Computational complexityIn computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem. The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory.
Computational complexity theoryIn theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used.
Complexity classIn computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements.
Parameterized complexityIn computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to multiple parameters of the input or output. The complexity of a problem is then measured as a function of those parameters. This allows the classification of NP-hard problems on a finer scale than in the classical setting, where the complexity of a problem is only measured as a function of the number of bits in the input.
Time complexityIn 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.
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.
Space complexityThe space complexity of an algorithm or a computer program is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is the memory required by an algorithm until it executes completely. This includes the memory space used by its inputs, called input space, and any other (auxiliary) memory it uses during execution, which is called auxiliary space. Similar to time complexity, space complexity is often expressed asymptotically in big O notation, such as etc.
Quantum complexity theoryQuantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational model based on quantum mechanics. It studies the hardness of computational problems in relation to these complexity classes, as well as the relationship between quantum complexity classes and classical (i.e., non-quantum) complexity classes. Two important quantum complexity classes are BQP and QMA.
Asymptotic computational complexityIn computational complexity theory, asymptotic computational complexity is the usage of asymptotic analysis for the estimation of computational complexity of algorithms and computational problems, commonly associated with the usage of the big O notation. With respect to computational resources, asymptotic time complexity and asymptotic space complexity are commonly estimated. Other asymptotically estimated behavior include circuit complexity and various measures of parallel computation, such as the number of (parallel) processors.
Reduction (complexity)In computability theory and computational complexity theory, a reduction is an algorithm for transforming one problem into another problem. A sufficiently efficient reduction from one problem to another may be used to show that the second problem is at least as difficult as the first. Intuitively, problem A is reducible to problem B, if an algorithm for solving problem B efficiently (if it existed) could also be used as a subroutine to solve problem A efficiently. When this is true, solving A cannot be harder than solving B.
Analysis of algorithmsIn computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that relates the size of an algorithm's input to the number of steps it takes (its time complexity) or the number of storage locations it uses (its space complexity). An algorithm is said to be efficient when this function's values are small, or grow slowly compared to a growth in the size of the input.
DimensionIn physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it - for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it - for example, both a latitude and longitude are required to locate a point on the surface of a sphere.
Dimensionality reductionDimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable (hard to control or deal with).
Curse of dimensionalityThe curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. The expression was coined by Richard E. Bellman when considering problems in dynamic programming. Dimensionally cursed phenomena occur in domains such as numerical analysis, sampling, combinatorics, machine learning, data mining and databases.
One-dimensional spaceIn physics and mathematics, a sequence of n numbers can specify a location in n-dimensional space. When n = 1, the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the position of each point on it can be described by a single number. In algebraic geometry there are several structures that are technically one-dimensional spaces but referred to in other terms. A field k is a one-dimensional vector space over itself.
Nonlinear dimensionality reductionNonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis.
Iterated logarithmIn computer science, the iterated logarithm of , written (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to . The simplest formal definition is the result of this recurrence relation: On the positive real numbers, the continuous super-logarithm (inverse tetration) is essentially equivalent: i.e. the base b iterated logarithm is if n lies within the interval , where denotes tetration.
Four-dimensional spaceFour-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled x, y, and z).
Minkowski–Bouligand dimensionIn fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set in a Euclidean space , or more generally in a metric space . It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand. To calculate this dimension for a fractal , imagine this fractal lying on an evenly spaced grid and count how many boxes are required to cover the set.
