Metric spaceIn mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane.
Complete metric spaceIn mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below).
Competitive analysis (online algorithm)Competitive analysis is a method invented for analyzing online algorithms, in which the performance of an online algorithm (which must satisfy an unpredictable sequence of requests, completing each request without being able to see the future) is compared to the performance of an optimal offline algorithm that can view the sequence of requests in advance. An algorithm is competitive if its competitive ratio—the ratio between its performance and the offline algorithm's performance—is bounded.
Online algorithmIn computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from the beginning and is required to output an answer which solves the problem at hand. In operations research, the area in which online algorithms are developed is called online optimization.
Uniform spaceIn the mathematical field of topology, a uniform space is a topological space with additional structure that is used to define uniform properties, such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis. In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points.
Randomized algorithmA randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables.
Uniform continuityIn mathematics, a real function of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number , then there is a positive real number such that at any and in any function interval of the size .
Search algorithmIn computer science, a search algorithm is an algorithm designed to solve a search problem. Search algorithms work to retrieve information stored within particular data structure, or calculated in the search space of a problem domain, with either discrete or continuous values. Although search engines use search algorithms, they belong to the study of information retrieval, not algorithmics. The appropriate search algorithm to use often depends on the data structure being searched, and may also include prior knowledge about the data.
Uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .
Uniform normIn mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the , the , the , or, when the supremum is in fact the maximum, the . The name "uniform norm" derives from the fact that a sequence of functions \left{f_n\right} converges to f under the metric derived from the uniform norm if and only if f_n converges to f uniformly.
