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.

Monte Carlo algorithmIn computing, a Monte Carlo algorithm is a randomized algorithm whose output may be incorrect with a certain (typically small) probability. Two examples of such algorithms are the Karger–Stein algorithm and the Monte Carlo algorithm for minimum feedback arc set. The name refers to the Monte Carlo casino in the Principality of Monaco, which is well-known around the world as an icon of gambling. The term "Monte Carlo" was first introduced in 1947 by Nicholas Metropolis.BPP (complexity)In computational complexity theory, a branch of computer science, bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded by 1/3 for all instances. BPP is one of the largest practical classes of problems, meaning most problems of interest in BPP have efficient probabilistic algorithms that can be run quickly on real modern machines. $Las Vegas algorithm$ Las Vegas algorithmIn computing, a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure. However, the runtime of a Las Vegas algorithm differs depending on the input. The usual definition of a Las Vegas algorithm includes the restriction that the expected runtime be finite, where the expectation is carried out over the space of random information, or entropy, used in the algorithm.

AlgorithmIn mathematics and computer science, an algorithm () is any well-defined set of instructions that when followed terminates after a finite number of steps that comprise a solution to a given computational problem. Advanced algorithms may utilize loops and involve many conditionals that decide the next step based on the inputs provided, resulting in long sequences of steps before halting, but all algorithms terminate by definition. $Pseudorandom number generator$ Pseudorandom number generatorA pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generated sequence is not truly random, because it is completely determined by an initial value or state, usually called the PRNG's seed (which may be based on truly random values).Universal hashingIn mathematics and computing, universal hashing (in a randomized algorithm or data structure) refers to selecting a hash function at random from a family of hash functions with a certain mathematical property (see definition below). This guarantees a low number of collisions in expectation, even if the data is chosen by an adversary. Many universal families are known (for hashing integers, vectors, strings), and their evaluation is often very efficient.P (complexity)In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. Cobham's thesis holds that P is the class of computational problems that are "efficiently solvable" or "tractable".

RandomnessIn common usage, randomness is the apparent or actual lack of definite patterns or predictability in information. A random sequence of events, symbols or steps often has no and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if there is a known probability distribution, the frequency of different outcomes over repeated events (or "trials") is predictable.

Paul ErdősPaul Erdős (Hungarian: Erdős Pál [ˈɛrdøːʃ ˈpaːl]; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. Erdős pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory.

QuicksortQuicksort is an efficient, general-purpose sorting algorithm. Quicksort was developed by British computer scientist Tony Hoare in 1959 and published in 1961. It is still a commonly used algorithm for sorting. Overall, it is slightly faster than merge sort and heapsort for randomized data, particularly on larger distributions. Quicksort is a divide-and-conquer algorithm.

Discrete uniform distributionIn probability theory and statistics, the discrete uniform distribution is a symmetricprobability distribution wherein each of some finite whole numbern of outcome values are equally likely to be observed. Thus every one of the n outcome values has equal probability 1/n. Intuitively, a discrete uniform distribution is "a known, finite number of outcomes all equally likely to happen."

Feedback arc setIn graph theory and graph algorithms, a feedback arc set or feedback edge set in a directed graph is a subset of the edges of the graph that contains at least one edge out of every cycle in the graph. Removing these edges from the graph breaks all of the cycles, producing an acyclic subgraph of the given graph, often called a directed acyclic graph.

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