algorithms

An algorithm is a finite, well-defined set of instructions that, given input, produces output and terminates after a finite number of steps. Each instruction is unambiguous, and the sequence is designed to be executable by a computer or carried out by a person following the rules. Algorithms are the basis of computer programs and of many systematic procedures used in mathematics, data processing, and everyday tasks.

The study of algorithms combines design methods with analysis of resource use. Their history stretches from ancient procedures for arithmetic to the formal theories developed in the 20th century by researchers such as Turing, Church, and von Neumann, which established formal models of computation and the framework for complexity and optimization.

Algorithms are evaluated by efficiency and correctness. Time complexity measures how the running time grows with input size, often expressed with Big-O notation. Space complexity measures memory use. Analyses consider worst-case, average-case, and sometimes best-case scenarios.

Algorithms are categorized by approach and domain. Common families include brute-force, divide and conquer, dynamic programming, greedy, and graph algorithms. Classic examples include sorting (quicksort, mergesort), searching (binary search), shortest path (Dijkstra, Bellman–Ford), and minimum spanning tree methods (Prim, Kruskal).

Applications span computing, databases, optimization, numerical simulation, cryptography, and artificial intelligence. Limitations arise from fundamental results such as undecidability and from problems believed not to admit efficient general solutions, known as NP-hard problems. The study of algorithms continues to evolve as new models of computation and new data challenges emerge.