Minimum Convex Partitions and Maximum Empty Polytopes
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Iterated Nearest Neighbors and Finding Minimal Polytopes* 1
Iterated Nearest Neighb ors and Finding Minimal Polytop es y z David Eppstein Je Erickson Abstract Weintro duce a new metho d for nding several typ es of optimal k -p oint sets, minimizing p erimeter, diameter, circumradius, and related measures, by testing sets of the O (k ) nearest neighb ors to each p oint. We argue that this is b etter in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the rst time to eciently maintain minimal sets as new p oints are inserted, to generalize our algorithms to higher dimensions, to nd minimal convex k -vertex p olygons and p olytop es, and to improvemany previous results. Weachievemany of our results via a new algorithm for nding rectilinear nearest neighb ors in the plane in time O (n log n + kn). We also demonstrate a related technique for nding minimum area k -p oint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of p oints. A generalization of this technique also allows us to nd minimum volume and b oundary measure sets in arbitrary dimensions. 1 Intro duction Anumb er of recent pap ers have discussed problems of selecting, from a set of n p oints, the k points optimizing some particular criterion [2, 14, 20]. Criteria that have b een studied include diameter [2], variance [2], area of the convex hull [20], convex hull p erimeter [14,20], and rectilinear diameter and p erimeter [2]. -
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. -
Scenechecker: Boosting Scenario Verification Using Symmetry
SceneChecker: Boosting Scenario Verification using Symmetry Abstractions? Hussein Sibai , Yangge Li , and Sayan Mitra University of Illinois at Urbana-Champaign Coordinated Science Laboratory {sibai2,li213,mitras}@illinois.edu Abstract. We present SceneChecker, a tool for verifying scenarios involving vehicles executing complex plans in large cluttered workspaces. SceneChecker converts the scenario verification problem to a standard hybrid system verification problem, and solves it effectively by exploiting structural properties in the plan and the vehicle dynamics. SceneChecker uses symmetry abstractions, a novel refine- ment algorithm, and importantly, is built to boost the performance of any existing reachability analysis tool as a plug-in subroutine. We evaluated SceneChecker on several scenarios involving ground and aerial vehicles with nonlinear dynamics and neural network controllers, employing different kinds of symmetries, using different reachability subroutines, and following plans with hundreds of way- points in complex workspaces. Compared to two leading tools, DryVR and Flow*, SceneChecker shows 14× average speedup in verification time, even while using those very tools as reachability subroutines1. Keywords: Hybrid systems · Safety verification · Symmetry. 1 Introduction Remarkable progress has been made in safety verification of hybrid and cyber-physical systems in the last decade [2,3,4,5,6,7,8,9]. The methods and tools developed have been applied to check safety of aerospace, medical, and autonomous vehicle control systems [4,5,10,11]. The next barrier in making these techniques usable for more com- plex applications is to deal with what is colloquially called the scenario verification problem. A key part of the scenario verification problem is to check that a vehicle or an agent can execute a plan through a complex environment. -
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. -
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. -
Optimal Payments in Dominant-Strategy Mechanisms for Single-Parameter Domains
Optimal Payments in Dominant-Strategy Mechanisms for Single-Parameter Domains Victor Naroditskiy Maria Polukarov Nicholas R. Jennings School of Electronics and Computer Science, University of Southampton, United Kingdom fvn,mp3,[email protected] Abstract We study dominant-strategy mechanisms in allocation domains where agents have one-dimensional types and quasi-linear utilities. Taking an allocation function as an input, we present an algorithmic technique for finding opti- mal payments in a class of mechanism design problems, including utilitarian and egalitarian allocation of homogeneous items with nondecreasing marginal costs. Our results link optimality of payment functions to a geometric condi- tion involving triangulations of polytopes. When this condition is satisfied, we constructively show the existence of an optimal payment function that is piecewise linear in agent types. 1. Introduction A celebrated positive result in dominant-strategy implementation is the class of Groves mechanism [10] for quasi-linear environments. For unrestricted type spaces, the Groves class describes all deterministic dominant-strategy mechanisms [21]. When types are given by single numbers|in single-parameter domains|it characterizes all efficient mechanisms. Mechanisms within the Groves class differ from one another in the amounts of payments they col- lect from the agents. Until recently, most of the attention in the literature was given to a single Groves mechanism called VCG.1 Our work develops a 1Also known as the Vickrey-Clarke-Groves, the pivotal, or the Clarke mechanism. technique for finding the best mechanism from the Groves class (and a more general class that includes non-efficient mechanisms) for a given objective. To see the need to optimize over mechanisms in the Groves class, consider allocating free items among a group of participants each having a private value for receiving an item (or, more generally, scenarios with no residual claimant absorbing the surplus or covering the deficit). -
A Concentration Result of Estimating Phi-Divergence Using Data
A Concentration Result of Estimating Phi-Divergence using Data Dependent Partition Fengqiao Luo1 and Sanjay Mehrotra2 1,2Department of Industrial Engineering and Management Science, Northwestern University September 18, 2018 Abstract Estimation of the φ-divergence between two unknown probability distributions using empirical data is a fundamental problem in information theory and statistical learning. We consider a multi-variate generalization of the data dependent partitioning method for estimating divergence between the two unknown distributions. Under the assumption that the distribution satisfies a power law of decay, we provide a convergence rate result for this method on the number of samples and hyper-rectangles required to ensure the estimation error is bounded by a given level with a given probability. 1 Introduction d Let P and Q be two probability distributions on defined on (R , Rd ), where Rd is the Borel measure on B B Rd. The φ-divergence of Q from P is defined as: dP Dφ(P Q)= φ dQ. (1) || Rd dQ Z The φ-divergence family includes the Kullback-Leibler (KL) divergence (Kullback and Leibler, 1951), Hellinger distance (Nikulin, 2001), total variation distance, χ2-divergence, α-divergence among others. Many other information-theoretic quantities such as entropy and mutual information can be formulated as special cases of φ-divergence. When the distributions P and Q are unknown, the estimate of D (P Q) based on the i.i.d. φ || samples from P and Q is a fundamental problem in information theory and statistics. The estimation of arXiv:1801.00852v1 [math.PR] 2 Jan 2018 φ-divergence as well as entropy and mutual information has many important applications. -
30 POLYGONS Joseph O’Rourke, Subhash Suri, and Csaba D
30 POLYGONS Joseph O'Rourke, Subhash Suri, and Csaba D. T´oth INTRODUCTION Polygons are one of the fundamental building blocks in geometric modeling, and they are used to represent a wide variety of shapes and figures in computer graph- ics, vision, pattern recognition, robotics, and other computational fields. By a polygon we mean a region of the plane enclosed by a simple cycle of straight line segments; a simple cycle means that nonadjacent segments do not intersect and two adjacent segments intersect only at their common endpoint. This chapter de- scribes a collection of results on polygons with both combinatorial and algorithmic flavors. After classifying polygons in the opening section, Section 30.2 looks at sim- ple polygonizations, Section 30.3 covers polygon decomposition, and Section 30.4 polygon intersection. Sections 30.5 addresses polygon containment problems and Section 30.6 touches upon a few miscellaneous problems and results. 30.1 POLYGON CLASSIFICATION Polygons can be classified in several different ways depending on their domain of application. In chip-masking applications, for instance, the most commonly used polygons have their sides parallel to the coordinate axes. GLOSSARY Simple polygon: A closed region of the plane enclosed by a simple cycle of straight line segments. Convex polygon: The line segment joining any two points of the polygon lies within the polygon. Monotone polygon: Any line orthogonal to the direction of monotonicity inter- sects the polygon in a single connected piece. Star-shaped polygon: The entire polygon is visible from some point inside the polygon. Orthogonal polygon: A polygon with sides parallel to the (orthogonal) coordi- nate axes. -
Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles
Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles Monika Henzinger Faculty of Computer Science, University of Vienna, Vienna, Austria [email protected] Stefan Neumann Faculty of Computer Science, University of Vienna, Vienna, Austria [email protected] Andreas Wiese Department of Industrial Engineering, Universidad de Chile, Santiago, Chile [email protected] Abstract Independent set is a fundamental problem in combinatorial optimization. While in general graphs the problem is essentially inapproximable, for many important graph classes there are approximation algorithms known in the offline setting. These graph classes include interval graphs and geometric intersection graphs, where vertices correspond to intervals/geometric objects and an edge indicates that the two corresponding objects intersect. We present dynamic approximation algorithms for independent set of intervals, hypercubes and hyperrectangles in d dimensions. They work in the fully dynamic model where each update inserts or deletes a geometric object. All our algorithms are deterministic and have worst-case update times that are polylogarithmic for constant d and ε > 0, assuming that the coordinates of all input objects are in [0,N]d and each of their edges has length at least 1. We obtain the following results: For weighted intervals, we maintain a (1 + ε)-approximate solution. For d-dimensional hypercubes we maintain a (1 + ε)2d-approximate solution in the unweighted case and a O(2d)-approximate solution in the weighted case. Also, we show that for maintaining an unweighted (1 + ε)-approximate solution one needs polynomial update time for d ≥ 2 if the ETH holds. For weighted d-dimensional hyperrectangles we present a dynamic algorithm with approximation ratio (1 + ε) logd−1 N. -
Approximate Polytope Membership Queries and Applications
UNIVERSITÉ CÔTE D’AZUR Approximate Polytope Membership Queries and Applications Mémoire d’habilitation à diriger des recherches Spécialité Informatique Guilherme Dias da Fonseca Université Clermont Auvergne, LIMOS, and INRIA Sophia Antipolis [email protected] Jean Cardinal Université Libre de Bruxelles Rapporteur Timothy Chan University of Illinois at UC Rapporteur Nabil Mustafa ESIEE Paris Rapporteur Jean-Daniel Boissonnat INRIA, Sophia Antipolis Garant Victor Chepoi Aix-Marseille Université Examinateur Bruno Martin Université de Nice Sophia Antipolis Examinateur David M. Mount University of Maryland Examinateur June 8, 2018 Abstract In the polytope membership problem, we are given a convex polytope P Rd for constant d 2, and the objective is to preprocess P into a data structure so that,⊂ given any query ≥ point q Rd, it is possible to determine efficiently whether q P . We consider this problem2 in an approximate setting. Given an approximation parameter2 " > 0, an "- approximate polytope membership query returns a positive result if q P , a negative result if the distance from q to P is greater than " diam(P ), and it may2 return either result otherwise. · We presented the first data structures especially tailored to approximate polytope (d 1)=4 membership. Initially, we showed how to answer queries in O(1=" − ) time using (d 1)=2 optimal O(1=" − ) storage. Later, we improved the analysis of the same data structure (d 1)=8 to O(log(1=")=" − ) query time for the same optimal storage. Switching to a different approach, we finally obtained an optimal data structure with O(log(1=")) query time (d 1)=2 and O(1=" − ) storage. -
Limiting Polytope Geometry for Rigid Rods, Disks, and Spheres Frank H
Journal of Statistical Physics, VoL 1, No. 1, 1969 Limiting Polytope geometry for Rigid Rods, Disks, and Spheres Frank H. Stillinger, Jr., 1 and Zevi W. Salsburg ~ Received February 13, 1969 The available configuration space for finite systems of rigid particles separates into equivalent disconnected regions if those systems are highly compressed. This paper presents a study of the geometric properties of the limiting high-compression regions (polytopes) for rods, disks, and spheres. The molecular distribution functions represent cross sections through the convex polytopes, and for that reason they are obliged to exhibit single-peak behavior by the Brtinn-Minkowski inequality. We demonstrate that increasing system dimensionality implies tendency toward nearest-neighbor particle-pair localization away from contact. The relation between the generalized Euler theorem for the limiting polytopes and cooperative "jamming" of groups of particles is explored. A connection is obtained between the moments of inertia of the polytopes (regarded as solid homogeneous bodies) and crystal elastic properties. Finally, we provide a list of unsolved problems in this geometrical many-body theory. KEY WORDS: Rigid spheres; Rigid disks; Rigid rods; Elasticity; High pressure; Polytopes; Convexity; Crystal anharmonicity; Pair correlation functions; Multidi- mensional geometry; Crystalline order; Crystal defects. 1. INTRODUCTION The hard-sphere system has been one of the most important and convenient models available to the theoretician during the development of modern statistical mechanics. Its attractiveness is largely due to easy visualization of energetically allowed particle configurations. Also, the relevant mathematical analysis, while generally far from trivial, is still relatively simpler than corresponding analysis for other molecular models which specify "softer" collisions. -
Fat Polygonal Partitions with Applications to Visualization and Embeddings∗
Fat Polygonal Partitions with Applications to Visualization and Embeddings∗ Mark de Bergy Krzysztof Onakz Anastasios Sidiropoulosx December 13, 2013 Abstract Let be a rooted and weighted tree, where the weight of any node is equal to the sum of the weightsT of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partitionT necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular par- titions in Rd. We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a polylog(∆)-approximation algorithm for embedding n-point ultra- metrics into Rd with minimum distortion, where ∆ denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points.