Algorithm

In 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.

In contrast to algorithms, heuristics might be applied to problems for which it is not possible for an algorithmic solution to exist, e.g., because the problem is undecidable or there is otherwise no way to define the correct or optimal result. For example, although social media recommender systems are erroneously called "algorithms" in popular media, they are accurately classified as heuristics, because there is no objective criteria by which a recommendation produced by such a system can be judged as "correct".

As an effective method, an algorithm can be expressed within both a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and input, a computation occurs at each step, eventually producing output and terminating. Randomized algorithms incorporate random inputs which can be used to simulate non-deterministic output even though the transitions between states are deterministic.

Etymology

Around 825 AD, Persian scientist and polymath al-Khwarizmi wrote kitāb al-ḥisāb al-hindī ("Book of Indian computation") and kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ("Addition and subtraction in Indian arithmetic"). In the early 12th century, Latin translations of these texts involving the Hindu–Arabic numeral system and arithmetic appeared, for example Liber Alghoarismi de practica arismetrice, attributed to John of Seville, and Liber Algoritmi de numero Indorum, attributed to Adelard of Bath. Here, alghoarismi or algoritmi is the Latinization of Al-Khwarizmi's name; the text starts with the phrase Dixit Algoritmi, or "Thus spoke Al-Khwarizmi".