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Collision Problem
The r-to-1 collision problem is an important theoretical problem in complexity theory, quantum computing, and computational mathematics. The collision problem most often refers to the 2-to-1 version: given n even and a function f:\,\\rightarrow\, we are promised that f is either 1-to-1 or 2-to-1. We are only allowed to make queries about the value of f(i) for any i\in\. The problem then asks how many such queries we need to make to determine with certainty whether f is 1-to-1 or 2-to-1. Classical solutions Deterministic Solving the 2-to-1 version deterministically requires \frac+1 queries, and in general distinguishing r-to-1 functions from 1-to-1 functions requires \frac + 1 queries. This is a straightforward application of the pigeonhole principle: if a function is r-to-1, then after \frac + 1 queries we are guaranteed to have found a collision. If a function is 1-to-1, then no collision exists. Thus, \frac + 1 queries suffice. If we are unlucky, then the first n/r quer ... [...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 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. 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 gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though current quantum computers may be too small to outperform usual (classical) computers for practical applications, larger realizations are believed to be capable of solving certain computational problems, such as integer factorization (which underlies RSA encryption), substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science. There are several models of quantum computation with the most widely used being quantum circuits. Other models include the quantum Turing machine, quantum annealing, and adiabatic quantum computation. Most models are based on the quantum bit, or "qubit", which is somewhat analogous to the bit in classical computation. A qubit can be in a 1 or 0 quantum s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Mathematics
Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.National Science Foundation, Division of Mathematical ScienceProgram description PD 06-888 Computational Mathematics 2006. Retrieved April 2007. A large part of computational mathematics consists roughly of using mathematics for allowing and improving computer computation in areas of science and engineering where mathematics are useful. This involves in particular algorithm design, computational complexity, numerical methods and computer algebra. Computational mathematics refers also to the use of computers for mathematics itself. This includes mathematical experimentation for establishing conjectures (particularly in number theory), the use of computers for proving theorems (for example the four color theorem), and the design and use of proof assistants. Areas of computational mathematics Computational mathematics emerged as a distinct part of applied ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-to-one Function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pigeonhole Principle
In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there must be at least two right-handed gloves, or at least two left-handed gloves, because there are three objects, but only two categories of handedness to put them into. This seemingly obvious statement, a type of counting argument, can be used to demonstrate possibly unexpected results. For example, given that the population of London is greater than the maximum number of hairs that can be present on a human's head, then the pigeonhole principle requires that there must be at least two people in London who have the same number of hairs on their heads. Although the pigeonhole principle appears as early as 1624 in a book attributed to Jean Leurechon, it is commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birthday Paradox
In probability theory, the birthday problem asks for the probability that, in a set of randomly chosen people, at least two will share a birthday. The birthday paradox is that, counterintuitively, the probability of a shared birthday exceeds 50% in a group of only 23 people. The birthday paradox is a veridical paradox: it appears wrong, but is in fact true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday, this result is made more intuitive by considering that the comparisons of birthdays will be made between every possible pair of individuals. With 23 individuals, there are (23 × 22) / 2 = 253 pairs to consider, much more than half the number of days in a year. Real-world applications for the birthday problem include a cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of finding a collision for a hash function, as well a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BHT Algorithm
In quantum computing, the Brassard-Høyer-Tapp algorithm or BHT algorithm is a quantum algorithm that solves the collision problem. In this problem, one is given ''n'' and an ''r''-to-1 function f:\,\\rightarrow\ and needs to find two inputs that ''f'' maps to the same output. The BHT algorithm only makes O(n^) queries to ''f'', which matches the lower bound of \Omega(n^) in the black box model. The algorithm was discovered by Gilles Brassard, Peter Høyer, and Alain Tapp in 1997. It uses Grover's algorithm, which was discovered the year before. Algorithm Intuitively, the algorithm combines the square root speedup from the birthday paradox In probability theory, the birthday problem asks for the probability that, in a set of randomly chosen people, at least two will share a birthday. The birthday paradox is that, counterintuitively, the probability of a shared birthday exceeds 5 ... using (classical) randomness with the square root speedup from Grover's (quantum) algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grover's Algorithm
In quantum computing, Grover's algorithm, also known as the quantum search algorithm, refers to a quantum algorithm for unstructured search that finds with high probability the unique input to a black box function that produces a particular output value, using just O(\sqrt) evaluations of the function, where N is the size of the function's domain. It was devised by Lov Grover in 1996. The analogous problem in classical computation cannot be solved in fewer than O(N) evaluations (because, on average, one has to check half of the domain to get a 50% chance of finding the right input). Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani proved that any quantum solution to the problem needs to evaluate the function \Omega(\sqrt) times, so Grover's algorithm is asymptotically optimal. Since classical algorithms for NP-complete problems require exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution for un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithms
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of space and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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