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Kinetic Closest Pair
A kinetic closest pair data structure is a kinetic data structure that maintains the closest pair of points, given a set ''P'' of ''n'' points that are moving continuously with time in a metric space. While many Kinetic data structure#Performance, efficient algorithms were known in the static case, they proved hard to Kinetic data structure#Certificates Approach, kinetize, so new static algorithms were developed to solve this problem. 2D case Approach 1 The simplest kinetic approach for maintenance of the closest pair is to use variants of the Delaunay triangulations. Consider a hexagon and partition it into six equilateral triangles, and then create a Delaunay triangulation based on each equilateral triangle, as each one is a convex shape. The union of these six Delaunay triangulations, so called Equilateral Delaunay graph (EDG), is a supergraph for the nearest neighbor graph (NNG); the endpoints of the edge with minimum length in EDG gives the closest pair. It is straightf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Data Structure
A kinetic data structure is a data structure used to track an attribute of a geometric system that is moving continuously. For example, a kinetic convex hull data structure maintains the convex hull of a group of n moving points. The development of kinetic data structures was motivated by computational geometry problems involving physical objects in continuous motion, such as collision or visibility detection in robotics, animation or computer graphics. Overview Kinetic data structures are used on systems where there is a set of values that are changing as a function of time, in a known fashion. So the system has some values, and for each value v, it is known that v=f(t). Kinetic data structures allow queries on a system at the current virtual time t, and two additional operations: *\textrm(t): Advances the system to time t. *\textrm(v,f(t)): Alters the trajectory of value v to f(t), as of the current time. Additional operations may be supported. For example, kinetic data stru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closest Pair Of Points
The closest pair of points problem or closest pair problem is a problem of computational geometry: given n points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean plane was among the first geometric problems that were treated at the origins of the systematic study of the computational complexity of geometric algorithms. Time bounds Randomized algorithms that solve the problem in linear time are known, in Euclidean spaces whose dimension is treated as a constant for the purposes of asymptotic analysis. This is significantly faster than the O(n^2) time (expressed here in big O notation) that would be obtained by a naive algorithm of finding distances between all pairs of points and selecting the smallest. It is also possible to solve the problem without randomization, in random-access machine models of computation with unlimited memory that allow the use of the floor function, in near-linear O(n\log\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Data Structure
A kinetic data structure is a data structure used to track an attribute of a geometric system that is moving continuously. For example, a kinetic convex hull data structure maintains the convex hull of a group of n moving points. The development of kinetic data structures was motivated by computational geometry problems involving physical objects in continuous motion, such as collision or visibility detection in robotics, animation or computer graphics. Overview Kinetic data structures are used on systems where there is a set of values that are changing as a function of time, in a known fashion. So the system has some values, and for each value v, it is known that v=f(t). Kinetic data structures allow queries on a system at the current virtual time t, and two additional operations: *\textrm(t): Advances the system to time t. *\textrm(v,f(t)): Alters the trajectory of value v to f(t), as of the current time. Additional operations may be supported. For example, kinetic data stru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delaunay Triangulation
In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum of all the angles of the triangles in the triangulation; they tend to avoid sliver triangles. The triangulation is named after Boris Delaunay for his work on this topic from 1934. For a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case). For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors. By considering circumscribed spheres, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symposium On Computational Geometry
The International Symposium on Computational Geometry (SoCG) is an academic conference in computational geometry. It was founded in 1985, and was originally sponsored by the SIGACT and SIGGRAPH Special Interest Groups of the Association for Computing Machinery (ACM). It dissociated from the ACM in 2014, motivated by the difficulties of organizing ACM conferences outside the United States and by the possibility of turning to an open-access system of publication. Since 2015 the conference proceedings have been published by the Leibniz International Proceedings in Informatics Dagstuhl is a computer science research center in Germany, located in and named after a district of the town of Wadern, Merzig-Wadern, Saarland. Location Following the model of the mathematical center at Oberwolfach, the center is installed in ... instead of by the ACM. Since 2019 the conference has been organized under the auspices of the newly-formed Society for Computational Geometry. A 2010 assessment of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Principal properties Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: *The area is A=\frac a^2, *The perimeter is p=3a\,\! *The radius of the circumscribed circle is R = \frac *The radius of the inscribed circle is r=\frac a or r=\frac *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is h=\frac a Denoting the radius of the circumscribed circle as ''R'', we can determine using trigonometry that: *The area of the triangle is \mathrm=\fracR^2 Many of these quantities have simple r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nearest Neighbor Graph
The nearest neighbor graph (NNG) is a directed graph defined for a set of points in a metric space, such as the Euclidean distance in the plane. The NNG has a vertex for each point, and a directed edge from ''p'' to ''q'' whenever ''q'' is a nearest neighbor of ''p'', a point whose distance from ''p'' is minimum among all the given points other than ''p'' itself. In many uses of these graphs, the directions of the edges are ignored and the NNG is defined instead as an undirected graph. However, the nearest neighbor relation is not a symmetric one, i.e., ''p'' from the definition is not necessarily a nearest neighbor for ''q''. In theoretical discussions of algorithms a kind of general position is often assumed, namely, the nearest (k-nearest) neighbor is unique for each object. In implementations of the algorithms it is necessary to bear in mind that this is not always the case. For situations in which it is necessary to make the nearest neighbor for each object unique, the set '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Tournament
A Kinetic Tournament is a kinetic data structure that functions as a priority queue for elements whose priorities change as a continuous function of time. It is implemented analogously to a "tournament" between elements to determine the "winner" (maximum or minimum element), with the Kinetic data structure#Certificates Approach, certificates enforcing the winner of each "match" in the tournament. It supports the usual priority queue operations - ''insert'', ''delete'' and ''find-max''. They are often used as components of other kinetic data structures, such as kinetic closest pair. Implementation A kinetic tournament is organized in a binary tree-like structure, where the leaves contain the elements, and each internal node contains the larger (or smaller) of the elements in its child nodes. Thus, the Tree (data structure), root of the tree contains the maximum (or minimum) element at a given time. The validity of the structure is enforced by creating a certificate at each node ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Closest Pair Preliminaries
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and entertainment * Kinetic art, a form of art involving mechanical and/or random movement, including optical illusions. * ''Kinetic'', the 13th episode of the first season of the TV series ''Smallville'' * ''Kinetic'' (comics), a comic by Allan Heinberg and Kelley Pucklett * "Kinetic" (song), a song by Radiohead Companies * Kinetic Engineering Limited, Indian automotive manufacturer * Kinetic Group, Australian-based public transport company Technology * "Kinetic", Seiko's trademark for its automatic quartz technology * The ''Kinetic camera system'' by Birt Acres (1854–1918), photographer and film pioneer * Kinetic projectile Military terminology * Kinetic military action See also * * * Kinetics (other) * Dynamics (disam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Sorted List
A kinetic sorted list is a kinetic data structure for maintaining a list of points under motion in sorted order. It is used as a kinetic predecessor data structure, and as a component in more complex kinetic data structures such as kinetic closest pair. Implementation This data structure maintains a list of the elements in sorted order, with the certificates enforcing the order between adjacent elements. When a certificate fails, the concerned elements are swapped. Then at most three certificates must be updated, the certificate of the swapped pair, and the two certificates involving the swapped elements and the elements of the sorted list which directly precede and follow the swapped pair. For example, given a sorted list , the certificates will be [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Tree
In computer science, a range tree is an ordered tree data structure to hold a list of points. It allows all points within a given range to be reported efficiently, and is typically used in two or higher dimensions. Range trees were introduced by Jon Louis Bentley in 1979. Similar data structures were discovered independently by Lueker, Lee and Wong, and Willard. The range tree is an alternative to the ''k''-d tree. Compared to ''k''-d trees, range trees offer faster query times of (in Big O notation) O(\log^dn+k) but worse storage of O(n\log^ n), where ''n'' is the number of points stored in the tree, ''d'' is the dimension of each point and ''k'' is the number of points reported by a given query. Bernard Chazelle improved this to query time O(\log^ n + k) and space complexity O\left(n\left(\frac\right)^\right). Data structure A range tree on a set of 1-dimensional points is a balanced binary search tree on those points. The points stored in the tree are stored in the leav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Priority Queue
A Kinetic Priority Queue is an abstract kinetic data structure. It is a variant of a priority queue designed to maintain the maximum (or minimum) priority element (key-value pair) when the priority of every element is changing as a continuous function of time. Kinetic priority queues have been used as components of several kinetic data structures, as well as to solve some important non-kinetic problems such as the k-set problem and the connected red blue segments intersection problem. Implementations The operations supported are: * : create an empty kinetic priority queue * (or find-min): - return the (or for a ) value stored in the queue at the current virtual time . * : - insert a key into the kinetic queue at the current virtual time, whose value changes as a continuous function of time . * - delete a key at the current virtual time . There are several variants of kinetic priority queues, which support the same basic operations but have different performance guarante ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
