Algorithm

An algorithm is a finite set of well-defined instructions that takes input and produces output, intended to solve a particular problem or perform a computation. It can be described in natural language, pseudocode, or a programming language, and is independent of any specific implementation.

Core characteristics include finiteness, definiteness, and effectiveness. Each step must be clear and executable, the process must terminate after a finite number of steps, and the operations must be feasible to perform. Algorithms are typically defined by their inputs, outputs, and the rules that transform inputs into outputs. They can be deterministic, producing the same result from a given input every time, or non-deterministic in theory, allowing for multiple possible paths to a result.

Algorithms are commonly described and analyzed in several representations, such as natural language, pseudocode, or flowcharts. Familiar examples include the Euclidean algorithm for computing the greatest common divisor, binary search for quickly locating an item in a sorted list, and sorting algorithms like quicksort or mergesort. Graph-based algorithms, such as Dijkstra’s or Bellman–Ford, solve shortest-path problems, while dynamic programming techniques address optimization tasks.

Complexity analysis measures how resource usage grows with input size, typically time and space. Big-O notation expresses upper bounds on this growth and guides algorithm selection and optimization. The field balances correctness, efficiency, and scalability, often involving trade-offs between time and memory.

Historically, the concept traces to ancient mathematics, and the term derives from the name of the Persian mathematician al-Khwarizmi. Today, algorithms underpin modern computing, data processing, and many areas of science and engineering, while researchers continually refine their design and analysis.