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Asymptotic Time Complexity
In 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. Scope With respect to computational resources, asymptotic time complexity and asymptotic space complexity of computational algorithms and programs are commonly estimated. Other asymptotically estimated behavior include circuit complexity and various measures of parallel computation, such as the number of (parallel) processors. Since the ground-breaking 1965 paper by Juris Hartmanis and Richard E. Stearns and the 1979 book by Michael Garey and David S. Johnson on NP-completeness,Michael Garey, and David S. Johnson: ''Computers and Intractability: A Guide to the Theory of NP-Completeness.'' New York: W. H. Freeman & Co., 1979. the term "computational complexity" (of algorithms) has become commonly referred to as asymptotic comput ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. 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. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analysis Of Algorithms
In 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. Different inputs of the same size may cause the algorithm to have different behavior, so best, worst and average case descriptions might all be of practical interest. When not otherwise specified, the function describing the performance of an algorithm is usually an upper bound, determined from the worst case inputs to the algorithm. The term "analysis of algorithms" was coined by Donald Knuth. Algorithm analysis is an important part of a broa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Models 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] |
Nondeterministic Algorithm
In computer science and computer programming, a nondeterministic algorithm is an algorithm that, even for the same input, can exhibit different behaviors on different runs, as opposed to a deterministic algorithm. Different models of computation give rise to different reasons that an algorithm may be non-deterministic, and different ways to evaluate its performance or correctness: *A concurrent algorithm can perform differently on different runs due to a race condition A race condition or race hazard is the condition of an electronics, software, or other system where the system's substantive behavior is dependent on the sequence or timing of other uncontrollable events, leading to unexpected or inconsistent .... This can happen even with a single-threaded algorithm when it interacts with resources external to it. In general, such an algorithm is considered to perform correctly only when ''all'' possible runs produce the desired results. *A probabilistic algorithm's behavior ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomized Algorithm
A 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. There is a distinction between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for example Quicksort), and algorithms which have a chance of producing an incorrect result ( Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem. In common practice, randomized alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deterministic Algorithm
In computer science, a deterministic algorithm is an algorithm that, given a particular input, will always produce the same output, with the underlying machine always passing through the same sequence of states. Deterministic algorithms are by far the most studied and familiar kind of algorithm, as well as one of the most practical, since they can be run on real machines efficiently. Formally, a deterministic algorithm computes a mathematical function; a function has a unique value for any input in its domain, and the algorithm is a process that produces this particular value as output. Formal definition Deterministic algorithms can be defined in terms of a state machine: a ''state'' describes what a machine is doing at a particular instant in time. State machines pass in a discrete manner from one state to another. Just after we enter the input, the machine is in its ''initial state'' or ''start state''. If the machine is deterministic, this means that from this point onwar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Analysis Of Algorithms
In analysis of algorithms, probabilistic analysis of algorithms is an approach to estimate the computational complexity of an algorithm or a computational problem. It starts from an assumption about a probabilistic distribution of the set of all possible inputs. This assumption is then used to design an efficient algorithm or to derive the complexity of a known algorithm. This approach is not the same as that of probabilistic algorithms, but the two may be combined. For non-probabilistic, more specifically deterministic, algorithms, the most common types of complexity estimates are the average-case complexity and the almost-always complexity. To obtain the average-case complexity, given an input distribution, the expected time of an algorithm is evaluated, whereas for the almost-always complexity estimate, it is evaluated that the algorithm admits a given complexity estimate that almost surely holds. In probabilistic analysis of probabilistic (randomized) algorithms, the distributi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Worst Case Analysis
Worst case analysis was, from 1978 until 1986, a doctrine under which mandated that an environmental impact statement include such an analysis: It led to a 1989 SCOTUS decision, written by John Paul Stevens and reported in '' Robertson v. Methow Valley Citizens Council'',Ecology Law Quarterly Volume 18 Issue 1 Article 5 January 1991: Marion D. Miller"The National Environmental Policy Act and Judicial Review after Robertson v. Methow Valley Citizens Council and Marsh v. Oregon Natural Resources Council"/ref> after a decision by GOODWIN and FERGUSON, STEPHENS to reverse the Federal District Court of Oregon ruling that the Regional Forester did not violate any laws when he issued a special use permit for a ski resort development in a roadless area in Okanogan National Forest in Washington state. The Rehnquist Court The Rehnquist Court was the period in the history of the Supreme Court of the United States during which William Rehnquist served as Chief Justice. Rehnquist succee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tacit Assumption
A tacit assumption or implicit assumption is an assumption that underlies a logical argument, course of action, Decision-making, decision, or judgment that is not explicitly voiced nor necessarily understood by the decision maker or judge. These assumptions may be made based on personal life experiences, and are not consciously apparent in the decision making environment. These assumptions can be the source of apparent paradoxes, misunderstandings and resistance to change in human organizational behavior. Tacit assumptions in science often include the Elegance#In_philosophy_of_science, elegance of natural laws, and the Relationship between mathematics and physics, applicability of mathematics. See also * Assumption-based planning * Consensus reality * Hidden curriculum * Implicit attitude * Implicit cognition * Implicit leadership theory * Implicit memory * Implied consent * Leading question * Premise * Presupposition * Shattered assumptions theory * Subreption * Tacit knowledge * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big Theta
Big ''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 German mathematicians 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. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; one well-known example is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a member of a #Related asymptotic notations, family of notations invented by German mathematicians Paul Gustav Heinrich Bachmann, Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '':wikt:Ordnung#German, Ordnung'', meaning the order of approximation. In computer science, big O notation is used to Computational complexity theory, classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetic function, arithmetical function and a better understood approximation; one well-known exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
