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{{redirect|Algorythm|the album|Beyond Creation}}
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[[File:GCD through successive subtractions.svg|thumb|Flowchart of using successive subtractions to find the [[greatest common divisor]] of number ''r'' and ''s''|alt=In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again.]]
In [[mathematics]] and [[computer science]], an '''algorithm''' ({{IPAc-en|audio=en-us-algorithm.ogg|ˈ|æ|l|ɡ|ə|r|ɪ|ð|əm}}) is a finite sequence of [[Rigour#Mathematics|mathematically rigorous]] instructions, typically used to solve a class of specific [[Computational problem|problem]]s or to perform a [[computation]].<ref name=":0">{{Cite web|url=https://www.merriam-webster.com/dictionary/algorithm|title=Definition of ALGORITHM|work=Merriam-Webster Online Dictionary |language=en |access-date=2019-11-14 |archive-url=https://web.archive.org/web/20200214074446/https://www.merriam-webster.com/dictionary/algorithm |archive-date=February 14, 2020|url-status=live}}</ref> Algorithms are used as specifications for performing [[calculation]]s and [[data processing]]. More advanced algorithms can use [[Conditional (computer programming)|conditional]]s to divert the code execution through various routes (referred to as [[automated decision-making]]) and deduce valid [[inference]]s (referred to as [[automated reasoning]]).
In contrast, a [[Heuristic (computer science)|heuristic]] is an approach to solving problems without well-defined correct or optimal results.<ref name=":2">David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, {{isbn|1402030045}}</ref> For example, although social media [[recommender system]]s are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation.
As an [[effective method]], an algorithm can be expressed within a finite amount of space and time<ref name=":3">"Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2).</ref> and in a well-defined [[formal language]]<ref name=":4">Well defined concerning the agent that executes the algorithm: "There is a computing agent, usually human, which can react to the instructions and carry out the computations" (Rogers 1987:2).</ref> for calculating a [[Function (mathematics)|function]].<ref>"an algorithm is a procedure for computing a ''function'' (concerning some chosen notation for integers) ... this limitation (to numerical functions) results in no loss of generality", (Rogers 1987:1).</ref> Starting from an initial state and initial input (perhaps [[Empty string|empty]]),<ref>"An algorithm has [[zero]] or more inputs, i.e., [[Quantity|quantities]] which are given to it initially before the algorithm begins" (Knuth 1973:5).</ref> the instructions describe a computation that, when [[Execution (computing)|execute]]d, proceeds through a finite<ref>"A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method{{'"}} (Knuth 1973:5).</ref> number of well-defined successive states, eventually producing "output"<ref>"An algorithm has one or more outputs, i.e., quantities which have a specified relation to the inputs" (Knuth 1973:5).</ref> and terminating at a final ending state. The transition from one state to the next is not necessarily [[deterministic]]; some algorithms, known as [[randomized algorithm]]s, incorporate random input.<ref>Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without the use of continuous methods or analog devices ... carried forward deterministically, without resort to random methods or devices, e.g., dice" (Rogers 1987:2).</ref>
== Etymology ==
Around 825 AD, Persian scientist and polymath [[Al-Khwarizmi|Muḥammad ibn Mūsā al-Khwārizmī]] 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 Algorismi de numero Indorum'', attributed to [[Adelard of Bath]].<ref name=":1">Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247</ref> Here, ''alghoarismi'' or ''algorismi'' is the [[Latinisation of names|Latinization]] of Al-Khwarizmi's name;<ref name=":0"
The word ''[[algorism]]'' in English came to mean the use of place-value notation in calculations; it occurs in the ''[[Ancrene Wisse]]'' from circa 1225.<ref>{{cite web|url=https://www.oed.com/dictionary/algorism_n?tl=true|title=algorism|work=Oxford English Dictionary|access-date=2025-05-18}}</ref> By the time [[Geoffrey Chaucer]] wrote ''[[The Canterbury Tales]]'' in the late 14th century, he used a variant of the same word in describing ''augrym stones'', stones used for place-value calculation.<ref>{{cite web|url=https://chaucer.fas.harvard.edu/pages/millers-prologue-and-tale|title=The Miller's Tale|at=Line 3210|first=Geoffrey|last=Chaucer}}</ref><ref>{{cite book|title=A Glossary of Tudor and Stuart Words: Especially from the Dramatists|editor-first=Anthony Lawson|editor-last=Mayhew|first=Walter William|last=Skeat|publisher=Clarendon Press|year=1914|contribution=agrim, agrum|pages=5–6|contribution-url=https://books.google.com/books?id=z58YAAAAIAAJ&pg=PA5}}</ref> In the 15th century, under the influence of the Greek word ἀριθμός (''arithmos'', "number"; ''cf.'' "arithmetic"), the Latin word was altered to ''algorithmus''.<ref>{{cite book
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=== Ancient algorithms ===
Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book |last1=Chabert |first1=Jean-Luc |title=A History of Algorithms: From the Pebble to the Microchip |date=2012 |publisher=Springer Science & Business Media |isbn=9783642181924 |pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book |last1=Sriram |first1=M. S. |editor1-last=Emch |editor1-first=Gerard G. |editor2-last=Sridharan |editor2-first=R. |editor3-last=Srinivas |editor3-first=M. D. |title=Contributions to the History of Indian Mathematics |date=2005 |publisher=Springer |isbn=978-93-86279-25-5 |page=153 |chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 |language=en |chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal |last=Zaslavsky |first=Claudia |date=1970 |title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria |url=https://www.jstor.org/stable/3027363 |journal=The Two-Year College Mathematics Journal |volume=1 |issue=2 |pages=76–99 |doi=10.2307/3027363 |jstor=3027363 |issn=0049-4925|url-access=subscription }}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book|last=Cooke|first=Roger L.|title=The History of Mathematics: A Brief Course|date=2005|publisher=John Wiley & Sons|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal |date=1999 |editor-last=Chabert |editor-first=Jean-Luc |title=A History of Algorithms |url=https://link.springer.com/book/10.1007/978-3-642-18192-4 |journal=SpringerLink |language=en |doi=10.1007/978-3-642-18192-4|isbn=978-3-540-63369-3 |url-access=subscription }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book |last1=Dooley |first1=John F. |title=A Brief History of Cryptology and Cryptographic Algorithms |date=2013 |publisher=Springer Science & Business Media |isbn=9783319016283 |pages=12–3}}</ref>
The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty|Hammurabi dynasty]] {{Circa|1800|1600 BC|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal |last1=Knuth |first1=Donald E. |date=1972 |title=Ancient Babylonian Algorithms |url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf |url-status=dead |journal=Commun. ACM |volume=15 |issue=7 |pages=671–677 |doi=10.1145/361454.361514 |issn=0001-0782 |s2cid=7829945 |archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf |archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]]. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book |last=Aaboe |first=Asger |author-link=Asger Aaboe |title=Episodes from the Early History of Astronomy |date=2001 |publisher=Springer |isbn=978-0-387-95136-2 |place=New York |pages=40–62}}</ref>
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To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to [[Fast Fourier transform|FFT]] algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging.<ref>{{cite web| title=Better Math Makes Faster Data Networks| author=Gillian Conahan| date=January 2013| url=http://discovermagazine.com/2013/jan-feb/34-better-math-makes-faster-data-networks| publisher=discovermagazine.com| access-date=May 13, 2014| archive-url=https://web.archive.org/web/20140513212427/http://discovermagazine.com/2013/jan-feb/34-better-math-makes-faster-data-networks| archive-date=May 13, 2014| url-status=live}}</ref> In general, speed improvements depend on special properties of the problem, which are very common in practical applications.<ref name="Hassanieh12">Haitham Hassanieh, [[Piotr Indyk]], Dina Katabi, and Eric Price, "[http://siam.omnibooksonline.com/2012SODA/data/papers/500.pdf ACM-SIAM Symposium On Discrete Algorithms (SODA)] {{webarchive|url=https://web.archive.org/web/20130704180806/http://siam.omnibooksonline.com/2012SODA/data/papers/500.pdf |date=July 4, 2013 }}, Kyoto, January 2012. See also the [http://groups.csail.mit.edu/netmit/sFFT/ sFFT Web Page] {{Webarchive|url=https://web.archive.org/web/20120221145740/http://groups.csail.mit.edu/netmit/sFFT/ |date=February 21, 2012 }}.</ref> Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.
{{Main|Best, worst and average case}}
The best case of an algorithm refers to the scenario or input for which the algorithm or data structure takes the least time and resources to complete its tasks.<ref>{{Cite web |title=Best Case |url=https://xlinux.nist.gov/dads/HTML/bestcase.html |access-date=29 May 2025 |website=Dictionary of Algorithms and Data Structures |publisher=National Institute of Standards and Technology (NIST) |agency=National Institute of Standards and Technology}}</ref> The worst case of an algorithm is the case that causes the algorithm or data structure to consume the maximum period of time and computational resources.<ref>{{Cite web |title=worst case |url=https://xlinux.nist.gov/dads/HTML/worstcase.html |access-date=29 May 2025 |website=Dictionary of Algorithms and Data Structures |publisher=National Institute of Standards and Technology (NIST) |agency=National Institute of Standards and Technology (NIST)}}</ref>
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{{wikibooks|Algorithms}}
{{Wikiversity department}}
{{Commons category
* {{springer|title=Algorithm|id=p/a011780|mode=cs1}}
* {{MathWorld | urlname=Algorithm | title=Algorithm}}
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