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Gap-Hamming Problem
In communication complexity, the gap-Hamming problem asks, if Alice and Bob are each given a (potentially different) string, what is the minimal number of bits that they need to exchange in order for Alice to approximately compute the Hamming distance between their strings. The solution to the problem roughly states that, if Alice and Bob are each given a string, then any communication protocol used to compute the Hamming distance between their strings does (asymptotically) no better than Bob sending his whole string to Alice. More specifically, if Alice and Bob are each given n-bit strings, there exists no communication protocol that lets Alice compute the hamming distance between their strings to within \pm\sqrt using less than \Omega(n) bits. The gap-Hamming problem has applications to proving lower bounds for many streaming algorithms, including moment frequency estimation and entropy estimation. Formal statement In this problem, Alice and Bob each receive a string, x \in \^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communication Complexity
In theoretical computer science, communication complexity studies the amount of communication required to solve a problem when the input to the problem is distributed among two or more parties. The study of communication complexity was first introduced by Andrew Yao in 1979, while studying the problem of computation distributed among several machines. The problem is usually stated as follows: two parties (traditionally called Alice and Bob) each receive a (potentially different) n- bit string x and y. The goal is for Alice to compute the value of a certain function, f(x, y), that depends on both x and y, with the least amount of communication between them. While Alice and Bob can always succeed by having Bob send his whole n-bit string to Alice (who then computes the function f), the idea here is to find clever ways of calculating ''f'' with fewer than n bits of communication. Note that, unlike in computational complexity theory, communication complexity is not concerned with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alice And Bob
Alice and Bob are fictional characters commonly used as placeholders in discussions about cryptography, cryptographic systems and Cryptographic protocol, protocols, and in other science and engineering literature where there are several participants in a thought experiment. The Alice and Bob characters were created by Ron Rivest, Adi Shamir, and Leonard Adleman in their 1978 paper "A Method for Obtaining Digital Signatures and Public-key Cryptosystems". Subsequently, they have become common archetypes in many scientific and engineering fields, such as quantum cryptography, game theory and physics. As the use of Alice and Bob became more widespread, additional characters were added, sometimes each with a particular meaning. These characters do not have to refer to people; they refer to generic agents which might be different computers or even different programs running on a single computer. Overview Alice and Bob are the names of fictional characters used for convenience and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Distance
In information theory, the Hamming distance between two String (computer science), strings or vectors of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to change one string into the other, or equivalently, the minimum number of ''errors'' that could have transformed one string into the other. In a more general context, the Hamming distance is one of several string metrics for measuring the edit distance between two sequences. It is named after the American mathematician Richard Hamming. A major application is in coding theory, more specifically to block codes, in which the equal-length strings are Vector space, vectors over a finite field. Definition The Hamming distance between two equal-length strings of symbols is the number of positions at which the corresponding symbols are different. Examples The symbols may be letters, bits, or decimal digits, am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communication Protocol
A communication protocol is a system of rules that allows two or more entities of a communications system to transmit information via any variation of a physical quantity. The protocol defines the rules, syntax, semantics (computer science), semantics, and synchronization of communication and possible Error detection and correction, error recovery methods. Protocols may be implemented by Computer hardware, hardware, software, or a combination of both. Communicating systems use well-defined formats for exchanging various messages. Each message has an exact meaning intended to elicit a response from a range of possible responses predetermined for that particular situation. The specified behavior is typically independent of how it is to be Implementation, implemented. Communication protocols have to be agreed upon by the parties involved. To reach an agreement, a protocol may be developed into a technical standard. A programming language describes the same for computations, so there ... [...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] |
Concentration Inequality
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', ''number concentration'', and ''volume concentration''. The concentration can refer to any kind of chemical mixture, but most frequently refers to solutes and solvents in solutions. The molar (amount) concentration has variants, such as normal concentration and osmotic concentration. Dilution is reduction of concentration, e.g. by adding solvent to a solution. The verb to concentrate means to increase concentration, the opposite of dilute. Etymology ''Concentration-'', ''concentratio'', action or an act of coming together at a single place, bringing to a common center, was used in post-classical Latin in 1550 or earlier, similar terms attested in Italian (1589), Spanish (1589), English (1606), French (1632). Qualitative description Often in informal, non-techn ... [...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] |
Quantum Complexity Theory
Quantum 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. Background A complexity class is a collection of computational problems that can be solved by a computational model under certain resource constraints. For instance, the complexity class P is defined as the set of problems solvable by a Turing machine in polynomial time. Similarly, quantum complexity classes may be defined using quantum models of computation, such as the quantum circuit model or the equivalent quantum Turing machine. One of the main aims of quantum complexity theory is to find out how these classes relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
