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[[File:GCD through successive subtractions.svg|thumb|Flowchart of using successive subtractions to find the [[greatest common divisor]] of number ''r'' and ''s''|alt=In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again.]]

In [[mathematics]] and [[computer science]], an '''algorithm''' ({{IPAc-en|audio=en-us-algorithm.ogg|ˈ|æ|l|ɡ|ə|r|ɪ|ð|əm}}) is a finite sequence of [[Rigour#Mathematics|mathematically rigorous]] instructions, typically used to solve a class of specific [[Computational problem|problem]]s or to perform a [[computation]].<ref name=":0">{{Cite web|url=https://www.merriam-webster.com/dictionary/algorithm|title=Definition of ALGORITHM|work=Merriam-Webster Online Dictionary |language=en |access-date=2019-11-14 |archive-url=https://web.archive.org/web/20200214074446/https://www.merriam-webster.com/dictionary/algorithm |archive-date=February 14, 2020|url-status=live}}</ref> Algorithms are used as specifications for performing [[calculation]]s and [[data processing]]. More advanced algorithms can use [[Conditional (computer programming)|conditional]]s to divert the code execution through various routes (referred to as [[automated decision-making]]) and deduce valid [[inference]]s (referred to as [[automated reasoning]]).

In contrast, a [[Heuristic (computer science)|heuristic]] is an approach to solving problems that do not havewithout well-defined correct or optimal results.<ref name=":2">David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, {{isbn|1402030045}}</ref> For example, although social media [[recommender system]]s are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation.

As an [[effective method]], an algorithm can be expressed within a finite amount of space and time<ref name=":3">"Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2).</ref> and in a well-defined [[formal language]]<ref name=":4">Well defined concerning the agent that executes the algorithm: "There is a computing agent, usually human, which can react to the instructions and carry out the computations" (Rogers 1987:2).</ref> for calculating a [[Function (mathematics)|function]].<ref>"an algorithm is a procedure for computing a ''function'' (concerning some chosen notation for integers) ... this limitation (to numerical functions) results in no loss of generality", (Rogers 1987:1).</ref> Starting from an initial state and initial input (perhaps [[Empty string|empty]]),<ref>"An algorithm has [[zero]] or more inputs, i.e., [[Quantity|quantities]] which are given to it initially before the algorithm begins" (Knuth 1973:5).</ref> the instructions describe a computation that, when [[Execution (computing)|execute]]d, proceeds through a finite<ref>"A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method{{'"}} (Knuth 1973:5).</ref> number of well-defined successive states, eventually producing "output"<ref>"An algorithm has one or more outputs, i.e., quantities which have a specified relation to the inputs" (Knuth 1973:5).</ref> and terminating at a final ending state. The transition from one state to the next is not necessarily [[deterministic]]; some algorithms, known as [[randomized algorithm]]s, incorporate random input.<ref>Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without the use of continuous methods or analog devices ... carried forward deterministically, without resort to random methods or devices, e.g., dice" (Rogers 1987:2).</ref>

== Etymology ==

Around 825 AD, Persian scientist and polymath [[Al-Khwarizmi|Muḥammad ibn Mūsā al-Khwārizmī]] wrote ''kitāb al-ḥisāb al-hindī'' ("Book of Indian computation") and ''kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī'' ("Addition and subtraction in Indian arithmetic").<ref name=":0" /> In the early 12th century, Latin translations of said al-Khwarizmithese texts involving the [[Hindu–Arabic numeral system]] and [[arithmetic]] appeared, for example ''Liber Alghoarismi de practica arismetrice'', attributed to [[John of Seville]], and ''Liber Algorismi de numero Indorum'', attributed to [[Adelard of Bath]].<ref name=":1">Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247</ref> HerebyHere, ''alghoarismi'' or ''algorismi'' is the [[Latinisation of names|Latinization]] of Al-Khwarizmi's name;<ref name=":0" /> the text starts with the phrase ''Dixit Algorismi'', or "Thus spoke Al-Khwarizmi".<ref name=":2" /> Around 1230, the English word ''[[algorism]]'' is attested and then by [[Geoffrey Chaucer|Chaucer]] in 1391, English adopted the French term.<ref name=":3" /><ref name=":4" />{{Clarification needed|date=April 2024}} In the 15th century, under the influence of the Greek word ἀριθμός (''arithmos'', "number"; ''cf.'' "arithmetic"), the Latin word was altered to ''algorithmus''.{{Citation needed|date=April 2024}}

The word ''[[algorism]]'' in English came to mean the use of place-value notation in calculations; it occurs in the ''[[Ancrene Wisse]]'' from circa 1225.<ref>{{cite web|url=https://www.oed.com/dictionary/algorism_n?tl=true|title=algorism|work=Oxford English Dictionary|access-date=2025-05-18}}</ref> By the time [[Geoffrey Chaucer]] wrote ''[[The Canterbury Tales]]'' in the late 14th century, he used a variant of the same word in describing ''augrym stones'', stones used for place-value calculation.<ref>{{cite web|url=https://chaucer.fas.harvard.edu/pages/millers-prologue-and-tale|title=The Miller's Tale|at=Line 3210|first=Geoffrey|last=Chaucer}}</ref><ref>{{cite book|title=A Glossary of Tudor and Stuart Words: Especially from the Dramatists|editor-first=Anthony Lawson|editor-last=Mayhew|first=Walter William|last=Skeat|publisher=Clarendon Press|year=1914|contribution=agrim, agrum|pages=5–6|contribution-url=https://books.google.com/books?id=z58YAAAAIAAJ&pg=PA5}}</ref> In the 15th century, under the influence of the Greek word ἀριθμός (''arithmos'', "number"; ''cf.'' "arithmetic"), the Latin word was altered to ''algorithmus''.<ref>{{cite book

| last = Grabiner | first = Judith V. | author-link = Judith Grabiner

| editor-last = Matthews | editor-first = Michael R.

| contribution = The role of mathematics in liberal arts education

| date = December 2013

| doi = 10.1007/978-94-007-7654-8_25

| isbn = 9789400776548

| pages = 793–836

| publisher = Springer

| title = International Handbook of Research in History, Philosophy and Science Teaching}}</ref> By 1596, this form of the word was used in English, as ''algorithm'', by [[Thomas Hood (mathematician)|Thomas Hood]].<ref>{{cite web|url=https://www.oed.com/dictionary/algorithm_n|title=algorithm|work=Oxford English Dictionary|access-date=2025-05-18}}</ref>

== Definition ==

{{For|a detailed presentation of the various points of view on the definition of "algorithm"|Algorithm characterizations}}

One informal definition is "a set of rules that precisely defines a sequence of operations",<ref>Stone 1973:4</ref>{{request quotation sfnp| reason = Stone (1972) suggests on page 4: "...any sequence of instructions that a robot can obey, is called an algorithm"|date1971|p=July 20208}} which would include all [[computer program]]s (including programs that do not perform numeric calculations), and any prescribed [[bureaucratic]] procedure<ref>

{{cite book |last1=Simanowski |first1=Roberto |author-link1=Roberto Simanowski |url=https://books.google.com/books?id=RJV5DwAAQBAJ |title=The Death Algorithm and Other Digital Dilemmas |date=2018 |publisher=MIT Press |isbn=9780262536370 |series=Untimely Meditations |volume=14 |___location=Cambridge, Massachusetts |page=147 |translator1-last=Chase |translator1-first=Jefferson |quote=[...] the next level of abstraction of central bureaucracy: globally operating algorithms. |access-date=27 May 2019 |archive-url=https://web.archive.org/web/20191222120705/https://books.google.com/books?id=RJV5DwAAQBAJ |archive-date=December 22, 2019 |url-status=live}}

</ref>

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=== Ancient algorithms ===

Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book |last1=Chabert |first1=Jean-Luc |title=A History of Algorithms: From the Pebble to the Microchip |date=2012 |publisher=Springer Science & Business Media |isbn=9783642181924 |pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book |last1=Sriram |first1=M. S. |editor1-last=Emch |editor1-first=Gerard G. |editor2-last=Sridharan |editor2-first=R. |editor3-last=Srinivas |editor3-first=M. D. |title=Contributions to the History of Indian Mathematics |date=2005 |publisher=Springer |isbn=978-93-86279-25-5 |page=153 |chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 |language=en |chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal |last=Zaslavsky |first=Claudia |date=1970 |title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria |url=https://www.jstor.org/stable/3027363 |journal=The Two-Year College Mathematics Journal |volume=1 |issue=2 |pages=76–99 |doi=10.2307/3027363 |jstor=3027363 |issn=0049-4925|url-access=subscription }}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book|last=Cooke|first=Roger L.|title=The History of Mathematics: A Brief Course|date=2005|publisher=John Wiley & Sons|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal |date=1999 |editor-last=Chabert |editor-first=Jean-Luc |title=A History of Algorithms |url=https://link.springer.com/book/10.1007/978-3-642-18192-4 |journal=SpringerLink |language=en |doi=10.1007/978-3-642-18192-4|isbn=978-3-540-63369-3 |url-access=subscription }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book |last1=Dooley |first1=John F. |title=A Brief History of Cryptology and Cryptographic Algorithms |date=2013 |publisher=Springer Science & Business Media |isbn=9783319016283 |pages=12–3}}</ref>

The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty|Hammurabi dynasty]] {{Circa|1800|1600 BC|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal |last1=Knuth |first1=Donald E. |date=1972 |title=Ancient Babylonian Algorithms |url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf |url-status=dead |journal=Commun. ACM |volume=15 |issue=7 |pages=671–677 |doi=10.1145/361454.361514 |issn=0001-0782 |s2cid=7829945 |archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf |archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]]. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book |last=Aaboe |first=Asger |author-link=Asger Aaboe |title=Episodes from the Early History of Astronomy |date=2001 |publisher=Springer |isbn=978-0-387-95136-2 |place=New York |pages=40–62}}</ref>

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==== Weight-driven clocks ====

Bolter credits the invention of the weight-driven clock as "the key invention [of [[Europe in the middle ages|Europe in the Middle Ages]]]," specifically the [[verge escapement]] mechanism<ref>Bolter 1984:24</ref> producing the tick and tock of a mechanical clock. "The accurate automatic machine"<ref>Bolter 1984:26</ref> led immediately to "mechanical [[automata theory|automata]]" in the 13th century and "computational machines"—the [[difference engine|difference]] and [[analytical engine]]s of [[Charles Babbage]] and [[Ada Lovelace]] in the mid-19th century.<ref>Bolter 1984:33–34, 204–206.</ref> Lovelace designed the first algorithm intended for processing on a computer, Babbage's analytical engine, which is the first device considered a real [[Turing-complete]] computer instead of just a [[calculator]]. Although athe full implementation of Babbage's second device was not realized for decades after her lifetime, Lovelace has been called "history's first programmer".

==== Electromechanical relay ====

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==Representations==

Algorithms can be expressed in many kinds of notation, including [[natural languages]], [[pseudocode]], [[flowchart]]s, [[DRAKON|drakon-chart]]s, [[programming languages]] or [[control table]]s (processed by [[Interpreter (computing)|interpreter]]s). 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.

=== Turing machines ===

There are many possible representations and [[Turing machine]] programs can be expressed as a sequence of machine tables (see [[finite-state machine]], [[state-transition table]], and [[control table]] for more), as flowcharts and drakon-charts (see [[state diagram]] for more), as a form of rudimentary [[machine code]] or [[assembly code]] 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.<ref name=":5">Sipser 2006:157</ref> A high-level description describes the qualities of the algorithm itself, ignoring how it is implemented on the Turing machine.<ref name=":5" /> An implementation description describes the general manner in which the machine moves its head and stores data in order to carry out the algorithm, but does not give exact states.<ref name=":5" /> In the most detail, a formal description gives the exact state table and list of transitions of the Turing machine.<ref name=":5" />

=== Flowchart representation ===

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{{Main|Empirical algorithmics|Profiling (computer programming)|Program optimization}}

The [[analysis of algorithms|analysis, and study of algorithm]]s is a discipline of [[computer science]]. Algorithms are often studied abstractly, without referencing any specific [[programming language]] or implementation. Algorithm analysis resembles other mathematical disciplines as it focuses on the algorithm's properties, not implementation. [[Pseudocode]] is typical for analysis as it is a simple and general representation. Most algorithms are implemented on particular hardware/software platforms and their [[algorithmic efficiency]] 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. [[Benchmark (computing)|Benchmark]]s may be used to compare before/after potential improvements to an algorithm after program optimization.

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To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to [[Fast Fourier transform|FFT]] algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging.<ref>{{cite web| title=Better Math Makes Faster Data Networks| author=Gillian Conahan| date=January 2013| url=http://discovermagazine.com/2013/jan-feb/34-better-math-makes-faster-data-networks| publisher=discovermagazine.com| access-date=May 13, 2014| archive-url=https://web.archive.org/web/20140513212427/http://discovermagazine.com/2013/jan-feb/34-better-math-makes-faster-data-networks| archive-date=May 13, 2014| url-status=live}}</ref> In general, speed improvements depend on special properties of the problem, which are very common in practical applications.<ref name="Hassanieh12">Haitham Hassanieh, [[Piotr Indyk]], Dina Katabi, and Eric Price, "[http://siam.omnibooksonline.com/2012SODA/data/papers/500.pdf ACM-SIAM Symposium On Discrete Algorithms (SODA)] {{webarchive|url=https://web.archive.org/web/20130704180806/http://siam.omnibooksonline.com/2012SODA/data/papers/500.pdf |date=July 4, 2013 }}, Kyoto, January 2012. See also the [http://groups.csail.mit.edu/netmit/sFFT/ sFFT Web Page] {{Webarchive|url=https://web.archive.org/web/20120221145740/http://groups.csail.mit.edu/netmit/sFFT/ |date=February 21, 2012 }}.</ref> Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.

=== Best Case and Worst Case ===

{{Main|Best, worst and average case}}

The best case of an algorithm refers to the scenario or input for which the algorithm or data structure takes the least time and resources to complete its tasks.<ref>{{Cite web |title=Best Case |url=https://xlinux.nist.gov/dads/HTML/bestcase.html |access-date=29 May 2025 |website=Dictionary of Algorithms and Data Structures |publisher=National Institute of Standards and Technology (NIST) |agency=National Institute of Standards and Technology}}</ref> The worst case of an algorithm is the case that causes the algorithm or data structure to consume the maximum period of time and computational resources.<ref>{{Cite web |title=worst case |url=https://xlinux.nist.gov/dads/HTML/worstcase.html |access-date=29 May 2025 |website=Dictionary of Algorithms and Data Structures |publisher=National Institute of Standards and Technology (NIST) |agency=National Institute of Standards and Technology (NIST)}}</ref>

== Design ==

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=== By implementation ===

; Recursion

: A [[recursive algorithm]] invokes itself repeatedly until meeting a termination condition, and is a common [[functional programming]] method. [[Iteration|Iterative]] algorithms use repetitions such as [[Program loops|loop]]s or data structures like [[Stack (data structure)|stack]]s to solve problems. Problems may be suited for one implementation or the other. The [[Tower of Hanoi]] 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.

; Serial, parallel or distributed

: 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 [[parallel algorithm|parallel]] or [[distributed algorithm|distributed]] 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.

; Deterministic or non-deterministic

: [[Deterministic algorithm]]s solve the problem with exact decisiondecisions at every step; whereas [[non-deterministic algorithm]]s solve problems via guessing. Guesses are typically made more accurate through the use of [[heuristics]].

; Exact or approximate

: While many algorithms reach an exact solution, [[approximation algorithm]]s seek an approximation that is close to the true solution. Such algorithms have practical value for many hard problems. For example, the [[Knapsack problem]], 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.<ref>{{Cite book|url=https://www.springer.com/us/book/9783540402862|title=Knapsack Problems {{!}} Hans Kellerer {{!}} Springer|language=en|isbn=978-3-540-40286-2|publisher=Springer|year=2004|doi=10.1007/978-3-540-24777-7|access-date=September 19, 2017|archive-url=https://web.archive.org/web/20171018181055/https://www.springer.com/us/book/9783540402862|archive-date=October 18, 2017|url-status=live|last1=Kellerer|first1=Hans|last2=Pferschy|first2=Ulrich|last3=Pisinger|first3=David|s2cid=28836720 }}</ref>

; Quantum algorithm

: [[Quantum algorithm]]s run on a realistic model of [[quantum computation]]. The term is usually used for those algorithms whichthat seem inherently quantum or use some essential feature of [[Quantum computing]] such as [[quantum superposition]] or [[quantum entanglement]].

=== By design paradigm ===

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: 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.

; Divide and conquer

: A [[divide-and-conquer algorithm]] repeatedly reduces a problem to one or more smaller instances of itself (usually [[recursion|recursively]]) until the instances are small enough to solve easily. [[mergesort|Merge sorting]] is an example of divide and conquer, where an unordered list canis berepeatedly dividedsplit into segmentssmaller containinglists, onewhich itemare sorted in the same way and sortingthen ofmerged.<ref>{{cite entirebook|title=Algorithm listDesign: canFoundations, beAnalysis, obtainedand byInternet mergingExamples|first1=Michael theT.|last1=Goodrich|first2=Roberto|last2=Tamassia|publisher=John segmentsWiley & Sons|year=2001|isbn=9780471383659|contribution=5.2 ADivide and Conquer|page=263}}</ref> In a simpler variant of divide and conquer is called a[[prune and search]] or ''decrease-and-conquer algorithm'', which solves one smaller instance of itself, and usesdoes thenot solutionrequire toa solvemerge the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithmsstep.{{Citation neededsfnp|Goodrich|Tamassia|2001|dateloc=October4.7.1 2024Prune-and-search|p=245}} An example of a decreaseprune and conquersearch algorithm is the [[binary search algorithm]].

; Search and enumeration

: Many problems (such as playing [[Chess|ches]]s) can be modelled as problems on [[graph theory|graph]]s. A [[graph exploration algorithm]] specifies rules for moving around a graph and is useful for such problems. This category also includes [[search algorithm]]s, [[branch and bound]] enumeration, and [[backtracking]].

;[[Randomized algorithm]]

: 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 [[randomness]].<ref>For instance, the [[volume]] of a [[convex polytope]] (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 journal

| last1 = Dyer | first1 = Martin

| last2 = Frieze | first2 = Alan

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# [[Las Vegas algorithm]]s always return the correct answer, but their running time is only probabilistically bound, e.g. [[Zero-error Probabilistic Polynomial time|ZPP]].

; [[Reduction (complexity)|Reduction of complexity]]

: This technique transforms difficult problems into better-known problems solvable with (hopefully) [[asymptotically optimal]] algorithms. The goal is to find a reducing algorithm whose [[Computational complexity theory|complexity]] is not dominated by the resulting reduced algorithms. For example, one [[selection algorithm]] 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 ''[[Transform and conquer algorithm|transform and conquer]]''.

; [[Back tracking]]

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

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: When a problem shows optimal substructures—meaning the optimal solution can be constructed from optimal solutions to subproblems—and [[overlapping subproblem]]s, meaning the same subproblems are used to solve many different problem instances, a quicker approach called ''dynamic programming'' avoids recomputing solutions. For example, [[Floyd–Warshall algorithm]], the shortest path between a start and goal vertex in a weighted [[graph (discrete mathematics)|graph]] can be found using the shortest path to the goal from all adjacent vertices. Dynamic programming and [[memoization]] 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.

; The greedy method

: [[Greedy algorithm]]s, 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 [[local optimum|local optima]]. The most popular use of greedy algorithms is finding minimal spanning trees of graphs without negative cycles. [[Huffman coding|Huffman Tree]], [[kruskal's algorithm|Kruskal]], [[Prim's algorithm|Prim]], [[Sollin's algorithm|Sollin]] are greedy algorithms that can solve this optimization problem.

;The heuristic method

:In [[optimization problem]]s, [[heuristic algorithm]]s 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 [[local search (optimization)|local search]], [[tabu search]], [[simulated annealing]], and [[genetic algorithm]]s. 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 [[approximation algorithm]].

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* [[Abstract machine]]

* [[ALGOL]]

* [[Logic programming#Algorithm = Logic + Control|Algorithm = Logic + Control]]

* [[Algorithm aversion]]

* [[Algorithm engineering]]

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* {{cite book| last = Sipser| first = Michael| title = Introduction to the Theory of Computation| year = 2006| publisher = PWS Publishing Company| isbn = 978-0-534-94728-6| url = https://archive.org/details/introductiontoth00sips}}

* {{cite book |last1=Sober |first1=Elliott |last2=Wilson |first2=David Sloan |year=1998 |title=Unto Others: The Evolution and Psychology of Unselfish Behavior |url=https://archive.org/details/untoothersevolut00sobe |url-access=registration |___location=Cambridge |publisher=Harvard University Press|isbn=9780674930469 }}

* {{Cite book|last=Stone|first=Harold S.|title=Introduction to Computer Organization and Data Structures|edition=1972|publisher=McGraw-Hill, New York|isbn=978-0-07-061726-19780070617261|year=19721971}} 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.&nbsp;4).

* {{cite book| last = Tausworthe| first = Robert C| title = Standardized Development of Computer Software Part 1 Methods| year = 1977| publisher = Prentice–Hall, Inc.| ___location = Englewood Cliffs NJ| isbn = 978-0-13-842195-3 }}

* {{Cite journal|last=Turing|first=Alan M.|author-link=A. M. Turing|title=On Computable Numbers, With An Application to the Entscheidungsproblem|journal=[[Proceedings of the London Mathematical Society]]|series=Series 2|volume=42|pages= 230–265 |year=1936–37|doi=10.1112/plms/s2-42.1.230 |s2cid=73712 }}. Corrections, ibid, vol. 43(1937) pp.&nbsp;544–546. Reprinted in ''The Undecidable'', p.&nbsp;116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.

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{{wikibooks|Algorithms}}

{{Wikiversity department}}

{{Commons category|Algorithms}}

* {{springer|title=Algorithm|id=p/a011780|mode=cs1}}

* {{MathWorld | urlname=Algorithm | title=Algorithm}}