygon with no visibility crossings. We answer a question posed by Devadoss et al. (2012) by presenting a polygon that cannot be deformed via continuous visibility-decreasing motion into a deflated polygon. We show that the least n for which there exists such an n-gon is seven. In order to demonstrate non-deflatability, we use a new combinatorial structure for polygons, the directed dual, which encodes the visibility properties of deflated polygons. We also show that any two deflated polygons with the same directed dual can be deformed, one into the other, through a visibility-preserving defor- mation. 1. Introduction Much work has been done on visibilities of polygons [6, 9] as well as on their convexification, including work on convexification through continuous motions [4]. Devadoss et al. [5] combine these two areas in asking the follow- ing two questions: (1) Can every polygon be convexified through a deforma- tion in which visibilities monotonically increase? (2) Can every polygon be deflated (i.e. lose all its visibility crossings) through a deformation in which visibilities monotonically decrease? The first of these questions was answered in the affirmative at CCCG 2011 by Aichholzer et al. [2]. In this paper, we resolve the second question in the negative by presenting a non-deflatable polygon, shown in Figure 10A. While it is possible to use ad hoc arguments to demonstrate the non-deflatability of this polygon, we develop a combinatorial structure, the directed dual, that allows us to prove non-deflatability for this and other examples using only combinatorial arguments.