Convex hulls of finite sets of points in two and three dimensions
TL;DR: The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls to ensure optimal time complexity within a multiplicative constant.
Abstract: The convex hulls of sets of n points in two and three dimensions can be determined with O(n log n) operations. The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls. Since any convex hull algorithm requires at least O(n log n) operations, the time complexity of the proposed algorithms is optimal within a multiplicative constant.
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TL;DR: The Voronoi diagram as discussed by the authors divides the plane according to the nearest-neighbor points in the plane, and then divides the vertices of the plane into vertices, where vertices correspond to vertices in a plane.
Abstract: Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor
4,236 citations
Book•
01 Jan 1987TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.
2,284 citations
TL;DR: In this article, the authors propose an approach based on characterizing the position and orientation of an object as a single point in a configuration space, in which each coordinate represents a degree of freedom in the position or orientation of the object.
Abstract: This paper presents algorithms for computing constraints on the position of an object due to the presence of ther objects. This problem arises in applications that require choosing how to arrange or how to move objects without collisions. The approach presented here is based on characterizing the position and orientation of an object as a single point in a configuration space, in which each coordinate represents a degree of freedom in the position or orientation of the object. The configurations forbidden to this object, due to the presence of other objects, can then be characterized as regions in the configuration space, called configuration space obstacles. The paper presents algorithms for computing these configuration space obstacles when the objects are polygons or polyhedra.
1,996 citations
Proceedings Article•
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TL;DR: A new neural architecture to learn the conditional probability of an output sequence with elements that are discrete tokens corresponding to positions in an input sequence using a recently proposed mechanism of neural attention, called Ptr-Nets, which improves over sequence-to-sequence with input attention, but also allows it to generalize to variable size output dictionaries.
Abstract: We introduce a new neural architecture to learn the conditional probability of an output sequence with elements that are discrete tokens corresponding to positions in an input sequence. Such problems cannot be trivially addressed by existent approaches such as sequence-to-sequence [1] and Neural Turing Machines [2], because the number of target classes in each step of the output depends on the length of the input, which is variable. Problems such as sorting variable sized sequences, and various combinatorial optimization problems belong to this class. Our model solves the problem of variable size output dictionaries using a recently proposed mechanism of neural attention. It differs from the previous attention attempts in that, instead of using attention to blend hidden units of an encoder to a context vector at each decoder step, it uses attention as a pointer to select a member of the input sequence as the output. We call this architecture a Pointer Net (Ptr-Net). We show Ptr-Nets can be used to learn approximate solutions to three challenging geometric problems - finding planar convex hulls, computing Delaunay triangulations, and the planar Travelling Salesman Problem - using training examples alone. Ptr-Nets not only improve over sequence-to-sequence with input attention, but also allow us to generalize to variable size output dictionaries. We show that the learnt models generalize beyond the maximum lengths they were trained on. We hope our results on these tasks will encourage a broader exploration of neural learning for discrete problems.
1,949 citations
Book•
01 Jul 1990
TL;DR: Algorithms for computing constraints on the position of an object due to the presence of ther objects, which arises in applications that require choosing how to arrange or how to move objects without collisions are presented.
1,641 citations
References
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Book•
01 Jan 1974
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Abstract: From the Publisher:
With this text, you gain an understanding of the fundamental concepts of algorithms, the very heart of computer science. It introduces the basic data structures and programming techniques often used in efficient algorithms. Covers use of lists, push-down stacks, queues, trees, and graphs. Later chapters go into sorting, searching and graphing algorithms, the string-matching algorithms, and the Schonhage-Strassen integer-multiplication algorithm. Provides numerous graded exercises at the end of each chapter.
0201000296B04062001
9,262 citations
TL;DR: P can be chosen to I&E the centroid oC the triangle formed by X, y and z and Express each si E S in polar coordinates th origin P and 8 = 0 in the direction of zu~ arhitnry fixed half-line L from P.
1,741 citations
"Convex hulls of finite sets of poin..." refers background in this paper
...Graham [5] for a set of points in the plane....
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TL;DR: A Steiner minimal tree for given points in the plane is a tree which interconnects these points using lines of shortest possible total length as mentioned in this paper, where the length of the shortest possible line is chosen.
Abstract: A Steiner minimal tree for given points $A_1 , \cdots ,A_n $ in the plane is a tree which interconnects these points using lines of shortest possible total length. In order to achieve minimum lengt...
946 citations
TL;DR: The problem of finding all maximal elements of V with respect to the partial ordering is considered and the computational com- plexity of the problem is defined to be the number of required comparisons of two components and is denoted by Cd(n).
Abstract: H. T. KUNG Carnegze-Mellon Un~verszty, P2ttsburgh, Pennsylvanza F. LUCCIO Unwerszht d~ P~sa, P~sa, Italy F. P. PREPARATA University of Ilhno~s, Urbana, Illinois ASSTRACT. Let U1 , U2, . . . , Ud be totally ordered sets and let V be a set of n d-dimensional vectors In U~ X Us. . X Ud . A partial ordering is defined on V in a natural way The problem of finding all maximal elements of V with respect to the partial ordering ~s considered The computational com- plexity of the problem is defined to be the number of required comparisons of two components and is denoted by Cd(n). It is tnwal that C~(n) = n - 1 and C,~(n) _ flog2 n!l for d _> 2
856 citations
"Convex hulls of finite sets of poin..." refers background or methods in this paper
...The arguments presented are similar to those developed in connection with finding the maxima of a set of vectors ~[8, 11]), which is a problem related to the one being investigated....
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...the same computat ion model was adopted in [8] and [11]....
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...The bound is based on a previous result concerning maxima of vectors [8, 11] and on a connection between the two problems which Shamos attributes to A....
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692 citations