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Algorithm

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</style><div role="note" class="hatnote navigation-not-searchable">"Algorithms" redirects here. For the subfield of computer science, see <a href="Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a>. For other uses, see <a href="Algorithm_(disambiguation)" class="mw-disambig" title="Algorithm (disambiguation)">Algorithm (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Algorythm" redirects here. For the album, see <a href="Beyond_Creation" title="Beyond Creation">Beyond Creation</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Computer_science" title="Computer science">computer science</a>, an algorithm (/ˈælɡərɪðəm/ <span data-nosnippet="" id="ooui-php-1" class="noexcerpt ext-phonos-PhonosButton ext-phonos-PhonosButton-emptylabel oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/7\/7f\/En-us-algorithm.ogg\/En-us-algorithm.ogg.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"En-us-algorithm.ogg"},"classes":["noexcerpt","ext-phonos-PhonosButton","ext-phonos-PhonosButton-emptylabel"]}"><a role="button" tabindex="0" href="https://upload.wikimedia.org/wikipedia/commons/transcoded/7/7f/En-us-algorithm.ogg/En-us-algorithm.ogg.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"></a>ⓘ) is a finite sequence of <a href="Rigour#Mathematics" title="Rigour">mathematically rigorous</a> instructions, typically used to solve a class of specific <a href="Computational_problem" title="Computational problem">problems</a> or to perform a <a href="Computation" title="Computation">computation</a>.<a href="#cite_note-:0-1">[1]</a> Algorithms are used as specifications for performing <a href="Calculation" title="Calculation">calculations</a> and <a href="Data_processing" title="Data processing">data processing</a>. More advanced algorithms can use <a href="Conditional_(computer_programming)" title="Conditional (computer programming)">conditionals</a> to divert the code execution through various routes (referred to as <a href="Automated_decision-making" title="Automated decision-making">automated decision-making</a>) and deduce valid <a href="Inference" title="Inference">inferences</a> (referred to as <a href="Automated_reasoning" title="Automated reasoning">automated reasoning</a>).

In contrast, a <a href="Heuristic_(computer_science)" title="Heuristic (computer science)">heuristic</a> is an approach to solving problems without well-defined correct or optimal results.<a href="#cite_note-:2-2">[2]</a> For example, although social media <a href="Recommender_system" title="Recommender system">recommender systems</a> are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation.

As an <a href="Effective_method" title="Effective method">effective method</a>, an algorithm can be expressed within a finite amount of space and time<a href="#cite_note-:3-3">[3]</a> and in a well-defined <a href="Formal_language" title="Formal language">formal language</a><a href="#cite_note-:4-4">[4]</a> for calculating a <a href="Function_(mathematics)" title="Function (mathematics)">function</a>.<a href="#cite_note-5">[5]</a> Starting from an initial state and initial input (perhaps <a href="Empty_string" title="Empty string">empty</a>),<a href="#cite_note-6">[6]</a> the instructions describe a computation that, when <a href="Execution_(computing)" title="Execution (computing)">executed</a>, proceeds through a finite<a href="#cite_note-7">[7]</a> number of well-defined successive states, eventually producing "output"<a href="#cite_note-8">[8]</a> and terminating at a final ending state. The transition from one state to the next is not necessarily <a href="Deterministic" class="mw-redirect" title="Deterministic">deterministic</a>; some algorithms, known as <a href="Randomized_algorithm" title="Randomized algorithm">randomized algorithms</a>, incorporate random input.<a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2></div>

Around 825 AD, Persian scientist and polymath <a href="Al-Khwarizmi" title="Al-Khwarizmi">Muḥammad ibn Mūsā al-Khwārizmī</a> 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 <a href="Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> and <a href="Arithmetic" title="Arithmetic">arithmetic</a> appeared, for example Liber Alghoarismi de practica arismetrice, attributed to <a href="John_of_Seville" title="John of Seville">John of Seville</a>, and Liber Algorismi de numero Indorum, attributed to <a href="Adelard_of_Bath" title="Adelard of Bath">Adelard of Bath</a>.<a href="#cite_note-:1-10">[10]</a> Here, alghoarismi or algorismi is the <a href="Latinisation_of_names" title="Latinisation of names">Latinization</a> of Al-Khwarizmi's name;<a href="#cite_note-:0-1">[1]</a> the text starts with the phrase Dixit Algorismi, or "Thus spoke Al-Khwarizmi".<a href="#cite_note-:2-2">[2]</a>

The word <a href="Algorism" title="Algorism">algorism</a> in English came to mean the use of place-value notation in calculations; it occurs in the <a href="Ancrene_Wisse" title="Ancrene Wisse">Ancrene Wisse</a> from circa 1225.<a href="#cite_note-11">[11]</a> By the time <a href="Geoffrey_Chaucer" title="Geoffrey Chaucer">Geoffrey Chaucer</a> wrote <a href="The_Canterbury_Tales" title="The Canterbury Tales">The Canterbury Tales</a> in the late 14th century, he used a variant of the same word in describing augrym stones, stones used for place-value calculation.<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a> In the 15th century, under the influence of the Greek word ἀριθμός (arithmos, "number"; cf. "arithmetic"), the Latin word was altered to algorithmus.<a href="#cite_note-14">[14]</a> By 1596, this form of the word was used in English, as algorithm, by <a href="Thomas_Hood_(mathematician)" title="Thomas Hood (mathematician)">Thomas Hood</a>.<a href="#cite_note-15">[15]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<div role="note" class="hatnote navigation-not-searchable">For a detailed presentation of the various points of view on the definition of "algorithm", see <a href="Algorithm_characterizations" title="Algorithm characterizations">Algorithm characterizations</a>.</div>

One informal definition is "a set of rules that precisely defines a sequence of operations",<a href="#cite_note-FOOTNOTEStone19718-16">[16]</a> which would include all <a href="Computer_program" title="Computer program">computer programs</a> (including programs that do not perform numeric calculations), and any prescribed <a href="Bureaucratic" class="mw-redirect" title="Bureaucratic">bureaucratic</a> procedure<a href="#cite_note-17">[17]</a>

or <a href="Cookbook" title="Cookbook">cook-book</a> <a href="Recipe" title="Recipe">recipe</a>.<a href="#cite_note-18">[18]</a> In general, a program is an algorithm only if it stops eventually<a href="#cite_note-19">[19]</a>—even though <a href="Infinite_loop#Intentional_looping" title="Infinite loop">infinite loops</a> may sometimes prove desirable. <a href="#CITEREFBoolosJeffrey1999">Boolos, Jeffrey & 1974, 1999</a> define an algorithm to be an explicit set of instructions for determining an output, that can be followed by a computing machine or a human who could only carry out specific elementary operations on symbols.<a href="#cite_note-20">[20]</a>

Most algorithms are intended to be <a href="Implementation" title="Implementation">implemented</a> as <a href="Computer_program" title="Computer program">computer programs</a>. However, algorithms are also implemented by other means, such as in a <a href="Biological_neural_network" class="mw-redirect" title="Biological neural network">biological neural network</a> (for example, the <a href="Human_brain" title="Human brain">human brain</a> performing <a href="Arithmetic" title="Arithmetic">arithmetic</a> or an insect looking for food), in an <a href="Electrical_circuit" class="mw-redirect" title="Electrical circuit">electrical circuit</a>, or a mechanical device.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

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<div class="mw-heading mw-heading3"><h3 id="Ancient_algorithms">Ancient algorithms</h3></div>

Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in <a href="Babylonian_mathematics" title="Babylonian mathematics">Babylonian mathematics</a> (around 2500 BC),<a href="#cite_note-Springer_Science_&_Business_Media-21">[21]</a> <a href="Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian mathematics</a> (around 1550 BC),<a href="#cite_note-Springer_Science_&_Business_Media-21">[21]</a> <a href="Indian_mathematics" title="Indian mathematics">Indian mathematics</a> (around 800 BC and later),<a href="#cite_note-:6-22">[22]</a><a href="#cite_note-23">[23]</a> the Ifa Oracle (around 500 BC),<a href="#cite_note-24">[24]</a> <a href="Greek_mathematics" class="mw-redirect" title="Greek mathematics">Greek mathematics</a> (around 240 BC),<a href="#cite_note-Cooke2005-25">[25]</a> <a href="Chinese_mathematics" title="Chinese mathematics">Chinese mathematics (around 200 BC and later)</a>,<a href="#cite_note-26">[26]</a> and <a href="Arabic_mathematics" class="mw-redirect" title="Arabic mathematics">Arabic mathematics</a> (around 800 AD).<a href="#cite_note-Dooley-27">[27]</a>

The earliest evidence of algorithms is found in ancient <a href="Mesopotamia" title="Mesopotamia">Mesopotamian</a> mathematics. A <a href="Sumer" title="Sumer">Sumerian</a> clay tablet found in <a href="Shuruppak" title="Shuruppak">Shuruppak</a> near <a href="Baghdad" title="Baghdad">Baghdad</a> and dated to <abbr title="circa">c.</abbr> 2500 BC describes the earliest <a href="Division_algorithm" title="Division algorithm">division algorithm</a>.<a href="#cite_note-Springer_Science_&_Business_Media-21">[21]</a> During the <a href="First_Babylonian_dynasty" class="mw-redirect" title="First Babylonian dynasty">Hammurabi dynasty</a> c. 1800 – c. 1600 BC, <a href="Babylonia" title="Babylonia">Babylonian</a> clay tablets described algorithms for computing formulas.<a href="#cite_note-28">[28]</a> Algorithms were also used in <a href="Babylonian_astronomy" title="Babylonian astronomy">Babylonian astronomy</a>. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<a href="#cite_note-29">[29]</a>

Algorithms for arithmetic are also found in ancient <a href="Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">Egyptian mathematics</a>, dating back to the <a href="Rhind_Mathematical_Papyrus" title="Rhind Mathematical Papyrus">Rhind Mathematical Papyrus</a> c. 1550 BC.<a href="#cite_note-Springer_Science_&_Business_Media-21">[21]</a> Algorithms were later used in ancient <a href="Hellenistic_mathematics" class="mw-redirect" title="Hellenistic mathematics">Hellenistic mathematics</a>. Two examples are the <a href="Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">Sieve of Eratosthenes</a>, which was described in the <a href="Introduction_to_Arithmetic" class="mw-redirect" title="Introduction to Arithmetic">Introduction to Arithmetic</a> by <a href="Nicomachus" title="Nicomachus">Nicomachus</a>,<a href="#cite_note-30">[30]</a><a href="#cite_note-Cooke2005-25">[25]</a>: Ch 9.2 and the <a href="Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a>, which was first described in <a href="Euclid's_Elements" title="Euclid's Elements">Euclid's Elements</a> (c. 300 BC).<a href="#cite_note-Cooke2005-25">[25]</a>: Ch 9.1 Examples of ancient Indian mathematics included the <a href="Shulba_Sutras" title="Shulba Sutras">Shulba Sutras</a>, the <a href="Kerala_school_of_astronomy_and_mathematics" title="Kerala school of astronomy and mathematics">Kerala School</a>, and the <a href="Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta" title="Brāhmasphuṭasiddhānta">Brāhmasphuṭasiddhānta</a>.<a href="#cite_note-:6-22">[22]</a>

The first cryptographic algorithm for deciphering encrypted code was developed by <a href="Al-Kindi" title="Al-Kindi">Al-Kindi</a>, a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of <a href="Cryptanalysis" title="Cryptanalysis">cryptanalysis</a> by <a href="Frequency_analysis" title="Frequency analysis">frequency analysis</a>, the earliest codebreaking algorithm.<a href="#cite_note-Dooley-27">[27]</a>

<div class="mw-heading mw-heading3"><h3 id="Computers">Computers</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Weight-driven_clocks">Weight-driven clocks</h4></div>

Bolter credits the invention of the weight-driven clock as "the key invention [of <a href="Europe_in_the_middle_ages" class="mw-redirect" title="Europe in the middle ages">Europe in the Middle Ages</a>]," specifically the <a href="Verge_escapement" title="Verge escapement">verge escapement</a> mechanism<a href="#cite_note-31">[31]</a> producing the tick and tock of a mechanical clock. "The accurate automatic machine"<a href="#cite_note-32">[32]</a> led immediately to "mechanical <a href="Automata_theory" title="Automata theory">automata</a>" in the 13th century and "computational machines"—the <a href="Difference_engine" title="Difference engine">difference</a> and <a href="Analytical_engine" title="Analytical engine">analytical engines</a> of <a href="Charles_Babbage" title="Charles Babbage">Charles Babbage</a> and <a href="Ada_Lovelace" title="Ada Lovelace">Ada Lovelace</a> in the mid-19th century.<a href="#cite_note-33">[33]</a> Lovelace designed the first algorithm intended for processing on a computer, Babbage's analytical engine, which is the first device considered a real <a href="Turing-complete" class="mw-redirect" title="Turing-complete">Turing-complete</a> computer instead of just a <a href="Calculator" title="Calculator">calculator</a>. Although the full implementation of Babbage's second device was not realized for decades after her lifetime, Lovelace has been called "history's first programmer".

<div class="mw-heading mw-heading4"><h4 id="Electromechanical_relay">Electromechanical relay</h4></div>

Bell and Newell (1971) write that the <a href="Jacquard_loom" class="mw-redirect" title="Jacquard loom">Jacquard loom</a>, a precursor to <a href="Hollerith_card" class="mw-redirect" title="Hollerith card">Hollerith cards</a> (punch cards), and "telephone switching technologies" led to the development of the first computers.<a href="#cite_note-34">[34]</a> By the mid-19th century, the <a href="Telegraph" class="mw-redirect" title="Telegraph">telegraph</a>, the precursor of the telephone, was in use throughout the world. By the late 19th century, the <a href="Ticker_tape" title="Ticker tape">ticker tape</a> (<abbr title="circa">c.</abbr> 1870s) was in use, as were Hollerith cards (c. 1890). Then came the <a href="Teleprinter" title="Teleprinter">teleprinter</a> (c. 1910) with its punched-paper use of <a href="Baudot_code" title="Baudot code">Baudot code</a> on tape.

Telephone-switching networks of <a href="Relays" class="mw-redirect" title="Relays">electromechanical relays</a> were invented in 1835. These led to the invention of the digital adding device by <a href="George_Stibitz" title="George Stibitz">George Stibitz</a> in 1937. While working in Bell Laboratories, he observed the "burdensome" use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea... When the tinkering was over, Stibitz had constructed a binary adding device".<a href="#cite_note-35">[35]</a><a href="#cite_note-36">[36]</a>

<div class="mw-heading mw-heading3"><h3 id="Formalization">Formalization</h3></div>

In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the <a href="Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a> (decision problem) posed by <a href="David_Hilbert" title="David Hilbert">David Hilbert</a>. Later formalizations were framed as attempts to define "<a href="Effective_calculability" class="mw-redirect" title="Effective calculability">effective calculability</a>"<a href="#cite_note-37">[37]</a> or "effective method".<a href="#cite_note-38">[38]</a> Those formalizations included the <a href="Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a>–<a href="Jacques_Herbrand" title="Jacques Herbrand">Herbrand</a>–<a href="Stephen_Cole_Kleene" title="Stephen Cole Kleene">Kleene</a> recursive functions of 1930, 1934 and 1935, <a href="Alonzo_Church" title="Alonzo Church">Alonzo Church</a>'s <a href="Lambda_calculus" title="Lambda calculus">lambda calculus</a> of 1936, <a href="Emil_Post" class="mw-redirect" title="Emil Post">Emil Post</a>'s <a href="Formulation_1" class="mw-redirect" title="Formulation 1">Formulation 1</a> of 1936, and <a href="Alan_Turing" title="Alan Turing">Alan Turing</a>'s <a href="Turing_machines" class="mw-redirect" title="Turing machines">Turing machines</a> of 1936–37 and 1939.

<div class="mw-heading mw-heading2"><h2 id="Representations">Representations</h2></div>

Algorithms can be expressed in many kinds of notation, including <a href="Natural_languages" class="mw-redirect" title="Natural languages">natural languages</a>, <a href="Pseudocode" title="Pseudocode">pseudocode</a>, <a href="Flowchart" title="Flowchart">flowcharts</a>, <a href="DRAKON" title="DRAKON">drakon-charts</a>, <a href="Programming_languages" class="mw-redirect" title="Programming languages">programming languages</a> or <a href="Control_table" title="Control table">control tables</a> (processed by <a href="Interpreter_(computing)" title="Interpreter (computing)">interpreters</a>). Natural language expressions of algorithms tend to be verbose and ambiguous and are rarely used for complex or technical algorithms. Pseudocode, flowcharts, drakon-charts, and control tables are structured expressions of algorithms that avoid common ambiguities of natural language. Programming languages are primarily for expressing algorithms in a computer-executable form but are also used to define or document algorithms.

<div class="mw-heading mw-heading3"><h3 id="Turing_machines">Turing machines</h3></div>

There are many possible representations and <a href="Turing_machine" title="Turing machine">Turing machine</a> programs can be expressed as a sequence of machine tables (see <a href="Finite-state_machine" title="Finite-state machine">finite-state machine</a>, <a href="State-transition_table" title="State-transition table">state-transition table</a>, and <a href="Control_table" title="Control table">control table</a> for more), as flowcharts and drakon-charts (see <a href="State_diagram" title="State diagram">state diagram</a> for more), as a form of rudimentary <a href="Machine_code" title="Machine code">machine code</a> or <a href="Assembly_code" class="mw-redirect" title="Assembly code">assembly code</a> called "sets of quadruples", and more. Algorithm representations can also be classified into three accepted levels of Turing machine description: high-level description, implementation description, and formal description.<a href="#cite_note-:5-39">[39]</a> A high-level description describes the qualities of the algorithm itself, ignoring how it is implemented on the Turing machine.<a href="#cite_note-:5-39">[39]</a> An implementation description describes the general manner in which the machine moves its head and stores data to carry out the algorithm, but does not give exact states.<a href="#cite_note-:5-39">[39]</a> In the most detail, a formal description gives the exact state table and list of transitions of the Turing machine.<a href="#cite_note-:5-39">[39]</a>

<div class="mw-heading mw-heading3"><h3 id="Flowchart_representation">Flowchart representation</h3></div>

The graphical aid called a <a href="Flowchart" title="Flowchart">flowchart</a> offers a way to describe and document an algorithm (and a computer program corresponding to it). It has four primary symbols: arrows showing program flow, rectangles (SEQUENCE, GOTO), diamonds (IF-THEN-ELSE), and dots (OR-tie). Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure.

<div class="mw-heading mw-heading2"><h2 id="Algorithmic_analysis">Algorithmic analysis</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a></div>

It is often important to know how much time, storage, or other cost an algorithm may require. Methods have been developed for the analysis of algorithms to obtain such quantitative answers (estimates); for example, an algorithm that adds up the elements of a list of n numbers would have a time requirement of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation>

</semantics>

</math><img src="./34109fe397fdcff370079185bfdb65826cb5565a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}" loading="lazy">⁠, using <a href="Big_O_notation" title="Big O notation">big O notation</a>. The algorithm only needs to remember two values: the sum of all the elements so far, and its current position in the input list. If the space required to store the input numbers is not counted, it has a space requirement of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)}">

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<annotation encoding="application/x-tex">{\displaystyle O(1)}</annotation>

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</math><img src="./e66384bc40452c5452f33563fe0e27e803b0cc21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle O(1)}" loading="lazy">⁠, otherwise ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">

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<annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation>

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Different algorithms may complete the same task with a different set of instructions in less or more time, space, or '<a href="Algorithmic_efficiency" title="Algorithmic efficiency">effort</a>' than others. For example, a <a href="Binary_search" title="Binary search">binary search</a> algorithm (with cost ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\log n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(\log n)}</annotation>

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</math><img src="./aae0f22048ba6b7c05dbae17b056bfa16e21807d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle O(\log n)}" loading="lazy">⁠) outperforms a sequential search (cost ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">

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<annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Formal_versus_empirical">Formal versus empirical</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Empirical_algorithmics" title="Empirical algorithmics">Empirical algorithmics</a>, <a href="Profiling_(computer_programming)" title="Profiling (computer programming)">Profiling (computer programming)</a>, and <a href="Program_optimization" title="Program optimization">Program optimization</a></div>

The <a href="Analysis_of_algorithms" title="Analysis of algorithms">analysis, and study of algorithms</a> is a discipline of <a href="Computer_science" title="Computer science">computer science</a>. Algorithms are often studied abstractly, without referencing any specific <a href="Programming_language" title="Programming language">programming language</a> or implementation. Algorithm analysis resembles other mathematical disciplines as it focuses on the algorithm's properties, not implementation. <a href="Pseudocode" title="Pseudocode">Pseudocode</a> is typical for analysis as it is a simple and general representation. Most algorithms are implemented on particular hardware/software platforms and their <a href="Algorithmic_efficiency" title="Algorithmic efficiency">algorithmic efficiency</a> is tested using real code. The efficiency of a particular algorithm may be insignificant for many "one-off" problems but it may be critical for algorithms designed for fast interactive, commercial, or long-life scientific usage. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign.

Empirical testing is useful for uncovering unexpected interactions that affect performance. <a href="Benchmark_(computing)" title="Benchmark (computing)">Benchmarks</a> may be used to compare before/after potential improvements to an algorithm after program optimization.

Empirical tests cannot replace formal analysis, though, and are non-trivial to perform fairly.<a href="#cite_note-KriegelSchubert2016-40">[40]</a>

<div class="mw-heading mw-heading3"><h3 id="Execution_efficiency">Execution efficiency</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Algorithmic_efficiency" title="Algorithmic efficiency">Algorithmic efficiency</a></div>

To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to <a href="Fast_Fourier_transform" title="Fast Fourier transform">FFT</a> algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging.<a href="#cite_note-41">[41]</a> In general, speed improvements depend on special properties of the problem, which are very common in practical applications.<a href="#cite_note-Hassanieh12-42">[42]</a> Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.

<div class="mw-heading mw-heading3"><h3 id="Best_Case_and_Worst_Case">Best Case and Worst Case</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Best%2C_worst_and_average_case" title="Best, worst and average case">Best, worst and average case</a></div>

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.<a href="#cite_note-43">[43]</a> 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.<a href="#cite_note-44">[44]</a>

<div class="mw-heading mw-heading2"><h2 id="Design">Design</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a class="mw-selflink-fragment" href="#By_design_paradigm">Algorithm § By design paradigm</a></div>

Algorithm design is a method or mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories, such as <a href="Divide-and-conquer_algorithm" title="Divide-and-conquer algorithm">divide-and-conquer</a> or <a href="Dynamic_programming" title="Dynamic programming">dynamic programming</a> within <a href="Operation_research" class="mw-redirect" title="Operation research">operation research</a>. Techniques for designing and implementing algorithm designs are also called algorithm design patterns,<a href="#cite_note-45">[45]</a> with examples including the template method pattern and the decorator pattern. One of the most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the <a href="Big_O_notation" title="Big O notation">big O notation</a> is used to describe e.g., an algorithm's run-time growth as the size of its input increases.<a href="#cite_note-46">[46]</a>

<div class="mw-heading mw-heading3"><h3 id="Structured_programming">Structured programming</h3></div>

Per the <a href="Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a>, any algorithm can be computed by any <a href="Turing_complete" class="mw-redirect" title="Turing complete">Turing complete</a> model. Turing completeness only requires four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. However, Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "<a href="Spaghetti_code" title="Spaghetti code">spaghetti code</a>", a programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language".<a href="#cite_note-47">[47]</a> Tausworthe augments the three <a href="Structured_program_theorem" title="Structured program theorem">Böhm-Jacopini canonical structures</a>:<a href="#cite_note-48">[48]</a> SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.<a href="#cite_note-49">[49]</a> An additional benefit of a structured program is that it lends itself to <a href="Proof_of_correctness" class="mw-redirect" title="Proof of correctness">proofs of correctness</a> using <a href="Mathematical_induction" title="Mathematical induction">mathematical induction</a>.<a href="#cite_note-50">[50]</a>

<div class="mw-heading mw-heading2"><h2 id="Legal_status">Legal status</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Software_patent" title="Software patent">Software patent</a></div>

By themselves, algorithms are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), so algorithms are not patentable (as in <a href="Gottschalk_v._Benson" title="Gottschalk v. Benson">Gottschalk v. Benson</a>). However practical applications of algorithms are sometimes patentable. For example, in <a href="Diamond_v._Diehr" title="Diamond v. Diehr">Diamond v. Diehr</a>, the application of a simple <a href="Feedback" title="Feedback">feedback</a> algorithm to aid in the curing of <a href="Synthetic_rubber" title="Synthetic rubber">synthetic rubber</a> was deemed patentable. The <a href="Software_patent_debate" title="Software patent debate">patenting of software</a> is controversial,<a href="#cite_note-51">[51]</a> and there are criticized patents involving algorithms, especially <a href="Data_compression" title="Data compression">data compression</a> algorithms, such as <a href="Unisys" title="Unisys">Unisys</a>'s <a href="Graphics_Interchange_Format" class="mw-redirect" title="Graphics Interchange Format">LZW patent</a>. Additionally, some cryptographic algorithms have export restrictions (see <a href="Export_of_cryptography" title="Export of cryptography">export of cryptography</a>).

<div class="mw-heading mw-heading2"><h2 id="Classification">Classification</h2></div>

<div class="mw-heading mw-heading3"><h3 id="By_implementation">By implementation</h3></div>

<dl><dt>Recursion</dt>

<dd>A <a href="Recursive_algorithm" class="mw-redirect" title="Recursive algorithm">recursive algorithm</a> invokes itself repeatedly until meeting a termination condition and is a common <a href="Functional_programming" title="Functional programming">functional programming</a> method. <a href="Iteration" title="Iteration">Iterative</a> algorithms use repetitions such as <a href="Program_loops" class="mw-redirect" title="Program loops">loops</a> or data structures like <a href="Stack_(data_structure)" class="mw-redirect" title="Stack (data structure)">stacks</a> to solve problems. Problems may be suited for one implementation or the other. The <a href="Tower_of_Hanoi" title="Tower of Hanoi">Tower of Hanoi</a> is a puzzle commonly solved using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.</dd>

<dt>Serial, parallel or distributed</dt>

<dd>Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time on serial computers. Serial algorithms are designed for these environments, unlike <a href="Parallel_algorithm" title="Parallel algorithm">parallel</a> or <a href="Distributed_algorithm" title="Distributed algorithm">distributed</a> algorithms. Parallel algorithms take advantage of computer architectures where multiple processors can work on a problem at the same time. Distributed algorithms use multiple machines connected via a computer network. Parallel and distributed algorithms divide the problem into subproblems and collect the results back together. Resource consumption in these algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Some sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable, but some problems have no parallel algorithms and are called inherently serial problems.</dd>

<dt>Deterministic or non-deterministic</dt>

<dd><a href="Deterministic_algorithm" title="Deterministic algorithm">Deterministic algorithms</a> solve the problem with exact decisions at every step; whereas <a href="Non-deterministic_algorithm" class="mw-redirect" title="Non-deterministic algorithm">non-deterministic algorithms</a> solve problems via guessing. Guesses are typically made more accurate through the use of <a href="Heuristics" class="mw-redirect" title="Heuristics">heuristics</a>.</dd>

<dt>Exact or approximate</dt>

<dd>While many algorithms reach an exact solution, <a href="Approximation_algorithm" title="Approximation algorithm">approximation algorithms</a> seek an approximation that is close to the true solution. Such algorithms have practical value for many hard problems. For example, the <a href="Knapsack_problem" title="Knapsack problem">Knapsack problem</a>, where there is a set of items, and the goal is to pack the knapsack to get the maximum total value. Each item has some weight and some value. The total weight that can be carried is no more than some fixed number X. So, the solution must consider the weights of items as well as their value.<a href="#cite_note-52">[52]</a></dd>

<dt>Quantum algorithm</dt>

<dd><a href="Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a> run on a realistic model of <a href="Quantum_computation" class="mw-redirect" title="Quantum computation">quantum computation</a>. The term is usually used for those algorithms that seem inherently quantum or use some essential feature of <a href="Quantum_computing" title="Quantum computing">Quantum computing</a> such as <a href="Quantum_superposition" title="Quantum superposition">quantum superposition</a> or <a href="Quantum_entanglement" title="Quantum entanglement">quantum entanglement</a>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="By_design_paradigm">By design paradigm</h3></div>

Another way of classifying algorithms is by their design methodology or <a href="Algorithmic_paradigm" title="Algorithmic paradigm">paradigm</a>. Some common paradigms are:

<dl><dt><a href="Brute-force_search" title="Brute-force search">Brute-force</a> or exhaustive search</dt>

<dd>Brute force is a problem-solving method of systematically trying every possible option until the optimal solution is found. This approach can be very time-consuming, testing every possible combination of variables. It is often used when other methods are unavailable or too complex. Brute force can solve a variety of problems, including finding the shortest path between two points and cracking passwords.</dd>

<dt>Divide and conquer</dt>

<dd>A <a href="Divide-and-conquer_algorithm" title="Divide-and-conquer algorithm">divide-and-conquer algorithm</a> repeatedly reduces a problem to one or more smaller instances of itself (usually <a href="Recursion" title="Recursion">recursively</a>) until the instances are small enough to solve easily. <a href="Mergesort" class="mw-redirect" title="Mergesort">Merge sorting</a> is an example of divide and conquer, where an unordered list is repeatedly split into smaller lists, which are sorted in the same way and then merged.<a href="#cite_note-53">[53]</a> In a simpler variant of divide and conquer called <a href="Prune_and_search" title="Prune and search">prune and search</a> or decrease-and-conquer algorithm, which solves one smaller instance of itself, and does not require a merge step.<a href="#cite_note-FOOTNOTEGoodrichTamassia20012454.7.1_Prune-and-search-54">[54]</a> An example of a prune and search algorithm is the <a href="Binary_search_algorithm" class="mw-redirect" title="Binary search algorithm">binary search algorithm</a>.</dd>

<dt>Search and enumeration</dt>

<dd>Many problems (such as playing <a href="Chess" title="Chess">chess</a>) can be modelled as problems on <a href="Graph_theory" title="Graph theory">graphs</a>. A <a href="Graph_exploration_algorithm" class="mw-redirect" title="Graph exploration algorithm">graph exploration algorithm</a> specifies rules for moving around a graph and is useful for such problems. This category also includes <a href="Search_algorithm" title="Search algorithm">search algorithms</a>, <a href="Branch_and_bound" title="Branch and bound">branch and bound</a> enumeration, and <a href="Backtracking" title="Backtracking">backtracking</a>.</dd>

<dt><a href="Randomized_algorithm" title="Randomized algorithm">Randomized algorithm</a></dt>

<dd>Such algorithms make some choices randomly (or pseudo-randomly). They find approximate solutions when finding exact solutions may be impractical (see heuristic method below). For some problems, the fastest approximations must involve some <a href="Randomness" title="Randomness">randomness</a>.<a href="#cite_note-55">[55]</a> Whether randomized algorithms with <a href="P_(complexity)" title="P (complexity)">polynomial time complexity</a> can be the fastest algorithm for some problems is an open question known as the <a href="P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a>. There are two large classes of such algorithms:</dd></dl>

<ol><li><a href="Monte_Carlo_algorithm" title="Monte Carlo algorithm">Monte Carlo algorithms</a> return a correct answer with high probability. E.g. <a href="RP_(complexity)" title="RP (complexity)">RP</a> is the subclass of these that run in <a href="Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>.</li>

<li><a href="Las_Vegas_algorithm" title="Las Vegas algorithm">Las Vegas algorithms</a> always return the correct answer, but their running time is only probabilistically bound, e.g. <a href="Zero-error_Probabilistic_Polynomial_time" class="mw-redirect" title="Zero-error Probabilistic Polynomial time">ZPP</a>.</li></ol>

<dl><dt><a href="Reduction_(complexity)" title="Reduction (complexity)">Reduction of complexity</a></dt>

<dd>This technique transforms difficult problems into better-known problems solvable with (hopefully) <a href="Asymptotically_optimal" class="mw-redirect" title="Asymptotically optimal">asymptotically optimal</a> algorithms. The goal is to find a reducing algorithm whose <a href="Computational_complexity_theory" title="Computational complexity theory">complexity</a> is not dominated by the resulting reduced algorithms. For example, one <a href="Selection_algorithm" title="Selection algorithm">selection algorithm</a> finds the median of an unsorted list by first sorting the list (the expensive portion), and then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as <a href="Transform_and_conquer_algorithm" class="mw-redirect" title="Transform and conquer algorithm">transform and conquer</a>.</dd>

<dt><a href="Back_tracking" class="mw-redirect" title="Back tracking">Back tracking</a></dt>

<dd>In this approach, multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Optimization_problems">Optimization problems</h3></div>

For <a href="Optimization_problem" title="Optimization problem">optimization problems</a> there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following:

<dl><dt><a href="Linear_programming" title="Linear programming">Linear programming</a></dt>

<dd>When searching for optimal solutions to a linear function bound by linear equality and inequality constraints, the constraints can be used directly to produce optimal solutions. There are algorithms that can solve any problem in this category, such as the popular <a href="Simplex_algorithm" title="Simplex algorithm">simplex algorithm</a>.<a href="#cite_note-56">[56]</a> Problems that can be solved with linear programming include the <a href="Maximum_flow_problem" title="Maximum flow problem">maximum flow problem</a> for directed graphs. If a problem also requires that any of the unknowns be <a href="Integer" title="Integer">integers</a>, then it is classified in <a href="Integer_programming" title="Integer programming">integer programming</a>. A linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial, i.e., the solutions satisfy these restrictions anyway. In the general case, a specialized algorithm or an algorithm that finds approximate solutions is used, depending on the difficulty of the problem.</dd>

<dt><a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a></dt>

<dd>When a problem shows optimal substructures—meaning the optimal solution can be constructed from optimal solutions to subproblems—and <a href="Overlapping_subproblem" class="mw-redirect" title="Overlapping subproblem">overlapping subproblems</a>, meaning the same subproblems are used to solve many different problem instances, a quicker approach called dynamic programming avoids recomputing solutions. For example, <a href="Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall algorithm</a>, the shortest path between a start and goal vertex in a weighted <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> can be found using the shortest path to the goal from all adjacent vertices. Dynamic programming and <a href="Memoization" title="Memoization">memoization</a> go together. Unlike divide and conquer, dynamic programming subproblems often overlap. The difference between dynamic programming and simple recursion is the caching or memoization of recursive calls. When subproblems are independent and do not repeat, memoization does not help; hence dynamic programming is not applicable to all complex problems. Using memoization dynamic programming reduces the complexity of many problems from exponential to polynomial.</dd>

<dt>The greedy method</dt>

<dd><a href="Greedy_algorithm" title="Greedy algorithm">Greedy algorithms</a>, similarly to a dynamic programming, work by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution and improve it by making small modifications. For some problems, they always find the optimal solution but for others they may stop at <a href="Local_optimum" class="mw-redirect" title="Local optimum">local optima</a>. The most popular use of greedy algorithms is finding minimal spanning trees of graphs without negative cycles. <a href="Huffman_coding" title="Huffman coding">Huffman Tree</a>, <a href="Kruskal's_algorithm" title="Kruskal's algorithm">Kruskal</a>, <a href="Prim's_algorithm" title="Prim's algorithm">Prim</a>, <a href="Sollin's_algorithm" class="mw-redirect" title="Sollin's algorithm">Sollin</a> are greedy algorithms that can solve this optimization problem.</dd>

<dt>The heuristic method</dt>

<dd>In <a href="Optimization_problem" title="Optimization problem">optimization problems</a>, <a href="Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristic algorithms</a> find solutions close to the optimal solution when finding the optimal solution is impractical. These algorithms get closer and closer to the optimal solution as they progress. In principle, if run for an infinite amount of time, they will find the optimal solution. They can ideally find a solution very close to the optimal solution in a relatively short time. These algorithms include <a href="Local_search_(optimization)" title="Local search (optimization)">local search</a>, <a href="Tabu_search" title="Tabu search">tabu search</a>, <a href="Simulated_annealing" title="Simulated annealing">simulated annealing</a>, and <a href="Genetic_algorithm" title="Genetic algorithm">genetic algorithms</a>. Some, like simulated annealing, are non-deterministic algorithms while others, like tabu search, are deterministic. When a bound on the error of the non-optimal solution is known, the algorithm is further categorized as an <a href="Approximation_algorithm" title="Approximation algorithm">approximation algorithm</a>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="List_of_algorithms" title="List of algorithms">List of algorithms</a></div>

One of the simplest algorithms finds the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be described in plain English as:

High-level description:

<ol><li>If a set of numbers is empty, then there is no highest number.</li>

<li>Assume the first number in the set is the largest.</li>

<li>For each remaining number in the set: if this number is greater than the current largest, it becomes the new largest.</li>

<li>When there are no unchecked numbers left in the set, consider the current largest number to be the largest in the set.</li></ol>

(Quasi-)formal description:

Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in <a href="Pseudocode" title="Pseudocode">pseudocode</a> or <a href="Pidgin_code" title="Pidgin code">pidgin code</a>:

<pre>Algorithm LargestNumber

Input: A list of numbers L.

Output: The largest number in the list L.

</pre>

<pre>if L.size = 0 return null

largest ← L[0]

for each item in L, do

if item > largest, then

largest ← item

return largest

</pre>

<ul><li>"←" denotes <a href="Assignment_(computer_science)" title="Assignment (computer science)">assignment</a>. For instance, "largest ← item" means that the value of largest changes to the value of item.</li>

<li>"return" terminates the algorithm and outputs the following value.</li></ul>

</div>

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<ul><li><a href="Abstract_machine" title="Abstract machine">Abstract machine</a></li>

<li><a href="ALGOL" title="ALGOL">ALGOL</a></li>

<li><a href="Logic_programming#Algorithm_=_Logic_+_Control" title="Logic programming">Algorithm = Logic + Control</a></li>

<li><a href="Algorithm_aversion" title="Algorithm aversion">Algorithm aversion</a></li>

<li><a href="Algorithm_engineering" title="Algorithm engineering">Algorithm engineering</a></li>

<li><a href="Algorithm_characterizations" title="Algorithm characterizations">Algorithm characterizations</a></li>

<li><a href="Algorithmic_bias" title="Algorithmic bias">Algorithmic bias</a></li>

<li><a href="Algorithmic_composition" title="Algorithmic composition">Algorithmic composition</a></li>

<li><a href="Algorithmic_entities" title="Algorithmic entities">Algorithmic entities</a></li>

<li><a href="Algorithmic_synthesis" class="mw-redirect" title="Algorithmic synthesis">Algorithmic synthesis</a></li>

<li><a href="Algorithmic_technique" title="Algorithmic technique">Algorithmic technique</a></li>

<li><a href="Algorithmic_topology" class="mw-redirect" title="Algorithmic topology">Algorithmic topology</a></li>

<li><a href="Computational_mathematics" title="Computational mathematics">Computational mathematics</a></li>

<li><a href="Garbage_in%2C_garbage_out" title="Garbage in, garbage out">Garbage in, garbage out</a></li>

<li><a href="Introduction_to_Algorithms" title="Introduction to Algorithms">Introduction to Algorithms</a> (textbook)</li>

<li><a href="Government_by_algorithm" title="Government by algorithm">Government by algorithm</a></li>

<li><a href="List_of_algorithms" title="List of algorithms">List of algorithms</a></li>

<li><a href="List_of_algorithm_general_topics" title="List of algorithm general topics">List of algorithm general topics</a></li>

<li><a href="Medium_is_the_message" class="mw-redirect" title="Medium is the message">Medium is the message</a></li>

<li><a href="Regulation_of_algorithms" title="Regulation of algorithms">Regulation of algorithms</a></li>

<li><a href="Theory_of_computation" title="Theory of computation">Theory of computation</a>

<ul><li><a href="Computability_theory" title="Computability theory">Computability theory</a></li>

<li><a href="Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li></ul></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.merriam-webster.com/dictionary/algorithm">"Definition of ALGORITHM"</a>. Merriam-Webster Online Dictionary. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200214074446/https://www.merriam-webster.com/dictionary/algorithm">Archived</a> from the original on February 14, 2020. Retrieved November 14, 2019.</cite>

</li>

<li id="cite_note-:2-2">^ <a href="#cite_ref-:2_2-0">a</a> <a href="#cite_ref-:2_2-1">b</a> David A. Grossman, Ophir Frieder, Information Retrieval: Algorithms and Heuristics, 2nd edition, 2004, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1402030045</bdi>

</li>

<li id="cite_note-:3-3"><a href="#cite_ref-:3_3-0">^</a> "Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2).

</li>

<li id="cite_note-:4-4"><a href="#cite_ref-:4_4-0">^</a> 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).

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> "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).

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> "An algorithm has <a href="Zero" class="mw-redirect" title="Zero">zero</a> or more inputs, i.e., <a href="Quantity" title="Quantity">quantities</a> which are given to it initially before the algorithm begins" (Knuth 1973:5).

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> "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).

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> "An algorithm has one or more outputs, i.e., quantities which have a specified relation to the inputs" (Knuth 1973:5).

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> 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).

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<li id="cite_note-:1-10"><a href="#cite_ref-:1_10-0">^</a> Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.oed.com/dictionary/algorism_n?tl=true">"algorism"</a>. Oxford English Dictionary. Retrieved May 18, 2025.</cite>

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<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFChaucer" class="citation web cs1">Chaucer, Geoffrey. <a rel="nofollow" class="external text" href="https://chaucer.fas.harvard.edu/pages/millers-prologue-and-tale">"The Miller's Tale"</a>. Line 3210.</cite>

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<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFSkeat1914" class="citation book cs1">Skeat, Walter William (1914). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=z58YAAAAIAAJ&pg=PA5">"agrim, agrum"</a>. In Mayhew, Anthony Lawson (ed.). A Glossary of Tudor and Stuart Words: Especially from the Dramatists. Clarendon Press. pp. 5–6.</cite>

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<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFGrabiner2013" class="citation book cs1"><a href="Judith_Grabiner" title="Judith Grabiner">Grabiner, Judith V.</a> (December 2013). "The role of mathematics in liberal arts education". In Matthews, Michael R. (ed.). International Handbook of Research in History, Philosophy and Science Teaching. Springer. pp. 793–836. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-007-7654-8_25">10.1007/978-94-007-7654-8_25</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9789400776548</bdi>.</cite>

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<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.oed.com/dictionary/algorithm_n">"algorithm"</a>. Oxford English Dictionary. Retrieved May 18, 2025.</cite>

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<li id="cite_note-FOOTNOTEStone19718-16"><a href="#cite_ref-FOOTNOTEStone19718_16-0">^</a> <a href="#CITEREFStone1971">Stone (1971)</a>, p. 8.

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<cite id="CITEREFSimanowski2018" class="citation book cs1"><a href="Roberto_Simanowski" title="Roberto Simanowski">Simanowski, Roberto</a> (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RJV5DwAAQBAJ">The Death Algorithm and Other Digital Dilemmas</a>. Untimely Meditations. Vol. 14. Translated by Chase, Jefferson. Cambridge, Massachusetts: MIT Press. p. 147. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262536370</bdi>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191222120705/https://books.google.com/books?id=RJV5DwAAQBAJ">Archived</a> from the original on December 22, 2019. Retrieved May 27, 2019. <q>[...] the next level of abstraction of central bureaucracy: globally operating algorithms.</q></cite>

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<cite id="CITEREFDietrich1999" class="citation book cs1">Dietrich, Eric (1999). "Algorithm". In Wilson, Robert Andrew; Keil, Frank C. (eds.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-wt1aZrGXLYC">The MIT Encyclopedia of the Cognitive Sciences</a>. MIT Cognet library. Cambridge, Massachusetts: MIT Press (published 2001). p. 11. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262731447</bdi>. Retrieved July 22, 2020. <q>An algorithm is a recipe, method, or technique for doing something.</q></cite>

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<li id="cite_note-19"><a href="#cite_ref-19">^</a> Stone requires that "it must terminate in a finite number of steps" (Stone 1973:7–8).

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<li id="cite_note-20"><a href="#cite_ref-20">^</a> Boolos and Jeffrey 1974, 1999:19

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<li id="cite_note-Springer_Science_&_Business_Media-21">^ <a href="#cite_ref-Springer_Science_&_Business_Media_21-0">a</a> <a href="#cite_ref-Springer_Science_&_Business_Media_21-1">b</a> <a href="#cite_ref-Springer_Science_&_Business_Media_21-2">c</a> <a href="#cite_ref-Springer_Science_&_Business_Media_21-3">d</a> <cite id="CITEREFChabert2012" class="citation book cs1">Chabert, Jean-Luc (2012). A History of Algorithms: From the Pebble to the Microchip. Springer Science & Business Media. pp. 7–8. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783642181924</bdi>.</cite>

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<li id="cite_note-:6-22">^ <a href="#cite_ref-:6_22-0">a</a> <a href="#cite_ref-:6_22-1">b</a> <cite id="CITEREFSriram2005" class="citation book cs1">Sriram, M. S. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153">"Algorithms in Indian Mathematics"</a>. In Emch, Gerard G.; Sridharan, R.; Srinivas, M. D. (eds.). Contributions to the History of Indian Mathematics. Springer. p. 153. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-93-86279-25-5</bdi>.</cite>

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<li id="cite_note-23"><a href="#cite_ref-23">^</a> Hayashi, T. (2023, January 1). <a rel="nofollow" class="external text" href="https://www.britannica.com/biography/Brahmagupta">Brahmagupta</a>. Encyclopedia Britannica.

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<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFZaslavsky1970" class="citation journal cs1">Zaslavsky, Claudia (1970). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3027363">"Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria"</a>. The Two-Year College Mathematics Journal. 1 (2): 76–99. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3027363">10.2307/3027363</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0049-4925">0049-4925</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3027363">3027363</a>.</cite>

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<li id="cite_note-Cooke2005-25">^ <a href="#cite_ref-Cooke2005_25-0">a</a> <a href="#cite_ref-Cooke2005_25-1">b</a> <a href="#cite_ref-Cooke2005_25-2">c</a> <cite id="CITEREFCooke2005" class="citation book cs1">Cooke, Roger L. (2005). The History of Mathematics: A Brief Course. John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-118-46029-0</bdi>.</cite>

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<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFChabert1999" class="citation journal cs1">Chabert, Jean-Luc, ed. (1999). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-642-18192-4">"A History of Algorithms"</a>. SpringerLink. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-18192-4">10.1007/978-3-642-18192-4</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-63369-3</bdi>.</cite>

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<li id="cite_note-Dooley-27">^ <a href="#cite_ref-Dooley_27-0">a</a> <a href="#cite_ref-Dooley_27-1">b</a> <cite id="CITEREFDooley2013" class="citation book cs1">Dooley, John F. (2013). A Brief History of Cryptology and Cryptographic Algorithms. Springer Science & Business Media. pp. 12–3. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783319016283</bdi>.</cite>

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<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite id="CITEREFKnuth1972" class="citation journal cs1">Knuth, Donald E. (1972). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf">"Ancient Babylonian Algorithms"</a> (PDF). Commun. ACM. 15 (7): 671–677. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F361454.361514">10.1145/361454.361514</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0001-0782">0001-0782</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7829945">7829945</a>. Archived from <a rel="nofollow" class="external text" href="http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf">the original</a> (PDF) on December 24, 2012.</cite>

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<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFAaboe2001" class="citation book cs1"><a href="Asger_Aaboe" title="Asger Aaboe">Aaboe, Asger</a> (2001). Episodes from the Early History of Astronomy. New York: Springer. pp. 40–62. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-95136-2</bdi>.</cite>

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<li id="cite_note-30"><a href="#cite_ref-30">^</a> <cite id="CITEREFAst" class="citation web cs1">Ast, Courtney. <a rel="nofollow" class="external text" href="http://www.math.wichita.edu/history/men/eratosthenes.html">"Eratosthenes"</a>. Wichita State University: Department of Mathematics and Statistics. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html">Archived</a> from the original on February 27, 2015. Retrieved February 27, 2015.</cite>

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<li id="cite_note-31"><a href="#cite_ref-31">^</a> Bolter 1984:24

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<li id="cite_note-32"><a href="#cite_ref-32">^</a> Bolter 1984:26

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<li id="cite_note-33"><a href="#cite_ref-33">^</a> Bolter 1984:33–34, 204–206.

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<li id="cite_note-34"><a href="#cite_ref-34">^</a> Bell and Newell diagram 1971:39, cf. Davis 2000

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<li id="cite_note-35"><a href="#cite_ref-35">^</a> Melina Hill, Valley News Correspondent, A Tinkerer Gets a Place in History, Valley News West Lebanon NH, Thursday, March 31, 1983, p. 13.

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<li id="cite_note-36"><a href="#cite_ref-36">^</a> Davis 2000:14

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<li id="cite_note-37"><a href="#cite_ref-37">^</a> Kleene 1943 in Davis 1965:274

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<li id="cite_note-38"><a href="#cite_ref-38">^</a> Rosser 1939 in Davis 1965:225

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<li id="cite_note-:5-39">^ <a href="#cite_ref-:5_39-0">a</a> <a href="#cite_ref-:5_39-1">b</a> <a href="#cite_ref-:5_39-2">c</a> <a href="#cite_ref-:5_39-3">d</a> Sipser 2006:157

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<li id="cite_note-KriegelSchubert2016-40"><a href="#cite_ref-KriegelSchubert2016_40-0">^</a> <cite id="CITEREFKriegelSchubertZimek2016" class="citation journal cs1"><a href="Hans-Peter_Kriegel" title="Hans-Peter Kriegel">Kriegel, Hans-Peter</a>; Schubert, Erich; <a href="Arthur_Zimek" title="Arthur Zimek">Zimek, Arthur</a> (2016). "The (black) art of run-time evaluation: Are we comparing algorithms or implementations?". Knowledge and Information Systems. 52 (2): 341–378. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10115-016-1004-2">10.1007/s10115-016-1004-2</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0219-1377">0219-1377</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:40772241">40772241</a>.</cite>

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<li id="cite_note-41"><a href="#cite_ref-41">^</a> <cite id="CITEREFGillian_Conahan2013" class="citation web cs1">Gillian Conahan (January 2013). <a rel="nofollow" class="external text" href="http://discovermagazine.com/2013/jan-feb/34-better-math-makes-faster-data-networks">"Better Math Makes Faster Data Networks"</a>. discovermagazine.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140513212427/http://discovermagazine.com/2013/jan-feb/34-better-math-makes-faster-data-networks">Archived</a> from the original on May 13, 2014. Retrieved May 13, 2014.</cite>

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<li id="cite_note-Hassanieh12-42"><a href="#cite_ref-Hassanieh12_42-0">^</a> Haitham Hassanieh, <a href="Piotr_Indyk" title="Piotr Indyk">Piotr Indyk</a>, Dina Katabi, and Eric Price, "<a rel="nofollow" class="external text" href="http://siam.omnibooksonline.com/2012SODA/data/papers/500.pdf">ACM-SIAM Symposium On Discrete Algorithms (SODA)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130704180806/http://siam.omnibooksonline.com/2012SODA/data/papers/500.pdf">Archived</a> July 4, 2013, at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Kyoto, January 2012. See also the <a rel="nofollow" class="external text" href="http://groups.csail.mit.edu/netmit/sFFT/">sFFT Web Page</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120221145740/http://groups.csail.mit.edu/netmit/sFFT/">Archived</a> February 21, 2012, at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.

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<li id="cite_note-43"><a href="#cite_ref-43">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/bestcase.html">"Best Case"</a>. Dictionary of Algorithms and Data Structures. National Institute of Standards and Technology (NIST). National Institute of Standards and Technology. Retrieved May 29, 2025.</cite>

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<li id="cite_note-45"><a href="#cite_ref-45">^</a> <cite id="CITEREFGoodrichTamassia2002" class="citation book cs1"><a href="Michael_T._Goodrich" title="Michael T. Goodrich">Goodrich, Michael T.</a>; <a href="Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a> (2002). <a rel="nofollow" class="external text" href="http://ww3.algorithmdesign.net/ch00-front.html">Algorithm Design: Foundations, Analysis, and Internet Examples</a>. John Wiley & Sons, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-38365-9</bdi>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150428201622/http://ww3.algorithmdesign.net/ch00-front.html">Archived</a> from the original on April 28, 2015. Retrieved June 14, 2018.</cite>

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<li id="cite_note-46"><a href="#cite_ref-46">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-o-notation">"Big-O notation (article) | Algorithms"</a>. Khan Academy. Retrieved June 3, 2024.</cite>

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<li id="cite_note-47"><a href="#cite_ref-47">^</a> <a href="John_G._Kemeny" title="John G. Kemeny">John G. Kemeny</a> and <a href="Thomas_E._Kurtz" title="Thomas E. Kurtz">Thomas E. Kurtz</a> 1985 Back to Basic: The History, Corruption, and Future of the Language, Addison-Wesley Publishing Company, Inc. Reading, MA, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-13433-0</bdi>.

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<li id="cite_note-48"><a href="#cite_ref-48">^</a> Tausworthe 1977:101

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<li id="cite_note-49"><a href="#cite_ref-49">^</a> Tausworthe 1977:142

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<li id="cite_note-50"><a href="#cite_ref-50">^</a> Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff and Chapter 9.1

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<li id="cite_note-52"><a href="#cite_ref-52">^</a> <cite id="CITEREFKellererPferschyPisinger2004" class="citation book cs1">Kellerer, Hans; Pferschy, Ulrich; Pisinger, David (2004). <a rel="nofollow" class="external text" href="https://www.springer.com/us/book/9783540402862">Knapsack Problems | Hans Kellerer | Springer</a>. Springer. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-24777-7">10.1007/978-3-540-24777-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-40286-2</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:28836720">28836720</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171018181055/https://www.springer.com/us/book/9783540402862">Archived</a> from the original on October 18, 2017. Retrieved September 19, 2017.</cite>

</li>

<li id="cite_note-53"><a href="#cite_ref-53">^</a> <cite id="CITEREFGoodrichTamassia2001" class="citation book cs1">Goodrich, Michael T.; Tamassia, Roberto (2001). "5.2 Divide and Conquer". Algorithm Design: Foundations, Analysis, and Internet Examples. John Wiley & Sons. p. 263. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780471383659</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTEGoodrichTamassia20012454.7.1_Prune-and-search-54"><a href="#cite_ref-FOOTNOTEGoodrichTamassia20012454.7.1_Prune-and-search_54-0">^</a> <a href="#CITEREFGoodrichTamassia2001">Goodrich & Tamassia (2001)</a>, p. 245, 4.7.1 Prune-and-search.

</li>

<li id="cite_note-55"><a href="#cite_ref-55">^</a> For instance, the <a href="Volume" title="Volume">volume</a> of a <a href="Convex_polytope" title="Convex polytope">convex polytope</a> (described using a membership oracle) can be approximated to high accuracy by a randomized polynomial time algorithm, but not by a deterministic one: see <cite id="CITEREFDyerFriezeKannan1991" class="citation journal cs1">Dyer, Martin; Frieze, Alan; Kannan, Ravi (January 1991). "A Random Polynomial-time Algorithm for Approximating the Volume of Convex Bodies". J. ACM. 38 (1): 1–17. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.145.4600">10.1.1.145.4600</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F102782.102783">10.1145/102782.102783</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13268711">13268711</a>.</cite>

</li>

<a href="George_B._Dantzig" class="mw-redirect" title="George B. Dantzig">George B. Dantzig</a> and Mukund N. Thapa. 2003. Linear Programming 2: Theory and Extensions. Springer-Verlag.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

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</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em">

<ul><li><cite id="CITEREFAxt1959" class="citation journal cs1">Axt, P (1959). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1993169">"On a Subrecursive Hierarchy and Primitive Recursive Degrees"</a>. Transactions of the American Mathematical Society. 92 (1): 85–105. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1993169">10.2307/1993169</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1993169">1993169</a>.</cite></li>

<li>Bell, C. Gordon and Newell, Allen (1971), Computer Structures: Readings and Examples, McGraw–Hill Book Company, New York. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-004357-4</bdi>.</li>

<li><cite id="CITEREFBlassGurevich2003" class="citation journal cs1"><a href="Andreas_Blass" title="Andreas Blass">Blass, Andreas</a>; <a href="Yuri_Gurevich" title="Yuri Gurevich">Gurevich, Yuri</a> (2003). <a rel="nofollow" class="external text" href="http://research.microsoft.com/~gurevich/Opera/164.pdf">"Algorithms: A Quest for Absolute Definitions"</a> (PDF). Bulletin of European Association for Theoretical Computer Science. 81. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://research.microsoft.com/~gurevich/Opera/164.pdf">Archived</a> (PDF) from the original on October 9, 2022.</cite> Includes a bibliography of 56 references.</li>

<li><cite id="CITEREFBolter1984" class="citation book cs1">Bolter, David J. (1984). Turing's Man: Western Culture in the Computer Age (1984 ed.). Chapel Hill, NC: The University of North Carolina Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8078-1564-9</bdi>.</cite>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8078-4108-0</bdi></li>

<li><cite id="CITEREFBoolosJeffrey1999" class="citation book cs1"><a href="George_Boolos" title="George Boolos">Boolos, George</a>; <a href="Richard_Jeffrey" title="Richard Jeffrey">Jeffrey, Richard</a> (1999) [1974]. <a rel="nofollow" class="external text" href="https://archive.org/details/computabilitylog0000bool_r8y9">Computability and Logic</a> (4th ed.). Cambridge University Press, London. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-20402-6</bdi>.</cite>: cf. Chapter 3 Turing machines where they discuss "certain enumerable sets not effectively (mechanically) enumerable".</li>

<li><cite id="CITEREFBurgin2004" class="citation book cs1">Burgin, Mark (2004). Super-Recursive Algorithms. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-95569-8</bdi>.</cite></li>

<li>Campagnolo, M.L., <a href="Cris_Moore" class="mw-redirect" title="Cris Moore">Moore, C.</a>, and Costa, J.F. (2000) An analog characterization of the subrecursive functions. In Proc. of the 4th Conference on Real Numbers and Computers, Odense University, pp. 91–109</li>

<li><cite id="CITEREFChurch1936" class="citation journal cs1"><a href="Alonzo_Church" title="Alonzo Church">Church, Alonzo</a> (1936). "An Unsolvable Problem of Elementary Number Theory". American Journal of Mathematics. 58 (2): 345–363. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2371045">10.2307/2371045</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2371045">2371045</a>.</cite> Reprinted in The Undecidable, p. 89ff. The first expression of "Church's Thesis". See in particular page 100 (The Undecidable) where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc.</li>

<li><cite id="CITEREFChurch1936" class="citation journal cs1"><a href="Alonzo_Church" title="Alonzo Church">Church, Alonzo</a> (1936). "A Note on the Entscheidungsproblem". The Journal of Symbolic Logic. 1 (1): 40–41. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2269326">10.2307/2269326</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2269326">2269326</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:42323521">42323521</a>.</cite> <cite id="CITEREFChurch1936" class="citation journal cs1">Church, Alonzo (1936). "Correction to a Note on the Entscheidungsproblem". The Journal of Symbolic Logic. 1 (3): 101–102. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2269030">10.2307/2269030</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2269030">2269030</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5557237">5557237</a>.</cite> Reprinted in The Undecidable, p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes.</li>

<li><cite id="CITEREFDaffa'1977" class="citation book cs1">Daffa', Ali Abdullah al- (1977). The Muslim contribution to mathematics. London: Croom Helm. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-85664-464-1</bdi>.</cite></li>

<li><cite id="CITEREFDavis1965" class="citation book cs1"><a href="Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Davis, Martin</a> (1965). <a rel="nofollow" class="external text" href="https://archive.org/details/undecidablebasic0000davi">The Undecidable: Basic Papers On Undecidable Propositions, Unsolvable Problems and Computable Functions</a>. New York: Raven Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-43228-1</bdi>.</cite> Davis gives commentary before each article. Papers of <a href="G%C3%B6del" class="mw-redirect" title="Gödel">Gödel</a>, <a href="Alonzo_Church" title="Alonzo Church">Alonzo Church</a>, <a href="Alan_Turing" title="Alan Turing">Turing</a>, <a href="J._Barkley_Rosser" title="J. Barkley Rosser">Rosser</a>, <a href="Kleene" class="mw-redirect" title="Kleene">Kleene</a>, and <a href="Emil_Post" class="mw-redirect" title="Emil Post">Emil Post</a> are included; those cited in the article are listed here by author's name.</li>

<li><cite id="CITEREFDavis2000" class="citation book cs1"><a href="Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Davis, Martin</a> (2000). Engines of Logic: Mathematicians and the Origin of the Computer. New York: W.W. Nortion. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-393-32229-3</bdi>.</cite> Davis offers concise biographies of <a href="Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a>, <a href="George_Boole" title="George Boole">Boole</a>, <a href="Gottlob_Frege" title="Gottlob Frege">Frege</a>, <a href="Georg_Cantor" title="Georg Cantor">Cantor</a>, <a href="David_Hilbert" title="David Hilbert">Hilbert</a>, Gödel and Turing with <a href="John_von_Neumann" title="John von Neumann">von Neumann</a> as the show-stealing villain. Very brief bios of <a href="Joseph-Marie_Jacquard" class="mw-redirect" title="Joseph-Marie Jacquard">Joseph-Marie Jacquard</a>, <a href="Babbage" class="mw-redirect" title="Babbage">Babbage</a>, <a href="Ada_Lovelace" title="Ada Lovelace">Ada Lovelace</a>, <a href="Claude_Shannon" title="Claude Shannon">Claude Shannon</a>, <a href="Howard_Aiken" class="mw-redirect" title="Howard Aiken">Howard Aiken</a>, etc.</li>

<li> This article incorporates <a href="Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from <cite id="CITEREFPaul_E._Black" class="citation cs1">Paul E. Black. <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/algorithm.html">"algorithm"</a>. <a href="Dictionary_of_Algorithms_and_Data_Structures" class="mw-redirect" title="Dictionary of Algorithms and Data Structures">Dictionary of Algorithms and Data Structures</a>. <a href="NIST" class="mw-redirect" title="NIST">NIST</a>.</cite></li>

<li><cite id="CITEREFDean,_Tim2012" class="citation journal cs1">Dean, Tim (2012). <a rel="nofollow" class="external text" href="https://doi.org/10.4148%2Fbiyclc.v7i0.1775">"Evolution and moral diversity"</a>. Baltic International Yearbook of Cognition, Logic and Communication. 7. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4148%2Fbiyclc.v7i0.1775">10.4148/biyclc.v7i0.1775</a>.</cite></li>

<li><cite id="CITEREFDennett1995" class="citation book cs1"><a href="Daniel_Dennett" title="Daniel Dennett">Dennett, Daniel</a> (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/darwinsdangerous0000denn">Darwin's Dangerous Idea</a>. New York: Touchstone/Simon & Schuster. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/darwinsdangerous0000denn/page/32">32</a>–36. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-684-80290-9</bdi>.</cite></li>

<li><cite id="CITEREFDilson2007" class="citation book cs1">Dilson, Jesse (2007). <a rel="nofollow" class="external text" href="https://archive.org/details/abacusworldsfirs0000dils">The Abacus</a> ((1968, 1994) ed.). St. Martin's Press, NY. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-312-10409-2</bdi>.</cite>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-312-10409-X</bdi></li>

<li><a href="Yuri_Gurevich" title="Yuri Gurevich">Yuri Gurevich</a>, <a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.146.3017&rep=rep1&type=pdf">Sequential Abstract State Machines Capture Sequential Algorithms</a>, ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources.</li>

<li><cite id="CITEREFvan_Heijenoort2001" class="citation book cs1"><a href="Jean_van_Heijenoort" title="Jean van Heijenoort">van Heijenoort, Jean</a> (2001). From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931 ((1967) ed.). Harvard University Press, Cambridge. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-674-32449-7</bdi>.</cite>, 3rd edition 1976[?], <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-674-32449-8</bdi> (pbk.)</li>

<li><cite id="CITEREFHodges1983" class="citation book cs1"><a href="Andrew_Hodges" title="Andrew Hodges">Hodges, Andrew</a> (1983). <a href="Alan_Turing%3A_The_Enigma" title="Alan Turing: The Enigma">Alan Turing: The Enigma</a>. New York: <a href="Simon_and_Schuster" class="mw-redirect" title="Simon and Schuster">Simon and Schuster</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-671-49207-6</bdi>.</cite>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-671-49207-1</bdi>. Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.</li>

<li><cite id="CITEREFKleene1936" class="citation journal cs1"><a href="Stephen_Kleene" class="mw-redirect" title="Stephen Kleene">Kleene, Stephen C.</a> (1936). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140903092121/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002278499&L=1">"General Recursive Functions of Natural Numbers"</a>. Mathematische Annalen. 112 (5): 727–742. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01565439">10.1007/BF01565439</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120517999">120517999</a>. Archived from <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002278499&L=1">the original</a> on September 3, 2014. Retrieved September 30, 2013.</cite> Presented to the American Mathematical Society, September 1935. Reprinted in The Undecidable, p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper An Unsolvable Problem of Elementary Number Theory that proved the "decision problem" to be "undecidable" (i.e., a negative result).</li>

<li><cite id="CITEREFKleene1943" class="citation journal cs1"><a href="Stephen_Kleene" class="mw-redirect" title="Stephen Kleene">Kleene, Stephen C.</a> (1943). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990131">"Recursive Predicates and Quantifiers"</a>. Transactions of the American Mathematical Society. 53 (1): 41–73. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990131">10.2307/1990131</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990131">1990131</a>.</cite> Reprinted in The Undecidable, p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in Kleene 1952:300) and name it "Church's Thesis"(Kleene 1952:317) (i.e., the <a href="Church_thesis" class="mw-redirect" title="Church thesis">Church thesis</a>).</li>

<li><cite id="CITEREFKleene1991" class="citation book cs1"><a href="Kleene" class="mw-redirect" title="Kleene">Kleene, Stephen C.</a> (1991) [1952]. Introduction to Metamathematics (Tenth ed.). North-Holland Publishing Company. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7204-2103-3</bdi>.</cite></li>

<li><cite id="CITEREFKnuth1997" class="citation book cs1"><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1997). Fundamental Algorithms, Third Edition. Reading, Massachusetts: Addison–Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-89683-1</bdi>.</cite></li>

<li><cite id="CITEREFKnuth1969" class="citation book cs1"><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald</a> (1969). Volume 2/Seminumerical Algorithms, The Art of Computer Programming First Edition. Reading, Massachusetts: Addison–Wesley.</cite></li>

<li>Kosovsky, N.K. Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms, LSU Publ., Leningrad, 1981</li>

<li><cite id="CITEREFKowalski1979" class="citation journal cs1"><a href="Robert_Kowalski" title="Robert Kowalski">Kowalski, Robert</a> (1979). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F359131.359136">"Algorithm=Logic+Control"</a>. <a href="Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a>. 22 (7): 424–436. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F359131.359136">10.1145/359131.359136</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2509896">2509896</a>.</cite></li>

<li>A.A. Markov (1954) Theory of algorithms. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60-51085.]</li>

<li><cite id="CITEREFMinsky1967" class="citation book cs1"><a href="Marvin_Minsky" title="Marvin Minsky">Minsky, Marvin</a> (1967). <a rel="nofollow" class="external text" href="https://archive.org/details/computationfinit0000mins">Computation: Finite and Infinite Machines</a> (First ed.). Prentice-Hall, Englewood Cliffs, NJ. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-165449-5</bdi>.</cite> Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 Computability, Effective Procedures and Algorithms. Infinite machines.</li>

<li><cite id="CITEREFPost1936" class="citation journal cs1"><a href="Emil_Post" class="mw-redirect" title="Emil Post">Post, Emil</a> (1936). "Finite Combinatory Processes, Formulation I". The Journal of Symbolic Logic. 1 (3): 103–105. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2269031">10.2307/2269031</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2269031">2269031</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:40284503">40284503</a>.</cite> Reprinted in The Undecidable, pp. 289ff. Post defines a simple algorithmic-like process of a man writing marks or erasing marks and going from box to box and eventually halting, as he follows a list of simple instructions. This is cited by Kleene as one source of his "Thesis I", the so-called <a href="Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a>.</li>

<li><cite id="CITEREFRogers1987" class="citation book cs1">Rogers, Hartley Jr. (1987). Theory of Recursive Functions and Effective Computability. The MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-262-68052-3</bdi>.</cite></li>

<li><cite id="CITEREFRosser1939" class="citation journal cs1"><a href="J._B._Rosser" class="mw-redirect" title="J. B. Rosser">Rosser, J.B.</a> (1939). "An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem". Journal of Symbolic Logic. 4 (2): 53–60. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2269059">10.2307/2269059</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2269059">2269059</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:39499392">39499392</a>.</cite> Reprinted in The Undecidable, p. 223ff. Herein is Rosser's famous definition of "effective method": "...a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps... a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer" (p. 225–226, The Undecidable)</li>

<li><cite id="CITEREFSantos-Lang2015" class="citation book cs1">Santos-Lang, Christopher (2015). <a rel="nofollow" class="external text" href="http://grinfree.com/MoralEcology.pdf">"Moral Ecology Approaches to Machine Ethics"</a> (PDF). In van Rysewyk, Simon; Pontier, Matthijs (eds.). Machine Medical Ethics. Intelligent Systems, Control and Automation: Science and Engineering. Vol. 74. Switzerland: Springer. pp. 111–127. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-08108-3_8">10.1007/978-3-319-08108-3_8</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-08107-6</bdi>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://grinfree.com/MoralEcology.pdf">Archived</a> (PDF) from the original on October 9, 2022.</cite></li>

<li><cite id="CITEREFScott2009" class="citation book cs1">Scott, Michael L. (2009). Programming Language Pragmatics (3rd ed.). Morgan Kaufmann Publishers/Elsevier. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-374514-9</bdi>.</cite></li>

<li><cite id="CITEREFSipser2006" class="citation book cs1">Sipser, Michael (2006). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00sips">Introduction to the Theory of Computation</a>. PWS Publishing Company. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-94728-6</bdi>.</cite></li>

<li><cite id="CITEREFSoberWilson1998" class="citation book cs1">Sober, Elliott; Wilson, David Sloan (1998). <a rel="nofollow" class="external text" href="https://archive.org/details/untoothersevolut00sobe">Unto Others: The Evolution and Psychology of Unselfish Behavior</a>. Cambridge: Harvard University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780674930469</bdi>.</cite></li>

<li><cite id="CITEREFStone1971" class="citation book cs1">Stone, Harold S. (1971). Introduction to Computer Organization and Data Structures. McGraw-Hill, New York. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780070617261</bdi>.</cite> Cf. in particular the first chapter titled: Algorithms, Turing Machines, and Programs. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an algorithm" (p. 4).</li>

<li><cite id="CITEREFTausworthe1977" class="citation book cs1">Tausworthe, Robert C (1977). Standardized Development of Computer Software Part 1 Methods. Englewood Cliffs NJ: Prentice–Hall, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-842195-3</bdi>.</cite></li>

<li><cite id="CITEREFTuring1936–37" class="citation journal cs1 cs1-prop-year-range-abbreviated"><a href="A._M._Turing" class="mw-redirect" title="A. M. Turing">Turing, Alan M.</a> (1936–37). "On Computable Numbers, With An Application to the Entscheidungsproblem". <a href="Proceedings_of_the_London_Mathematical_Society" class="mw-redirect" title="Proceedings of the London Mathematical Society">Proceedings of the London Mathematical Society</a>. Series 2. 42: 230–265. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-42.1.230">10.1112/plms/s2-42.1.230</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:73712">73712</a>.</cite>. Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in The Undecidable, p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.</li>

<li><cite id="CITEREFTuring1939" class="citation journal cs1"><a href="A._M._Turing" class="mw-redirect" title="A. M. Turing">Turing, Alan M.</a> (1939). "Systems of Logic Based on Ordinals". Proceedings of the London Mathematical Society. 45: 161–228. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2-45.1.161">10.1112/plms/s2-45.1.161</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/21.11116%2F0000-0001-91CE-3">21.11116/0000-0001-91CE-3</a>.</cite> Reprinted in The Undecidable, pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton.</li>

<li><a href="United_States_Patent_and_Trademark_Office" title="United States Patent and Trademark Office">United States Patent and Trademark Office</a> (2006), <a rel="nofollow" class="external text" href="http://www.uspto.gov/web/offices/pac/mpep/documents/2100_2106_02.htm">2106.02 **>Mathematical Algorithms: 2100 Patentability</a>, Manual of Patent Examining Procedure (MPEP). Latest revision August 2006</li></ul>

</div>

<ul><li>Zaslavsky, C. (1970). Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria. The Two-Year College Mathematics Journal, 1(2), 76–99. <a rel="nofollow" class="external free" href="https://doi.org/10.2307/3027363">https://doi.org/10.2307/3027363</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFBellah1985" class="citation book cs1"><a href="Robert_N._Bellah" title="Robert N. Bellah">Bellah, Robert Neelly</a> (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XsUojihVZQcC">Habits of the Heart: Individualism and Commitment in American Life</a>. Berkeley: University of California Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-520-25419-0</bdi>.</cite></li>

<li><cite id="CITEREFBerlinski2001" class="citation book cs1">Berlinski, David (2001). <a rel="nofollow" class="external text" href="https://archive.org/details/adventofalgorith0000berl">The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer</a>. Harvest Books. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-15-601391-8</bdi>.</cite></li>

<li><cite id="CITEREFChabert1999" class="citation book cs1">Chabert, Jean-Luc (1999). A History of Algorithms: From the Pebble to the Microchip. Springer Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-63369-3</bdi>.</cite></li>

<li><cite id="CITEREFThomas_H._CormenCharles_E._LeisersonRonald_L._RivestClifford_Stein2009" class="citation book cs1">Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest; Clifford Stein (2009). Introduction To Algorithms (3rd ed.). MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-262-03384-8</bdi>.</cite></li>

<li><cite id="CITEREFHarel,_DavidFeldman,_Yishai2004" class="citation book cs1">Harel, David; Feldman, Yishai (2004). Algorithmics: The Spirit of Computing. Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-321-11784-7</bdi>.</cite></li>

<li><cite id="CITEREFHertzkeMcRorie1998" class="citation book cs1">Hertzke, Allen D.; McRorie, Chris (1998). "The Concept of Moral Ecology". In Lawler, Peter Augustine; McConkey, Dale (eds.). Community and Political Thought Today. Westport, CT: <a href="Praeger_Publishers" class="mw-redirect" title="Praeger Publishers">Praeger</a>.</cite></li>

<li>Jon Kleinberg, Éva Tardos(2006): Algorithm Design, Pearson/Addison-Wesley, ISBN 978-0-32129535-4</li>

<li><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a> (2000). <a rel="nofollow" class="external text" href="http://www-cs-faculty.stanford.edu/~uno/aa.html">Selected Papers on Analysis of Algorithms</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170701190647/http://www-cs-faculty.stanford.edu/~uno/aa.html">Archived</a> July 1, 2017, at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Stanford, California: Center for the Study of Language and Information.</li>

<li>Knuth, Donald E. (2010). <a rel="nofollow" class="external text" href="http://www-cs-faculty.stanford.edu/~uno/da.html">Selected Papers on Design of Algorithms</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170716225848/http://www-cs-faculty.stanford.edu/~uno/da.html">Archived</a> July 16, 2017, at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Stanford, California: Center for the Study of Language and Information.</li>

<li><cite id="CITEREFWallachAllen2008" class="citation book cs1">Wallach, Wendell; Allen, Colin (November 2008). Moral Machines: Teaching Robots Right from Wrong. US: Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-537404-9</bdi>.</cite></li>

<li><cite id="CITEREFBleakley,_Chris2020" class="citation book cs1">Bleakley, Chris (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3pr5DwAAQBAJ">Poems that Solve Puzzles: The History and Science of Algorithms</a>. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-885373-2</bdi>.</cite></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Look up <a href="https://en.wiktionary.org/wiki/Special:Search/algorithm" class="extiw external" title="wiktionary:Special:Search/algorithm">algorithm</a> in Wiktionary, the free dictionary.</div></div>

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<div class="side-box-text plainlist">Wikibooks has a book on the topic of: <a href="https://en.wikibooks.org/wiki/Algorithms" class="extiw external" title="wikibooks:Algorithms">Algorithms</a></div></div>

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<div class="side-box-text plainlist">At <a href="Wikiversity" title="Wikiversity">Wikiversity</a>, you can learn more and teach others about Algorithm at the <a href="https://en.wikiversity.org/wiki/Topic:Algorithm" class="extiw external" title="v:Topic:Algorithm">Department of Algorithm</a></div></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Algorithms" class="extiw external" title="commons:Category:Algorithms">Algorithms</a>.</div></div>

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<ul><li><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Algorithm">"Algorithm"</a>. <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>. <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>. 2001 [1994].</cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Algorithm.html">"Algorithm"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="https://www.nist.gov/dads/">Dictionary of Algorithms and Data Structures</a> – <a href="National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a></li></ul>

<dl><dt>Algorithm repositories</dt></dl>

<ul><li><a rel="nofollow" class="external text" href="http://www.cs.sunysb.edu/~algorith/">The Stony Brook Algorithm Repository</a> – <a href="State_University_of_New_York_at_Stony_Brook" class="mw-redirect" title="State University of New York at Stony Brook">State University of New York at Stony Brook</a></li>

<li><a rel="nofollow" class="external text" href="http://calgo.acm.org/">Collected Algorithms of the ACM</a> – <a href="Association_for_Computing_Machinery" title="Association for Computing Machinery">Associations for Computing Machinery</a></li>

<li><a rel="nofollow" class="external text" href="http://www-cs-staff.stanford.edu/~knuth/sgb.html">The Stanford GraphBase</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151206222112/http://www-cs-staff.stanford.edu/%7Eknuth/sgb.html">Archived</a> December 6, 2015, at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – <a href="Stanford_University" title="Stanford University">Stanford University</a></li></ul>

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<ul><li><a href="Algorithm_design" class="mw-redirect" title="Algorithm design">design</a></li>

<li><a href="Analysis_of_algorithms" title="Analysis of algorithms">analysis</a></li></ul></li>

<li><a href="Automata_theory" title="Automata theory">Automata theory</a></li>

<li><a href="Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li>

<li><a href="Coding_theory" title="Coding theory">Coding theory</a></li>

<li><a href="Computational_geometry" title="Computational geometry">Computational geometry</a></li>

<li><a href="Constraint_satisfaction_problem" title="Constraint satisfaction problem">Constraint satisfaction</a>

<ul><li><a href="Constraint_programming" title="Constraint programming">Constraint programming</a></li></ul></li>

<li><a href="Logic_in_computer_science" title="Logic in computer science">Computational logic</a></li>

<li><a href="Cryptography" title="Cryptography">Cryptography</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a></li>

<li><a href="Computational_statistics" title="Computational statistics">Statistics</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Mathematicalsoftware51" scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_software" title="Mathematical software">Mathematical software</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_arbitrary-precision_arithmetic_software" title="List of arbitrary-precision arithmetic software">Arbitrary-precision arithmetic</a></li>

<li><a href="List_of_finite_element_software_packages" title="List of finite element software packages">Finite element analysis</a></li>

<li><a href="Tensor_software" title="Tensor software">Tensor software</a></li>

<li><a href="List_of_interactive_geometry_software" title="List of interactive geometry software">Interactive geometry software</a></li>

<li><a href="List_of_optimization_software" title="List of optimization software">Optimization software</a></li>

<li><a href="List_of_statistical_software" title="List of statistical software">Statistical software</a></li>

<li><a href="List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical-analysis software</a></li>

<li><a href="List_of_numerical-analysis_software" title="List of numerical-analysis software">Numerical libraries</a></li>

<li><a href="Solver" title="Solver">Solvers</a></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Computer_algebra" title="Computer algebra">Computer algebra</a></li>

<li><a href="Computational_number_theory" title="Computational number theory">Computational number theory</a></li>

<li><a href="Combinatorics" title="Combinatorics">Combinatorics</a></li>

<li><a href="Graph_theory" title="Graph theory">Graph theory</a></li>

<li><a href="Discrete_geometry" title="Discrete geometry">Discrete geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approximation_theory" title="Approximation theory">Approximation theory</a></li>

<li><a href="Clifford_analysis" title="Clifford analysis">Clifford analysis</a>

<ul><li><a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li></ul></li>

<li><a href="Differential_equation" title="Differential equation">Differential equations</a>

<ul><li><a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a></li>

<li><a href="Partial_differential_equation" title="Partial differential equation">Partial differential equations</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equations</a></li></ul></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential geometry</a>

<ul><li><a href="Differential_form" title="Differential form">Differential forms</a></li>

<li><a href="Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li>

<li><a href="Geometric_analysis" title="Geometric analysis">Geometric analysis</a></li></ul></li>

<li><a href="Dynamical_system" title="Dynamical system">Dynamical systems</a>

<ul><li><a href="Chaos_theory" title="Chaos theory">Chaos theory</a></li>

<li><a href="Control_theory" title="Control theory">Control theory</a></li></ul></li>

<li><a href="Functional_analysis" title="Functional analysis">Functional analysis</a>

<ul><li><a href="Operator_algebra" title="Operator algebra">Operator algebra</a></li>

<li><a href="Operator_theory" title="Operator theory">Operator theory</a></li></ul></li>

<li><a href="Harmonic_analysis_(mathematics)" class="mw-redirect" title="Harmonic analysis (mathematics)">Harmonic analysis</a>

<ul><li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li></ul></li>

<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a>

<ul><li><a href="Exterior_algebra" title="Exterior algebra">Exterior</a></li>

<li><a href="Geometric_algebra" title="Geometric algebra">Geometric</a></li>

<li><a href="Tensor" title="Tensor">Tensor</a></li>

<li><a href="Vector_calculus#Vector_algebra" title="Vector calculus">Vector</a></li></ul></li>

<li><a href="Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a>

<ul><li><a href="Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior</a></li>

<li><a href="Geometric_calculus" title="Geometric calculus">Geometric</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li>

<li><a href="Vector_calculus" title="Vector calculus">Vector</a></li></ul></li>

<li><a href="Numerical_analysis" title="Numerical analysis">Numerical analysis</a>

<ul><li><a href="Numerical_linear_algebra" title="Numerical linear algebra">Numerical linear algebra</a></li>

<li><a href="Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">Numerical methods for ordinary differential equations</a></li>

<li><a href="Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">Numerical methods for partial differential equations</a></li>

<li><a href="Validated_numerics" title="Validated numerics">Validated numerics</a></li></ul></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Variational calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Probability_theory" title="Probability theory">Probability theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Probability_distribution" title="Probability distribution">Distributions</a> (<a href="Random_variable" title="Random variable">random variables</a>)</li>

<li><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a> / <a href="Stochastic_calculus" title="Stochastic calculus">analysis</a></li>

<li><a href="Functional_integration" title="Functional integration">Path integral</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Stochastic variational calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_physics" title="Mathematical physics">Mathematical physics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a>

<ul><li><a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a></li>

<li><a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a></li></ul></li>

<li><a href="Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">Field theory</a>

<ul><li><a href="Classical_field_theory" title="Classical field theory">Classical</a></li>

<li><a href="Conformal_field_theory" title="Conformal field theory">Conformal</a></li>

<li><a href="Effective_field_theory" title="Effective field theory">Effective</a></li>

<li><a href="Gauge_theory" title="Gauge theory">Gauge</a></li>

<li><a href="Quantum_field_theory" title="Quantum field theory">Quantum</a></li>

<li><a href="Statistical_field_theory" title="Statistical field theory">Statistical</a></li>

<li><a href="Topological_field_theory" class="mw-redirect" title="Topological field theory">Topological</a></li></ul></li>

<li><a href="Perturbation_theory" title="Perturbation theory">Perturbation theory</a>

<ul><li><a href="Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">in quantum mechanics</a></li></ul></li>

<li><a href="Potential_theory" title="Potential theory">Potential theory</a></li>

<li><a href="String_theory" title="String theory">String theory</a>

<ul><li><a href="Bosonic_string_theory" title="Bosonic string theory">Bosonic</a></li>

<li><a href="Topological_string_theory" title="Topological string theory">Topological</a></li></ul></li>

<li><a href="Supersymmetry" title="Supersymmetry">Supersymmetry</a>

<ul><li><a href="Supersymmetric_quantum_mechanics" title="Supersymmetric quantum mechanics">Supersymmetric quantum mechanics</a></li>

<li><a href="Supersymmetric_theory_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">Supersymmetric theory of stochastic dynamics</a></li></ul></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Algebraicstructures48" scope="row" class="navbox-group" style="width:1%"><a href="Algebraic_structure" title="Algebraic structure">Algebraic structures</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li>

<li><a href="Path_integral_formulation" title="Path integral formulation">Feynman integral</a></li>

<li><a href="Poisson_algebra" title="Poisson algebra">Poisson algebra</a></li>

<li><a href="Quantum_group" title="Quantum group">Quantum group</a></li>

<li><a href="Renormalization_group" title="Renormalization group">Renormalization group</a></li>

<li><a href="Particle_physics_and_representation_theory" title="Particle physics and representation theory">Representation theory</a></li>

<li><a href="Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li>

<li><a href="Superalgebra" title="Superalgebra">Superalgebra</a></li>

<li><a href="Supersymmetry_algebra" title="Supersymmetry algebra">Supersymmetry algebra</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Decision_theory" title="Decision theory">Decision sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Game_theory" title="Game theory">Game theory</a></li>

<li><a href="Operations_research" title="Operations research">Operations research</a></li>

<li><a href="Mathematical_optimization" title="Mathematical optimization">Optimization</a></li>

<li><a href="Social_choice_theory" title="Social choice theory">Social choice theory</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li>

<li><a href="Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li>

<li><a href="Mathematical_psychology" title="Mathematical psychology">Psychology</a></li>

<li><a href="Mathematical_sociology" title="Mathematical sociology">Sociology</a></li>

<li>"<a href="The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences" title="The Unreasonable Effectiveness of Mathematics in the Natural Sciences">The Unreasonable Effectiveness of Mathematics in the Natural Sciences</a>"</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mathematics" title="Mathematics">Mathematics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>

<ul><li><a href="Japan_Society_for_Industrial_and_Applied_Mathematics" title="Japan Society for Industrial and Applied Mathematics">Japan Society for Industrial and Applied Mathematics</a></li></ul></li>

<li><a href="Soci%C3%A9t%C3%A9_de_Math%C3%A9matiques_Appliqu%C3%A9es_et_Industrielles" title="Société de Mathématiques Appliquées et Industrielles">Société de Mathématiques Appliquées et Industrielles</a></li>

<li><a href="International_Council_for_Industrial_and_Applied_Mathematics" title="International Council for Industrial and Applied Mathematics">International Council for Industrial and Applied Mathematics</a></li>

<li>European Community on Computational Methods in Applied Sciences</li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li>Category</li>

<li><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a> / <a href="Topic_outline_of_mathematics" class="mw-redirect" title="Topic outline of mathematics">outline</a> / <a href="List_of_mathematics_topics" class="mw-redirect" title="List of mathematics topics">topics list</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Data_structures_and_algorithms145" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_structures_and_algorithms145" style="font-size:114%;margin:0 4em"><a href="Data_structure" title="Data structure">Data structures</a> and </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Data structures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Array_(data_structure)" title="Array (data structure)">Array</a></li>

<li><a href="Associative_array" title="Associative array">Associative array</a></li>

<li><a href="Binary_search_tree" title="Binary search tree">Binary search tree</a></li>

<li><a href="Fenwick_tree" title="Fenwick tree">Fenwick tree</a></li>

<li><a href="Graph_(abstract_data_type)" title="Graph (abstract data type)">Graph</a></li>

<li><a href="Hash_table" title="Hash table">Hash table</a></li>

<li><a href="Linked_list" title="Linked list">Linked list</a></li>

<li><a href="Queue_(abstract_data_type)" title="Queue (abstract data type)">Queue</a></li>

<li><a href="Segment_tree" title="Segment tree">Segment tree</a></li>

<li><a href="Stack_(abstract_data_type)" title="Stack (abstract data type)">Stack</a></li>

<li><a href="String_(computer_science)" title="String (computer science)">String</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algorithms and <a href="Algorithmic_paradigm" title="Algorithmic paradigm">algorithmic paradigms</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Backtracking" title="Backtracking">Backtracking</a></li>

<li><a href="Binary_search" title="Binary search">Binary search</a></li>

<li><a href="Breadth-first_search" title="Breadth-first search">Breadth-first search</a></li>

<li><a href="Brute-force_search" title="Brute-force search">Brute-force search</a></li>

<li><a href="Depth-first_search" title="Depth-first search">Depth-first search</a></li>

<li><a href="Divide-and-conquer_algorithm" title="Divide-and-conquer algorithm">Divide and conquer</a></li>

<li><a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li>

<li><a href="Graph_traversal" title="Graph traversal">Graph traversal</a></li>

<li><a href="Greedy_algorithm" title="Greedy algorithm">Greedy</a></li>

<li><a href="Hash_function" title="Hash function">Hash function</a></li>

<li><a href="Minimax" title="Minimax">Minimax</a></li>

<li><a href="Online_algorithm" title="Online algorithm">Online</a></li>

<li><a href="Randomized_algorithm" title="Randomized algorithm">Randomized</a></li>

<li><a href="Recursion_(computer_science)" title="Recursion (computer science)">Recursion</a></li>

<li><a href="Root-finding_algorithm" title="Root-finding algorithm">Root-finding</a></li>

<li><a href="Sorting_algorithm" title="Sorting algorithm">Sorting</a></li>

<li><a href="Streaming_algorithm" title="Streaming algorithm">Streaming</a></li>

<li><a href="Sweep_line_algorithm" title="Sweep line algorithm">Sweep line</a></li>

<li><a href="String-searching_algorithm" title="String-searching algorithm">String-searching</a></li>

<li><a href="Topological_sorting" title="Topological sorting">Topological sorting</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="List_of_data_structures" title="List of data structures">List of data structures</a></li>

<li><a href="List_of_algorithms" title="List of algorithms">List of algorithms</a></li></ul>

</div></td></tr></tbody></table></div>

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