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.
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.
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.
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.
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.
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.
FactorialIn mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: For example, The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah.
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.
Falling and rising factorialsIn mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (x)_n , where n is a non-negative integer.
Hermite polynomialsIn mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets for wavelet transform analysis probability, such as the Edgeworth series, as well as in connection with Brownian motion; combinatorics, as an example of an Appell sequence, obeying the umbral calculus; numerical analysis as Gaussian quadrature; physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term is present); systems theory in connection with nonlinear operations on Gaussian noise.
Productive efficiencyIn microeconomic theory, productive efficiency (or production efficiency) is a situation in which the economy or an economic system (e.g., bank, hospital, industry, country) operating within the constraints of current industrial technology cannot increase production of one good without sacrificing production of another good. In simple terms, the concept is illustrated on a production possibility frontier (PPF), where all points on the curve are points of productive efficiency.
Gamma functionIn mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.
Laguerre polynomialsIn mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's differential equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of where n is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin).
Double factorialIn mathematics, the double factorial of a number n, denoted by n!!, is the product of all the positive integers up to n that have the same parity (odd or even) as n. That is, Restated, this says that for even n, the double factorial is while for odd n it is For example, 9!! = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0!! = 1 as an empty product. The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as The sequence of double factorials for odd n = 1, 3, 5, 7, 9,...
Allocative efficiencyAllocative efficiency is a state of the economy in which production is aligned with consumer preferences; in particular, the set of outputs is chosen so as to maximize the wellbeing of society. This is achieved if every good or service is produced up until the last unit provides a marginal benefit to consumers equal to the marginal cost of production. In economics, allocative efficiency entails production at the point on the production possibilities frontier that is optimal for society.
Big O notationBig O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for Ordnung, meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows.
Bernoulli polynomialsIn mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). For the Bernoulli polynomials, the number of crossings of the x-axis in the unit interval does not go up with the degree.
PolynomialIn mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. Polynomials appear in many areas of mathematics and science.
Pareto efficiencyPareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related: Given an initial situation, a Pareto improvement is a new situation where some agents will gain, and no agents will lose.
