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Fixed-radius Near Neighbors
In computational geometry, the fixed-radius near neighbor problem is a variant of the nearest neighbor search problem. In the fixed-radius near neighbor problem, one is given as input a set of points in ''d''-dimensional Euclidean space and a fixed distance Δ. One must design a data structure that, given a query point ''q'', efficiently reports the points of the data structure that are within distance Δ of ''q''. The problem has long been studied; cites a 1966 paper by Levinthal that uses this technique as part of a system for visualizing molecular structures, and it has many other applications. Solution by rounding and hashing One method for solving the problem is to round the points to an integer lattice, scaled so that the distance between grid points is the desired distance Δ. A hash table can be used to find, for each input point, the other inputs that are mapped to nearby grid points, which can then be tested for whether their unrounded positions are actually within d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Nearest Neighbor Search
Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values. Formally, the nearest-neighbor (NN) search problem is defined as follows: given a set ''S'' of points in a space ''M'' and a query point ''q'' ∈ ''M'', find the closest point in ''S'' to ''q''. Donald Knuth in vol. 3 of '' The Art of Computer Programming'' (1973) called it the post-office problem, referring to an application of assigning to a residence the nearest post office. A direct generalization of this problem is a ''k''-NN search, where we need to find the ''k'' closest points. Most commonly ''M'' is a metric space and dissimilarity is expressed as a distance metric, which is symmetric and satisfies the triangle inequality. Even more common, ''M'' is take ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Integer Lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice (group), lattice in the Euclidean space whose lattice points are tuple, -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. Automorphism group The automorphism group (or group (mathematics), group of congruence relation, congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order of a group, order 2''n'' ''n''!. As a matrix group it is given by the set of all ''n'' × ''n'' signed permutation matrices. This group is group isomorphism, isomorphic to the semidirect product :(\mathbb Z_2)^n \rtimes S_n where the symmetric group ''S''''n'' acts on (Z2)''n'' by permutation (this is a classic example of a wreath product). For the square lattice, this is the group o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hash Table
In computer science, a hash table is a data structure that implements an associative array, also called a dictionary or simply map; an associative array is an abstract data type that maps Unique key, keys to Value (computer science), values. A hash table uses a hash function to compute an ''index'', also called a ''hash code'', into an array of ''buckets'' or ''slots'', from which the desired value can be found. During lookup, the key is hashed and the resulting hash indicates where the corresponding value is stored. A map implemented by a hash table is called a hash map. Most hash table designs employ an Perfect hash function, imperfect hash function. Hash collision, Hash collisions, where the hash function generates the same index for more than one key, therefore typically must be accommodated in some way. In a well-dimensioned hash table, the average time complexity for each lookup is independent of the number of elements stored in the table. Many hash table designs also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Constant Of Proportionality
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called ''coefficient of proportionality'' (or ''proportionality constant'') and its reciprocal is known as ''constant of normalization'' (or ''normalizing constant''). Two sequences are inversely proportional if corresponding elements have a constant product. Two functions f(x) and g(x) are ''proportional'' if their ratio \frac is a constant function. If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., (for details see Ratio). Proportionality is closely related to ''linearity''. Direct proportionality Given an independent variable ''x'' and a dependent variable ''y'', ''y'' is directly proportional to ''x'' if there is a positive constant ''k'' such that: : y = kx The relation is often ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Exponential Growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now. In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth. Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function f(x) = x^3 grows at an ever increasing rate, but is much slower than growing exponentially. For example, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Information Processing Letters
''Information Processing Letters'' is a peer review, peer-reviewed scientific journal in the field of computer science, published by Elsevier. The aim of the journal is to enable fast dissemination of results in the field of Data processing, information processing in the form of short papers. Submissions are limited to nine double-spaced pages. The scope of IPL covers fundamental aspects of information processing and computing. This naturally covers topics in the broadly understood field of theoretical computer science, including algorithms, formal languages and automata, computational complexity, computational logic, distributed and parallel algorithms, computational geometry, learning theory, computational number theory, computational biology, coding theory, theoretical cryptography, and applied discrete mathematics. Generally, submissions in all areas of scientific inquiry are considered, provided that they describe research contributions credibly motivated by applications to com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Indifference Graph
In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other.. Indifference graphs are also the intersection graphs of sets of unit intervals, or of properly nested intervals (intervals none of which contains any other one). Based on these two types of interval representations, these graphs are also called unit interval graphs or proper interval graphs; they form a subclass of the interval graphs. Equivalent characterizations The finite indifference graphs may be equivalently characterized as *The intersection graphs of unit intervals, *The intersection graphs of sets of intervals no two of which are nested (one containing the other),. *The claw-free graph, claw-free interval graphs, *The graphs that do not have an induced subgraph isomorphic to a Claw (graph theory), claw ''K''1,3, net (a triangle with a degree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Unit Disk Graph
In geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex for each disk in the family, and with an edge between two vertices whenever the corresponding vertices lie within a unit distance of each other. They are commonly formed from a Poisson point process, making them a simple example of a random structure. Definitions There are several possible definitions of the unit disk graph, equivalent to each other up to a choice of scale factor: * Unit disk graphs are the graphs formed from a collection of points in the Euclidean plane, with a vertex for each point and an edge connecting each pair of points whose distance is below a fixed threshold. * Unit disk graphs are the intersection graphs of equal-radius circles, or of equal-radius disks. These graphs have a vertex for each circle or disk, and an edge connecting each pair of circles or disks that have a nonempty intersection. * Un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
GPU Technology Conference
Nvidia GTC (GPU Technology Conference) is a global artificial intelligence (AI) conference for developers that brings together developers, engineers, researchers, inventors, and IT professionals. Topics focus on AI, computer graphics, data science, machine learning and autonomous machines. Each conference begins with a keynote from Nvidia CEO and founder Jensen Huang, followed by a variety of sessions and talks with experts from around the world. It originated in 2009 in San Jose, California, with an initial focus on the potential for solving computing challenges through GPUs. In recent years, the conference focus has shifted to various applications of artificial intelligence and deep learning, including: self-driving cars, healthcare, high performance computing, professional visualization, and Nvidia Deep Learning Institute (DLI) training. History The first GTC was held from September 30 to October 2, 2009 at the Fairmont San Jose hotel and attracted roughly 1,500 attendees. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cell Lists
Cell lists (also sometimes referred to as cell linked-lists) is a data structure in molecular dynamics simulations to find all atom pairs within a given cut-off distance of each other. These pairs are needed to compute the short-range non-bonded interactions in a system, such as Van der Waals force In molecular physics and chemistry, the van der Waals force (sometimes van der Waals' force) is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical elec ...s or the short-range part of the electrostatic interaction when using Ewald summation. Algorithm Cell lists work by subdividing the simulation domain into cells with an edge length greater than or equal to the cut-off radius of the interaction to be computed. The particles are sorted into these cells and the interactions are computed between particles in the same or neighbouring cells. In its most basic form, the non-bonded interactions f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
