"A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure."
This subfield focuses on the development of algorithms that use randomization to achieve faster or more efficient solutions, such as in graph coloring or optimization problems.
Probability theory: A strong understanding of probability theory is necessary to understand and analyze randomized algorithms, as they rely on random events.
Analysis of algorithms: Randomized algorithm analysis is more complex than deterministic algorithms, requiring techniques such as probabilistic analysis and expected values.
Monte Carlo algorithms: These are randomized algorithms that use random numbers to make approximations or calculations. Examples include Monte Carlo integration and Monte Carlo simulation.
Las Vegas algorithms: These are randomized algorithms that always produce a correct result, but their running time may vary. Examples include randomized quicksort and randomized selection.
Randomized data structures: These are data structures that use randomness to achieve better performance than deterministic data structures. Examples include randomized search trees and skip lists.
Markov chain Monte Carlo (MCMC) methods: These are a class of algorithms that use Markov chains to generate samples from probability distributions. They have applications in statistical physics, Bayesian inference, and machine learning.
Randomized rounding: This is a technique for converting fractional solutions of optimization problems into feasible integer solutions. It has applications in graph algorithms, linear programming, and network design.
Probabilistic graph algorithms: These are algorithms that use randomness to solve graph problems, such as graph coloring, maximum flow, and minimum cut.
Randomized cryptography: This is the study of using randomness to enhance the security of cryptographic protocols. Examples include probabilistic encryption and randomized hash functions.
Randomized network algorithms: These are algorithms that use randomness to solve network optimization problems, such as routing, load balancing, and network coding.
Monte Carlo algorithms: These algorithms use random sampling to solve a problem with a high probability of producing the correct output, but not necessarily always.
Las Vegas algorithms: These algorithms use randomization to guarantee that the solution produced is always correct, but may take more time than deterministic algorithms.
Markov chain Monte Carlo (MCMC) algorithms: These algorithms use a Markov chain to generate a sequence of samples from a probability distribution to solve problems in statistics and physics.
Randomized approximation algorithms: These algorithms use randomization to provide a solution that is close to the optimal solution for a given problem, while also being computationally efficient.
Randomized rounding algorithms: These algorithms use rounding and randomization to obtain a feasible solution to an optimization problem, often used in integer programming and graph theory.
Probabilistic data structures: These are data structures that use randomization to achieve space and time efficiency in storing and querying large amounts of data, such as Bloom filters and Count-min sketches.
Random walks: These are algorithms that use random processes to explore graphs and other data structures, often used in machine learning and computer networks.
Evolutionary algorithms: These are optimization algorithms that use principles of natural selection and genetics to generate and evaluate solutions to a problem.
Randomized quick sort: This is a popular sorting algorithm that uses randomization to improve the worst-case performance of quicksort, particularly when the input is partially sorted or when duplicate elements are present.
"The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior."
"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."
"One has to distinguish 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)."
"In some cases, probabilistic algorithms are the only practical means of solving a problem."
"In common practice, randomized algorithms are approximated using a pseudorandom number generator in place of a true source of random bits."
"such an implementation may deviate from the expected theoretical behavior and mathematical guarantees which may depend on the existence of an ideal true random number generator."
"The expected running time is finite (Las Vegas algorithms, for example Quicksort)."
"One has to distinguish between algorithms that use the random input so that they always terminate with the correct answer."
"The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior."
"algorithms which have a chance of producing an incorrect result (Monte Carlo algorithms)."
"The hope of achieving good performance in the 'average case' over all possible choices of random."
"In some cases, probabilistic algorithms are the only practical means of solving a problem."
"an ideal true random number generator."
"such an implementation may deviate from the expected theoretical behavior and mathematical guarantees."
"In common practice, randomized algorithms are approximated using a pseudorandom number generator."
"where the expected running time is finite (Las Vegas algorithms, for example Quicksort)."
"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."
"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."