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Kinetic Euclidean Minimum Spanning Tree
A kinetic Euclidean minimum spanning tree is a kinetic data structure that maintains the Euclidean minimum spanning tree (EMST) of a set ''P'' of ''n'' points that are moving continuously. For the set of points ''P'' in 2-dimensional space, there are two kinetic algorithms for maintenance of the EMST. Rahmati and Zarei build a kinetic data structure based on the kinetic Delaunay triangulation to handle updates to the EMST in polylog time per event. Their kinetic data structure handles O(n*m) events, where m is the number of all changes to the Delaunay triangulation of the moving points. Their kinetic approach can work well for maintenance of the minimum spanning tree (MST) of a planar graph whose edge weights are changing as a continuous function of time. Abam, Rahmati, and Zarei provide a significant improvement on exact kinetic maintenance on the Euclidean minimum spanning tree A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-di ... [...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] |
Euclidean Minimum Spanning Tree
A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system of line segments with the points as endpoints, minimizing the total length of the segments. In it, any two points can reach each other along a path through the line segments. It can be found as the minimum spanning tree of a complete graph with the points as vertices and the Euclidean distances between points as edge weights. The edges of the minimum spanning tree meet at angles of at least 60°, at most six to a vertex. In higher dimensions, the number of edges per vertex is bounded by the kissing number of tangent unit spheres. The total length of the edges, for points in a unit square, is at most proportional to the square root of the number of points. Each edge lies in an empty region of the plane, and these regions can be used to prove that the Euclidean minimum spanning tree is a subgraph of other geometric graphs includin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Delaunay Triangulation
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 (disambigu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polylogarithmic Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expressed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Spanning Tree
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components. There are many use cases for minimum spanning trees. One example is a telecommunications company trying to lay cable in a new neighborhood. If it is constrained to bury the cable only along certain paths (e.g. roads), then there would be a graph containing the points (e.g. houses) connected by those paths. Some of the paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weights. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
