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Strongly-polynomial Time
In computer science, a ''polynomial-time algorithm'' is generally speaking an algorithm whose running time is upper-bounded by some polynomial function of the input size. The definition naturally depends on the computational model, which determines how the ''running time'' is measured, and how the ''input size'' is measured. Two prominent computational models are the Turing-machine model and the arithmetic model. A strongly-polynomial time algorithm is polynomial in both models, whereas a weakly-polynomial time algorithm is polynomial only in the Turing machine model. The difference between strongly- and weakly-polynomial time is when the inputs to the algorithms consist of integer or rational numbers. It is particularly common in optimization. Computational models Two common computational models are the Turing-machine model and the arithmetic model: * In the arithmetic model, every real number requires a single memory cell, whereas in the Turing model the storage size o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Algorithm
In theoretical 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 genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm. The machine operates on an infinite memory tape divided into discrete mathematics, discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the Alphabet (formal languages), alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on the symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, or halts the computation. The choice of which replacement symbol to write, which direction to move the head, and whet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Model Of Computation
In computing, especially computational geometry, a real RAM (random-access machine) is a mathematical model of a computer that can compute with exact real numbers instead of the binary fixed-point or floating-point numbers used by most actual computers. The real RAM was formulated by Michael Ian Shamos in his 1978 Ph.D. dissertation. Model The "RAM" part of the real RAM model name stands for "random-access machine". This is a model of computing that resembles a simplified version of a standard computer architecture. It consists of a stored program, a computer memory unit consisting of an array of cells, and a central processing unit with a bounded number of registers. Each memory cell or register can store a real number. Under the control of the program, the real RAM can transfer real numbers between memory and registers, and perform arithmetic operations on the values stored in the registers. The allowed operations typically include addition, subtraction, multiplication, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization (mathematics)
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his ''Elements'' (). It is an example of an ''algorithm'', a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repeated Squaring
In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation. These can be of quite general use, for example in modular arithmetic or powering of matrices. For semigroups for which additive notation is commonly used, like elliptic curves used in cryptography, this method is also referred to as double-and-add. Basic method Recursive version The method is based on the observation that, for any integer n > 0, one has: x^n= \begin x \, ( x^)^, & \mbox n \mbox \\ (x^)^ , & \mbox n \mbox \end If the exponent is zero then the answer is 1. If the exponent is negative then we can reuse the previous formula by rewriting the value using a positive exponent. That is, x^n = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Model Of Computation
In computing, especially computational geometry, a real RAM (random-access machine) is a mathematical model of a computer that can compute with exact real numbers instead of the binary fixed-point or floating-point numbers used by most actual computers. The real RAM was formulated by Michael Ian Shamos in his 1978 Ph.D. dissertation. Model The "RAM" part of the real RAM model name stands for "random-access machine". This is a model of computing that resembles a simplified version of a standard computer architecture. It consists of a stored program, a computer memory unit consisting of an array of cells, and a central processing unit with a bounded number of registers. Each memory cell or register can store a real number. Under the control of the program, the real RAM can transfer real numbers between memory and registers, and perform arithmetic operations on the values stored in the registers. The allowed operations typically include addition, subtraction, multiplication, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Of Computation
In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how units of computations, memories, and communications are organized. The computational complexity of an algorithm can be measured given a model of computation. Using a model allows studying the performance of algorithms independently of the variations that are specific to particular implementations and specific technology. Categories Models of computation can be classified into three categories: sequential models, functional models, and concurrent models. Sequential models Sequential models include: * Finite-state machines * Post machines ( Post–Turing machines and tag machines). * Pushdown automata * Register machines ** Random-access machines * Turing machines * Decision tree model * External memory model Functional models Functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest common divisor of and is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, . In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor, etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see ''Polynomial greatest common divisor'') and other commutative rings (see ' below). Overview Definition The ''greatest common divisor'' (GCD) of integers and , at least one of which is nonzero, is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-polynomial Time
In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is a polynomial in the ''numeric value'' of the input (the largest integer present in the input)—but not necessarily in the ''length'' of the input (the number of bits required to represent it), which is the case for polynomial time algorithms.Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, 1979. In general, the numeric value of the input is exponential in the input length, which is why a pseudo-polynomial time algorithm does not necessarily run in polynomial time with respect to the input length. An NP-complete problem with known pseudo-polynomial time algorithms is called weakly NP-complete. An NP-complete problem is called strongly NP-complete if it is proven that it cannot be solved by a pseudo-polynomial time algorithm unless . The strong/weak kinds of NP-hardness are defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
