Randomized methods are a set of algorithms and techniques that involve the use of random numbers or randomness to solve computational problems. These methods are widely used in various areas of computer science and mathematics, including optimization, machine learning, and cryptography. The main idea behind randomized methods is to introduce randomness into the problem-solving process in order to achieve faster or more efficient solutions. This randomness can be used to explore a larger search space, quickly converge to a solution, or break symmetries that would otherwise impede progress. Randomized methods are often used when deterministic algorithms are either too slow or too complex to implement. They are particularly useful for problems with large data sets, complex constraints, or combinatorial structures that make traditional approaches impractical. Some common examples of randomized methods include Monte Carlo methods, randomized algorithms for graph problems, and randomized optimization techniques like genetic algorithms and simulated annealing. Overall, randomized methods provide a powerful and versatile toolset for solving a wide range of computational problems, offering efficient and robust solutions in many practical scenarios.