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{{Short description|Sequence of operations for a task}} | |||
] are often used to represent algorithms.]] | |||
{{Redirect|Algorithms|the subfield of computer science|Analysis of algorithms|other uses|Algorithm (disambiguation)}} | |||
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] 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 ] and ], an '''algorithm''' ({{IPAc-en|audio=en-us-algorithm.ogg|ˈ|æ|l|ɡ|ə|r|ɪ|ð|əm}}) is a finite sequence of ] instructions, typically used to solve a class of specific ]s or to perform a ].<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 ]s and ]. More advanced algorithms can use ]s to divert the code execution through various routes (referred to as ]) and deduce valid ]s (referred to as ]). | |||
In contrast, a ] is an approach to solving problems that do not have 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 ]s are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. | |||
In ] and ] an '''algorithm''' is a finite ] of well-defined instructions for accomplishing some task which, given an initial state, will ] in a corresponding recognizable end-state. Algorithms can be implemented by ]s. | |||
As an ], 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 ]<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 ].<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 ]),<ref>"An algorithm has ] or more inputs, i.e., ] which are given to it initially before the algorithm begins" (Knuth 1973:5).</ref> the instructions describe a computation that, when ]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 ]; some algorithms, known as ]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> | |||
Informally, the concept of an algorithm is often illustrated by the example of a ], although many algorithms are much more complex; algorithms often have steps that repeat (]) or require decisions (such as ] or ]). | |||
== Etymology == | |||
The concept of algorithms was formalized in ] by ]'s ] and ]'s ], starting the field of ]. | |||
Around 825 AD, Persian scientist and polymath ] 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-Khwarizmi texts involving the ] and ] appeared, for example ''Liber Alghoarismi de practica arismetrice'', attributed to ], and ''Liber Algorismi de numero Indorum'', attributed to ].<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> Hereby, ''alghoarismi'' or ''algorismi'' is the ] of Al-Khwarizmi's name; the text starts with the phrase ''Dixit Algorismi'', or "Thus spoke Al-Khwarizmi".<ref name=":2" /> Around 1230, the English word '']'' is attested and then by ] 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}} | |||
== Definition == | |||
Different algorithms may complete the same task with a different set of instructions in more or less time, space, or effort than others. For example, given two different recipes for making potato salad, one may have ''peel the potato'' before ''boil the potato'' while the other presents the steps in the reverse order, yet they both call for these steps to be repeated for all potatoes and end when the potato salad is ready to be eaten. <!-- poor example .. who would boil each potato separately? and making a salad in general requires no cooking ... --> | |||
{{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 | reason = Stone (1972) suggests on page 4: "...any sequence of instructions that a robot can obey, is called an algorithm"|date=July 2020}} which would include all ]s (including programs that do not perform numeric calculations), and any prescribed ] procedure<ref> | |||
== Formalization of algorithms == | |||
{{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> | |||
or ] ].<ref> | |||
{{cite book |last1=Dietrich |first1=Eric |url=https://books.google.com/books?id=-wt1aZrGXLYC |title=The MIT Encyclopedia of the Cognitive Sciences |publisher=MIT Press |year=1999 |isbn=9780262731447 |editor1-last=Wilson |editor1-first=Robert Andrew |series=MIT Cognet library |location=Cambridge, Massachusetts |publication-date=2001 |page=11 |chapter=Algorithm |quote=An algorithm is a recipe, method, or technique for doing something. |access-date=22 July 2020 |editor2-last=Keil |editor2-first=Frank C.}} | |||
</ref> In general, a program is an algorithm only if it stops eventually<ref>Stone requires that "it must terminate in a finite number of steps" (Stone 1973:7–8).</ref>—even though ]s may sometimes prove desirable. {{Harvtxt|Boolos|Jeffrey|1974, 1999|ref=CITEREFBoolosJeffrey1999}} 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''.''<ref>Boolos and Jeffrey 1974,1999:19</ref> | |||
Most algorithms are intended to be ]ed as ]s. However, algorithms are also implemented by other means, such as in a ] (for example, the ] performing ] or an insect looking for food), in an ], or a mechanical device. | |||
Algorithms are essential to the way ]s process information, because a ] is essentially an algorithm that tells the computer what specific steps to perform (in what specific order) in order to carry out a | |||
specified task, such as calculating employees’ paychecks or printing students’ report cards. Thus, an algorithm can be considered to be any sequence of operations which can be performed by a ] system. | |||
== History == | |||
Typically, when an algorithm is associated with processing information, data is read from an input source or device, written to an output sink or device, and/or stored for further use. Stored data is regarded as part of the internal state of the entity performing the algorithm. The state is stored in a ]. | |||
{{Missing information|1=section|2=20th and 21st century development of computer algorithms|date=October 2023}} | |||
=== Ancient algorithms === | |||
Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in ] (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> ] (around 1550 BC),<ref name="Springer Science & Business Media" /> ] (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). . 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}}</ref> ] (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> ],<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 }}</ref> and ] (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 ] mathematics. A ]ian clay tablet found in ] near ] and dated to {{Circa|2500 BC}} describes the earliest ].<ref name="Springer Science & Business Media" /> During the ] {{Circa|1800|1600 BC|lk=no}}, ]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 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> | |||
For any such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable). | |||
Algorithms for arithmetic are also found in ancient ], dating back to the ] {{Circa|1550 BC|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient ]. Two examples are the ], which was described in the '']'' by ],<ref>{{cite web |last=Ast |first=Courtney |title=Eratosthenes |url=http://www.math.wichita.edu/history/men/eratosthenes.html |url-status=live |archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html |archive-date=February 27, 2015 |access-date=February 27, 2015 |publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp|Ch 9.2}} and the ], which was first described in '']'' ({{circa|300 BC|lk=no}}).<ref name="Cooke2005" />{{rp|Ch 9.1}}Examples of ancient Indian mathematics included the ], the ], and the ].<ref name=":6" /> | |||
Because an algorithm is a precise list of precise steps, the order of computation will almost always be critical to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting 'from the top' and going 'down to the bottom', an idea that is described more formally by '']''. | |||
The first cryptographic algorithm for deciphering encrypted code was developed by ], a 9th-century Arab mathematician, in ''A Manuscript On Deciphering Cryptographic Messages''. He gave the first description of ] by ], the earliest codebreaking algorithm.<ref name="Dooley" /> | |||
So far, this discussion of the formalization of an algorithm has assumed the premises of ]. This is the most common conception, and it attempts to describe a task in discrete, 'mechanical' means. Unique to this conception of formalized algorithms is the ], setting the value of a variable. It derives from the intuition of ']' as a scratchpad. There is an example below of such an assignment. | |||
=== Computers === | |||
See ] and ] for alternate conceptions of what constitutes an algorithm. | |||
=== |
==== Weight-driven clocks ==== | ||
Bolter credits the invention of the weight-driven clock as "the key invention ]]," specifically the ] 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 ]" in the 13th century and "computational machines"—the ] and ]s of ] and ] 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 ] computer instead of just a ]. Although a 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 ==== | |||
Algorithms are not only implemented as ]s, but often also by other means, such as in a biological ] (for example, the ] implementing ] or an insect relocating food), in ]s, or in a mechanical device. | |||
Bell and Newell (1971) write that the ], a precursor to ]s (punch cards), and "telephone switching technologies" led to the development of the first computers.<ref>Bell and Newell diagram 1971:39, cf. Davis 2000</ref> By the mid-19th century, the ], the precursor of the telephone, was in use throughout the world. By the late 19th century, the ] ({{circa|1870s}}) was in use, as were Hollerith cards (c. 1890). Then came the ] ({{circa|1910|lk=no}}) with its punched-paper use of ] on tape. | |||
Telephone-switching networks of ] were invented in 1835. These led to the invention of the digital adding device by ] 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".<ref>Melina Hill, Valley News Correspondent, ''A Tinkerer Gets a Place in History'', Valley News West Lebanon NH, Thursday, March 31, 1983, p. 13.</ref><ref>Davis 2000:14</ref> | |||
The ] is one discipline of ], and is often practiced abstractly (without the use of a specific ] or other implementation). In this sense, it resembles other mathematical disciplines in that the analysis focuses on the underlying principles of the algorithm, and not on any particular implementation. One way to embody (or sometimes ''codify'') an algorithm is the writing of ]. | |||
=== Formalization === | |||
Some writers restrict the definition of ''algorithm'' to procedures that eventually finish. Others include procedures that could run forever without stopping, arguing that some entity may be required to carry out such permanent tasks. In the latter case, success can no longer be defined in terms of halting with a meaningful output. Instead, terms of success that allow for unbounded output sequences must be defined. For example, an algorithm that verifies if there are more zeros than ones in an infinite random binary sequence must run forever to be effective. If it is implemented correctly, however, the algorithm's output will be useful: for as long as it examines the sequence, the algorithm will give a positive response while the number of examined zeros outnumber the ones, and a negative response otherwise. Success for this algorithm could then be defined as eventually outputting only positive responses if there are actually more zeros than ones in the sequence, and in any other case outputting any mixture of positive and negative responses. | |||
]'s diagram from "]", the first published computer algorithm]] | |||
In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the ''] ''(decision problem) posed by ]. Later formalizations were framed as attempts to define "]"<ref>Kleene 1943 in Davis 1965:274</ref> or "effective method".<ref>Rosser 1939 in Davis 1965:225</ref> Those formalizations included the ]–]–] recursive functions of 1930, 1934 and 1935, ]'s ] of 1936, ]'s ] of 1936, and ]'s ] of 1936–37 and 1939. | |||
Summarizing the above discussion about what algorithm should consist. | |||
==Representations== | |||
* Zero or more Inputs | |||
Algorithms can be expressed in many kinds of notation, including ], ], ]s, ]s, ] or ]s (processed by ]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. | |||
* One or more Outputs | |||
* Finiteness or computability | |||
* Definitiveness or Preciseness | |||
== |
=== Turing machines === | ||
There are many possible representations and ] programs can be expressed as a sequence of machine tables (see ], ], and ] for more), as flowcharts and drakon-charts (see ] for more), as a form of rudimentary ] or ] 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 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 === | |||
One of the simplest algorithms is to find the largest number in an (unsorted) list of numbers. The solution necessarily requires looking at every number in the list, but only once at each. From this follows a simple algorithm: | |||
The graphical aid called 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. | |||
== Algorithmic analysis == | |||
# Look at each item in the list. If it is larger than any that has been seen so far, make a note of it. | |||
{{Main|Analysis of algorithms}} | |||
# The latest noted item is the largest in the list when the process is complete. | |||
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 {{tmath|O(n)}}, using ]. 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 {{tmath|O(1)}}, otherwise {{tmath|O(n)}} is required. | |||
And here is a more formal coding of the algorithm in ]: | |||
Different algorithms may complete the same task with a different set of instructions in less or more time, space, or ']' than others. For example, a ] algorithm (with cost {{tmath|O(\log n)}}) outperforms a sequential search (cost {{tmath|O(n)}} ) when used for ]s on sorted lists or arrays. | |||
'''Algorithm''' LargestNumber | |||
Input: A non-empty list of numbers ''L''. | |||
Output: The ''largest'' number in the list ''L''. | |||
''largest'' ← -∞ | |||
'''for each''' ''item'' '''in''' list ''L'', '''do''' | |||
'''if''' the ''item'' > ''largest'', '''then''' | |||
''largest'' ← the ''item'' | |||
'''return''' ''largest'' | |||
===Formal versus empirical=== | |||
Notes on notation: | |||
{{Main|Empirical algorithmics|Profiling (computer programming)|Program optimization}} | |||
*"←" is a loose shorthand for "changes to". For instance, with "''largest'' ← the ''item''", it means that the ''largest'' number found so far changes to this ''item''. | |||
*"'''return'''" terminates the algorithm and outputs the value listed behind it. | |||
The ]s is a discipline of ]. Algorithms are often studied abstractly, without referencing any specific ] or implementation. Algorithm analysis resembles other mathematical disciplines as it focuses on the algorithm's properties, not implementation. ] is typical for analysis as it is a simple and general representation. Most algorithms are implemented on particular hardware/software platforms and their ] 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. | |||
As it happens, most people who implement algorithms want to know how much of a particular resource (such as time or storage) a given algorithm requires. Methods have been developed for the ] to obtain such quantitative answers; for example, the algorithm above has a time requirement of O(''n''), using the ] with ''n'' as the length of the list. At all times the algorithm only needs to remember two values: the largest number found so far, and its current position in the input list. Therefore this algorithm has a space requirement of ''O(log n)'', since a number from 1 to ''n'' takes ''log n'' bits to store. (Note that the size of the inputs is not counted as space used by the algorithm.) | |||
Empirical testing is useful for uncovering unexpected interactions that affect performance. ]s may be used to compare before/after potential improvements to an algorithm after program optimization. | |||
For a more complex example see ], which also happens to be one of the oldest algorithms. | |||
Empirical tests cannot replace formal analysis, though, and are non-trivial to perform fairly.<ref name="KriegelSchubert2016">{{cite journal|last1=Kriegel|first1=Hans-Peter|author-link=Hans-Peter Kriegel|last2=Schubert|first2=Erich|last3=Zimek|first3=Arthur|author-link3=Arthur Zimek|title=The (black) art of run-time evaluation: Are we comparing algorithms or implementations?|journal=Knowledge and Information Systems|volume=52|issue=2|year=2016|pages=341–378|issn=0219-1377|doi=10.1007/s10115-016-1004-2|s2cid=40772241}}</ref> | |||
== |
=== Execution efficiency === | ||
{{Main|Algorithmic efficiency}} | |||
To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to ] 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, ], Dina Katabi, and Eric Price, " {{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 {{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. | |||
The word ''algorithm'' comes from the name of the 9th century ] mathematician ]. The word '']'' originally referred only to the rules of performing ] using ] but evolved via European Latin translation of al-Khwarizmi's name into ''algorithm'' by the 18th century. The word evolved to include all definite procedures for solving problems or performing tasks. | |||
== Design == | |||
The first case of an algorithm written for a ] was ]'s ] written in 1842, for which she is considered by many to be the world's first ]. However, since ] never completed his ] the algorithm was never implemented on it. | |||
{{See also|Algorithm#By design paradigm}} | |||
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 ] or ] within ]. Techniques for designing and implementing algorithm designs are also called algorithm design patterns,<ref>{{cite book |last1=Goodrich |first1=Michael T. |author1-link=Michael T. Goodrich |url=http://ww3.algorithmdesign.net/ch00-front.html |title=Algorithm Design: Foundations, Analysis, and Internet Examples |last2=Tamassia |first2=Roberto |author2-link=Roberto Tamassia |publisher=John Wiley & Sons, Inc. |year=2002 |isbn=978-0-471-38365-9 |access-date=June 14, 2018 |archive-url=https://web.archive.org/web/20150428201622/http://ww3.algorithmdesign.net/ch00-front.html |archive-date=April 28, 2015 |url-status=live}}</ref> 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 ] is used to describe e.g., an algorithm's run-time growth as the size of its input increases.<ref>{{Cite web |title=Big-O notation (article) {{!}} Algorithms |url=https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-o-notation |access-date=2024-06-03 |website=Khan Academy |language=en}}</ref> | |||
The lack of ] in the "well-defined procedure" definition of algorithms posed some difficulties for mathematicians and ]ians of the ] and early ]. This problem was largely solved with the description of the ], an abstract model of a ] formulated by ], and the demonstration that every method yet found for describing "well-defined procedures" advanced by other mathematicians could be emulated on a Turing machine (a statement known as the ]). | |||
=== Structured programming === | |||
Nowadays, a formal criterion for an algorithm is that it is a procedure that can be implemented on a completely specified Turing machine or one of the equivalent ]s. Turing's initial interest was in the ]: deciding when an algorithm describes a terminating procedure. In practical terms ] matters more: it includes the problems called ], which are generally presumed to take more than ] for any (deterministic) algorithm. NP denotes the class of decision problems that can be solved by a non-deterministic Turing machine in polynomial time. | |||
Per the ], any algorithm can be computed by any ] 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 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".<ref>] and ] 1985 ''Back to Basic: The History, Corruption, and Future of the Language'', Addison-Wesley Publishing Company, Inc. Reading, MA, {{ISBN|0-201-13433-0}}.</ref> Tausworthe augments the three ]:<ref>Tausworthe 1977:101</ref> SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.<ref>Tausworthe 1977:142</ref> An additional benefit of a structured program is that it lends itself to ]s using ].<ref>Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff and Chapter 9.1</ref> | |||
== |
== Legal status == | ||
{{see also|Software patent}} | |||
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 '']''). However practical applications of algorithms are sometimes patentable. For example, in '']'', the application of a simple ] algorithm to aid in the curing of ] was deemed patentable. The ] is controversial,<ref>{{Cite news |date=2013-05-16 |title=The Experts: Does the Patent System Encourage Innovation? |url=https://www.wsj.com/articles/SB10001424127887323582904578487200821421958 |access-date=2017-03-29 |work=] |issn=0099-9660}}</ref> and there are criticized patents involving algorithms, especially ] algorithms, such as ]'s ]. Additionally, some cryptographic algorithms have export restrictions (see ]). | |||
There are many ways to classify algorithms, and the merits of each classification have been the subject of ongoing debate. | |||
== Classification == | |||
=== By implementation === | |||
; Recursion | |||
: A ] invokes itself repeatedly until meeting a termination condition, and is a common ] method. ] algorithms use repetitions such as ]s or data structures like ]s to solve problems. Problems may be suited for one implementation or the other. The ] 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 ] or ] 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 | |||
: ]s solve the problem with exact decision at every step; whereas ]s solve problems via guessing. Guesses are typically made more accurate through the use of ]. | |||
; Exact or approximate | |||
: While many algorithms reach an exact solution, ]s seek an approximation that is close to the true solution. Such algorithms have practical value for many hard problems. For example, the ], 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 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 | |||
: ]s run on a realistic model of ]. The term is usually used for those algorithms which seem inherently quantum or use some essential feature of ] such as ] or ]. | |||
=== By design paradigm === | |||
One way to classify algorithms is by implementation means. | |||
Another way of classifying algorithms is by their design methodology or ]. Some common paradigms are: | |||
; ] or exhaustive search | |||
* '''Recursion''' vs. '''Iteration''': A ] is one that invokes (makes reference to) itself repeatedly until a certain condition matches, which is a method common to ]. Iterative algorithms use repetitive constructs like loops and possibly with data structures like stack to solve the problems. Some problems are naturally suited for one implementation to other. For example, ] is well understood in recursive implementation. Every recursive version has an equivalent (but possibly more complex) iterative version. | |||
: 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 ] repeatedly reduces a problem to one or more smaller instances of itself (usually ]) until the instances are small enough to solve easily. ] is an example of divide and conquer, where an unordered list can be divided into segments containing one item and sorting of entire list can be obtained by merging the segments. A simpler variant of divide and conquer is called a ''decrease-and-conquer algorithm'', which solves one smaller instance of itself, and uses the solution to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithms.{{Citation needed|date=October 2024}} An example of a decrease and conquer algorithm is the ]. | |||
; Search and enumeration | |||
: Many problems (such as playing ]s) can be modelled as problems on ]s. A ] specifies rules for moving around a graph and is useful for such problems. This category also includes ]s, ] enumeration, and ]. | |||
;] | |||
: 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 ].<ref>For instance, the ] of 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 journal | |||
| last1 = Dyer | first1 = Martin | |||
| last2 = Frieze | first2 = Alan | |||
| last3 = Kannan | first3 = Ravi | |||
| date = January 1991 | |||
| doi = 10.1145/102782.102783 | |||
| issue = 1 | |||
| journal = J. ACM | |||
| pages = 1–17 | |||
| title = A Random Polynomial-time Algorithm for Approximating the Volume of Convex Bodies | |||
| volume = 38| citeseerx = 10.1.1.145.4600| s2cid = 13268711 | |||
}}</ref> Whether randomized algorithms with ] can be the fastest algorithm for some problems is an open question known as the ]. There are two large classes of such algorithms: | |||
# ]s return a correct answer with high probability. E.g. ] is the subclass of these that run in ]. | |||
# ]s always return the correct answer, but their running time is only probabilistically bound, e.g. ]. | |||
; ] | |||
: This technique transforms difficult problems into better-known problems solvable with (hopefully) ] algorithms. The goal is to find a reducing algorithm whose ] is not dominated by the resulting reduced algorithms. For example, one ] finds the median of an unsorted list by first sorting the list (the expensive portion), then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as '']''. | |||
; ] | |||
: In this approach, multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution. | |||
=== Optimization problems === | |||
* '''Serial''' vs. '''Parallel''': Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time. Those computers are sometimes called serial computers. An algorithm designed for such an environment is called a serial algorithm, as opposed to ]s, which take advantage of computer architectures where several processors can work on a problem at the same time. Parallel algorithms divide the problem into more symmetrical or asymmetrical subproblems and pass them to many processors and put the results back together at one end. The resource consumption in parallel algorithms is both processor cycles on each processors and also the communication overhead between the processors. Sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Recursive algorithms are generally parallelizable. Some problems have no parallel algorithms, and are called inherently serial problems. Those problems cannot be solved faster by employing more processors. One such example is ] of a graph, which happens to be recursive, but can ''not'' be parallelized. | |||
For ]s 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: | |||
; ] | |||
* '''Deterministic''' vs. '''Approximate''' vs. '''Random''': Deterministic algorithms solve the problem with exact decision at every step of the algorithm. Random algorithms as their name suggests explore the search space randomly until the solution is found. The various heuristic algorithms would probably also fall into random category, as their name (e.g. a genetic algorithm) describes its implementation. Approximate algorithms follow either deterministic or random strategy and get to solution with some degree of confidence. Such algorithms have practical value for hard problems. | |||
: 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 ].<ref> | |||
] and Mukund N. Thapa. 2003. ''Linear Programming 2: Theory and Extensions''. Springer-Verlag.</ref> Problems that can be solved with linear programming include the ] for directed graphs. If a problem also requires that any of the unknowns be ], then it is classified in ]. 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. | |||
; ] | |||
: When a problem shows optimal substructures—meaning the optimal solution can be constructed from optimal solutions to subproblems—and ]s, meaning the same subproblems are used to solve many different problem instances, a quicker approach called ''dynamic programming'' avoids recomputing solutions. For example, ], the shortest path between a start and goal vertex in a weighted ] can be found using the shortest path to the goal from all adjacent vertices. Dynamic programming and ] 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 | |||
: ], 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 ]. The most popular use of greedy algorithms is finding minimal spanning trees of graphs without negative cycles. ], ], ], ] are greedy algorithms that can solve this optimization problem. | |||
;The heuristic method | |||
:In ]s, ]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 ], ], ], and ]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 ]. | |||
== Examples == | |||
=== Classification by design paradigm === | |||
{{Further|List of algorithms}} | |||
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: | |||
Another way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories will include many different types of algorithms. Some commonly found paradigms include: | |||
''High-level description:'' | |||
* '''Divide and conquer'''. A ] repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually ]), until the instances are small enough to solve easily. One such example of divide and conquer is merge sorting. Sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in conquer phase by merging them. A simpler variant of divide and conquer is called '''decrease and conquer algorithm''', that solves an identical subproblem and uses the solution of this subproblem to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so conquer stage will be more complex than decrease and conquer algorithms. An example of decrease and conquer algorithm is ]. | |||
# If a set of numbers is empty, then there is no highest number. | |||
* ''']'''. When a problem shows ], meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems, and ], meaning the same subproblems are used to solve many different problem instances, we can often solve the problem quickly using ''dynamic programming'', an approach that avoids recomputing solutions that have already been computed. For example, the shortest path to a goal from a vertex in a weighted ] can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and ] go together. One main difference between dynamic programming and divide and conquer is, subproblems are more or less independent in divide and conquer, where as repetition of subproblems occur in dynamic programming. The only difference between dynamic programming and straightforward recursion is in caching or memoization of recursive calls. Where subproblems are independent, memoization does not help, so dynamic programming is not a solution for all. By using memoization or maintaining a table of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity. | |||
# Assume the first number in the set is the largest. | |||
* '''The greedy method'''. A ] is similar to a ], but the difference is that solutions to the subproblems do not have to be known at each stage; instead a "greedy" choice can be made of what looks best for the moment. Difference between dynamic programming and greedy method is, it extends the solution with the best possible decision (not all feasible decisions) at a algorithmic stage based on the current local optimum and the best decision (not all possible decisions) made in previous stage. It is not exhaustive, and does not give accurate answer to many problems. But when it works, it will be the fastest method. The most popular greedy algorithm is finding the minimal spanning tree as given by ]. | |||
# For each remaining number in the set: if this number is greater than the current largest, it becomes the new largest. | |||
* '''Linear programming'''. When solving a problem using ], the program is put into a number of linear ] and then an attempt is made to maximize (or minimize) the inputs. Many problems (such as the ] for directed ]) can be stated in a linear programming way, and then be solved by a 'generic' algorithm such as the ]. A complex variant of linear programming is called integer programming, where the solution space is restricted to all integers. | |||
# When there are no unchecked numbers left in the set, consider the current largest number to be the largest in the set. | |||
* ''']''': It is another powerful technique in solving many problems by transforming one problem into another problem. For example, one ] for finding the median in an unsorted list is first translating this problem into sorting problem and finding the middle element in sorted list. The key of reduction algorithms is finding the simple transformation such that complexity of reduction algorithm does not dominate the complexity of reduced algorithm. | |||
* '''Search and enumeration'''. Many problems (such as playing ]) can be modeled as problems on ]. A ] specifies rules for moving around a graph and is useful for such problems. This category also includes the ]s and ]. | |||
* '''The probabilistic and heuristic paradigm'''. Algorithms belonging to this class fit the definition of an algorithm more loosely. | |||
# ]s are those that make some choices randomly (or pseudo-randomly); for some problems, it can in fact be proven that the fastest solutions must involve some ]. | |||
# ]s attempt to find solutions to problems by mimicking biological ]ary processes, with a cycle of random mutations yielding successive generations of "solutions". Thus, they emulate reproduction and "survival of the fittest". In ], this approach is extended to algorithms, by regarding the algorithm itself as a "solution" to a problem. Also there are | |||
# ] algorithms, whose general purpose is not to find a optimal solution, but an approximate solution where the time or resources to find a perfect solution are not practical. An example of this would be ], ], or ] algorithms, a class of heuristic probabilistic algorithms that vary the solution of a problem by a random amount. The name "simulated annealing" alludes to the metallurgic term meaning the heating and cooling of metal to achieve freedom from defects. The purpose of the random variance is to find close to globally optimal solutions rather than simply locally optimal ones, the idea being that the random element will be decreased as the algorithm settles down to a solution. | |||
''(Quasi-)formal description:'' | |||
=== Classification by field of study === | |||
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 ] or ]: | |||
{{algorithm-begin|name=LargestNumber}} | |||
Every field of science has its own problems and needs efficient algorithms. Some of these fields are overlapping with each other. Related problems in one field are often studied together. Some example classes are ]s, ]s, ]s, ], ], ], ], ], ], ], ] algorithms and ]. | |||
Input: A list of numbers ''L''. | |||
Output: The largest number in the list ''L''. | |||
'' |
'''if''' ''L.size'' = 0 '''return''' null | ||
''largest'' ← ''L'' | |||
'''for each''' ''item'' '''in''' ''L'', '''do''' | |||
'''if''' ''item'' > ''largest'', '''then''' | |||
''largest'' ← ''item'' | |||
'''return''' ''largest'' | |||
{{algorithm-end}} | |||
== See also == | |||
=== Classification by complexity === | |||
{{Portal|Mathematics|Computer programming}} | |||
{{div col|colwidth=22em}} | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* '']'' (textbook) | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
* ] | |||
** ] | |||
** ] | |||
{{div col end}} | |||
== Notes == | |||
Some algorithms complete in linear time, and some complete in exponential amount of time, and some never complete. One problem may have multiple algorithms, and some problems may have no algorithms. Some problems have no known efficient algorithms. There are also mappings from some problems to other problems. So computer scientists found it is suitable to classify the problems rather than algorithms into equivalance classes based on the complexity. | |||
{{Reflist}} | |||
== Bibliography == | |||
''See also:'' ''']es''' for more details. | |||
{{refbegin|30em}} | |||
* {{cite journal | last1 = Axt | first1 = P | year = 1959 | title = On a Subrecursive Hierarchy and Primitive Recursive Degrees | journal = Transactions of the American Mathematical Society | volume = 92 | issue = 1| pages = 85–105 | doi=10.2307/1993169| jstor = 1993169 | doi-access = free}} | |||
* Bell, C. Gordon and Newell, Allen (1971), ''Computer Structures: Readings and Examples'', McGraw–Hill Book Company, New York. {{ISBN|0-07-004357-4}}. | |||
* {{Cite journal|author1-link=Andreas Blass|first1=Andreas|last1=Blass|author2-link=Yuri Gurevich|first2=Yuri|last2=Gurevich|year=2003|url=http://research.microsoft.com/~gurevich/Opera/164.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://research.microsoft.com/~gurevich/Opera/164.pdf |archive-date=2022-10-09 |url-status=live|title=Algorithms: A Quest for Absolute Definitions|journal= Bulletin of European Association for Theoretical Computer Science|volume= 81}} Includes a bibliography of 56 references. | |||
* {{cite book| last = Bolter| first = David J.| title = Turing's Man: Western Culture in the Computer Age| edition = 1984| year = 1984| publisher = The University of North Carolina Press|location= Chapel Hill, NC| isbn = 978-0-8078-1564-9 }}, {{ISBN|0-8078-4108-0}} | |||
* {{cite book| last1 = Boolos| first1 = George| last2 = Jeffrey| first2 = Richard| title = Computability and Logic| url = https://archive.org/details/computabilitylog0000bool_r8y9| url-access = registration| edition = 4th| orig-year = 1974| year = 1999| publisher = Cambridge University Press, London| isbn = 978-0-521-20402-6| author1-link = George Boolos| author2-link = Richard Jeffrey }}: cf. Chapter 3 ''Turing machines'' where they discuss "certain enumerable sets not effectively (mechanically) enumerable". | |||
* {{cite book| last = Burgin| first = Mark| title = Super-Recursive Algorithms| year = 2004| publisher = Springer| isbn = 978-0-387-95569-8 }} | |||
* Campagnolo, M.L., ], 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 | |||
* {{Cite journal|last=Church|first=Alonzo|author-link=Alonzo Church|title=An Unsolvable Problem of Elementary Number Theory|journal=American Journal of Mathematics|volume=58|pages= 345–363|year=1936|doi=10.2307/2371045|issue=2|jstor=2371045}} 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. | |||
* {{Cite journal|last=Church|first=Alonzo|author-link=Alonzo Church|title=A Note on the Entscheidungsproblem|journal=The Journal of Symbolic Logic|volume=1|year=1936|pages=40–41|doi=10.2307/2269326|issue=1|jstor=2269326|s2cid=42323521 }} {{cite journal|last=Church|first=Alonzo|title=Correction to a Note on the Entscheidungsproblem|journal=The Journal of Symbolic Logic|volume=1|year=1936|pages=101–102|doi=10.2307/2269030|issue=3|jstor=2269030|s2cid=5557237 }} Reprinted in ''The Undecidable'', p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes. | |||
* {{cite book| last = Daffa'| first = Ali Abdullah al-| title = The Muslim contribution to mathematics| year = 1977| publisher = Croom Helm| location = London| isbn = 978-0-85664-464-1 }} | |||
* {{cite book| last = Davis| first = Martin| author-link = Martin Davis (mathematician)| title = The Undecidable: Basic Papers On Undecidable Propositions, Unsolvable Problems and Computable Functions| url = https://archive.org/details/undecidablebasic0000davi| url-access = registration| year = 1965| publisher = Raven Press| location = New York| isbn = 978-0-486-43228-1 }} Davis gives commentary before each article. Papers of ], ], ], ], ], and ] are included; those cited in the article are listed here by author's name. | |||
* {{cite book| last = Davis| first = Martin| author-link = Martin Davis (mathematician)| title = Engines of Logic: Mathematicians and the Origin of the Computer| year = 2000| publisher = W.W. Nortion| location = New York| isbn = 978-0-393-32229-3 }} Davis offers concise biographies of ], ], ], ], ], Gödel and Turing with ] as the show-stealing villain. Very brief bios of ], ], ], ], ], etc. | |||
* {{DADS|algorithm|algorithm}} | |||
* {{cite journal|title= Evolution and moral diversity |author=Dean, Tim |journal=Baltic International Yearbook of Cognition, Logic and Communication|year=2012|volume=7|doi=10.4148/biyclc.v7i0.1775 |doi-access=free}} | |||
* {{cite book| last = Dennett| first = Daniel| author-link = Daniel Dennett| title = Darwin's Dangerous Idea| pages = –36| year = 1995| publisher = Touchstone/Simon & Schuster| location = New York| isbn = 978-0-684-80290-9| url = https://archive.org/details/darwinsdangerous0000denn| url-access = registration}} | |||
* {{cite book| last = Dilson| first = Jesse| title = The Abacus| edition = (1968, 1994)| year = 2007| publisher = St. Martin's Press, NY| isbn = 978-0-312-10409-2| url = https://archive.org/details/abacusworldsfirs0000dils}}, {{ISBN|0-312-10409-X}} | |||
* ], , ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources. | |||
* {{cite book| last = van Heijenoort| first = Jean| author-link = Jean van Heijenoort| title = From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931| edition = (1967)| year = 2001| publisher = Harvard University Press, Cambridge| isbn = 978-0-674-32449-7 }}, 3rd edition 1976, {{ISBN|0-674-32449-8}} (pbk.) | |||
* {{cite book| last = Hodges| first = Andrew| author-link = Andrew Hodges| title = Alan Turing: The Enigma| year = 1983| publisher = ]| location = New York| isbn = 978-0-671-49207-6| title-link = Alan Turing: The Enigma}}, {{ISBN|0-671-49207-1}}. Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof. | |||
* {{Cite journal|last=Kleene|first=Stephen C.|author-link=Stephen Kleene|title=General Recursive Functions of Natural Numbers|journal=Mathematische Annalen|volume=112|pages=727–742|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002278499&L=1|year=1936|doi=10.1007/BF01565439|issue=5|s2cid=120517999|access-date=September 30, 2013|archive-url=https://web.archive.org/web/20140903092121/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002278499&L=1|archive-date=September 3, 2014|url-status=dead}} 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). | |||
* {{Cite journal|last=Kleene|first=Stephen C.|author-link=Stephen Kleene |title= Recursive Predicates and Quantifiers|journal= Transactions of the American Mathematical Society|volume=53|pages=41–73|year=1943 |doi= 10.2307/1990131|issue=1|jstor=1990131|doi-access=free}} 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 ]). | |||
* {{cite book| last = Kleene| first = Stephen C.| author-link = Kleene| title = Introduction to Metamathematics| edition = Tenth|year= 1991| orig-year = 1952| publisher = North-Holland Publishing Company| isbn = 978-0-7204-2103-3 }} | |||
* {{cite book| last = Knuth| first = Donald| author-link = Donald Knuth| title = Fundamental Algorithms, Third Edition| year = 1997| publisher = Addison–Wesley| location = Reading, Massachusetts| isbn = 978-0-201-89683-1 }} | |||
* {{Cite book|last=Knuth|first=Donald|author-link=Donald Knuth|title=Volume 2/Seminumerical Algorithms, The Art of Computer Programming First Edition|publisher=Addison–Wesley|location=Reading, Massachusetts|year=1969}} | |||
* Kosovsky, N.K. ''Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms'', LSU Publ., Leningrad, 1981 | |||
* {{Cite journal|last=Kowalski|first=Robert|author-link=Robert Kowalski|title=Algorithm=Logic+Control|journal=]|volume=22|issue=7|pages=424–436|year=1979|doi=10.1145/359131.359136|s2cid=2509896|doi-access=free}} | |||
* ] (1954) ''Theory of algorithms''. Imprint Moscow, Academy of Sciences of the USSR, 1954 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. | |||
* {{cite book| last = Minsky| first = Marvin| author-link = Marvin Minsky| title = Computation: Finite and Infinite Machines| url = https://archive.org/details/computationfinit0000mins| url-access = registration| edition = First| year = 1967| publisher = Prentice-Hall, Englewood Cliffs, NJ| isbn = 978-0-13-165449-5 }} Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 ''Computability, Effective Procedures and Algorithms. Infinite machines.'' | |||
* {{Cite journal|last=Post|first=Emil|author-link=Emil Post|title=Finite Combinatory Processes, Formulation I |journal=The Journal of Symbolic Logic |volume=1 |year=1936 |pages=103–105 |doi=10.2307/2269031 |issue=3 |jstor=2269031|s2cid=40284503 }} 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 ]. | |||
* {{Cite book|last=Rogers|first=Hartley Jr.|title=Theory of Recursive Functions and Effective Computability|publisher=The MIT Press|year=1987|isbn=978-0-262-68052-3}} | |||
* {{Cite journal|last=Rosser|first=J.B.|author-link=J. B. Rosser|title=An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem|journal=Journal of Symbolic Logic|volume= 4 |issue=2|year=1939|doi=10.2307/2269059|pages=53–60|jstor=2269059|s2cid=39499392 }} 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'') | |||
* {{cite book |last=Santos-Lang |first=Christopher |editor1-first=Simon |editor1-last=van Rysewyk |editor2-first=Matthijs |editor2-last=Pontier |title=Machine Medical Ethics |volume=74 |publisher=Springer | location=Switzerland | pages=111–127 | chapter=Moral Ecology Approaches to Machine Ethics| chapter-url=http://grinfree.com/MoralEcology.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://grinfree.com/MoralEcology.pdf |archive-date=2022-10-09 |url-status=live | doi=10.1007/978-3-319-08108-3_8|series=Intelligent Systems, Control and Automation: Science and Engineering |date=2015 |isbn=978-3-319-08107-6 }} | |||
* {{Cite book|last=Scott|first=Michael L.|title=Programming Language Pragmatics |edition=3rd |publisher=Morgan Kaufmann Publishers/Elsevier|year=2009|isbn=978-0-12-374514-9}} | |||
* {{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=978-0-07-061726-1|year=1972}} 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). | |||
* {{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=]|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. 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. | |||
* {{Cite journal|last=Turing|first=Alan M.|author-link=A. M. Turing|title=Systems of Logic Based on Ordinals|journal=Proceedings of the London Mathematical Society|volume=45|pages=161–228|year=1939|doi=10.1112/plms/s2-45.1.161|hdl=21.11116/0000-0001-91CE-3|hdl-access=free}} Reprinted in ''The Undecidable'', pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton. | |||
* ] (2006), , Manual of Patent Examining Procedure (MPEP). Latest revision August 2006 | |||
{{refend|30em}} | |||
* 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. https://doi.org/10.2307/3027363 | |||
==Further reading== | |||
== Legal issues == | |||
{{refbegin}} | |||
* {{cite book |last=Bellah |first=Robert Neelly |year=1985 |author-link=Robert N. Bellah |title=Habits of the Heart: Individualism and Commitment in American Life |location=Berkeley |isbn=978-0-520-25419-0 |publisher=University of California Press |url=https://books.google.com/books?id=XsUojihVZQcC }} | |||
Some countries allow algorithms to be ] when embodied in software or in hardware. Patents have long been a controversial issue (see, for example, the ]). | |||
* {{cite book |last=Berlinski |first=David |title=The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer |year=2001 |publisher=Harvest Books |isbn=978-0-15-601391-8 |url=https://archive.org/details/adventofalgorith0000berl }} | |||
* {{cite book |last=Chabert |first=Jean-Luc |title=A History of Algorithms: From the Pebble to the Microchip |year=1999 |publisher=Springer Verlag |isbn=978-3-540-63369-3}} | |||
Some countries do not allow certain algorithms, such as cryptographic algorithms, to be ] from that country. | |||
* {{cite book |author1=Thomas H. Cormen |author2=Charles E. Leiserson |author3=Ronald L. Rivest |author4=Clifford Stein |title=Introduction To Algorithms |edition=3rd |year=2009 |publisher=MIT Press |isbn=978-0-262-03384-8}} | |||
* {{cite book |author=Harel, David |author2=Feldman, Yishai |title=Algorithmics: The Spirit of Computing |year=2004 |publisher=Addison-Wesley |isbn=978-0-321-11784-7}} | |||
==See also== | |||
* {{cite book |last1=Hertzke |first1=Allen D. |last2=McRorie |first2=Chris |year=1998 |editor1-last=Lawler |editor1-first=Peter Augustine |editor2-last=McConkey |editor2-first=Dale |chapter=The Concept of Moral Ecology |title=Community and Political Thought Today |location=Westport, CT |publisher=] }} | |||
* ] | |||
* Jon Kleinberg, Éva Tardos(2006): ''Algorithm Design'', Pearson/Addison-Wesley, ISBN 978-0-32129535-4 | |||
* ] | |||
* ] (2000). '' {{Webarchive|url=https://web.archive.org/web/20170701190647/http://www-cs-faculty.stanford.edu/~uno/aa.html |date=July 1, 2017 }}''. Stanford, California: Center for the Study of Language and Information. | |||
* ] | |||
* Knuth, Donald E. (2010). '' {{Webarchive|url=https://web.archive.org/web/20170716225848/http://www-cs-faculty.stanford.edu/~uno/da.html |date=July 16, 2017 }}''. Stanford, California: Center for the Study of Language and Information. | |||
* ] | |||
* {{Cite book |first1=Wendell |last1=Wallach |first2=Colin |last2=Allen |date=November 2008 |title=Moral Machines: Teaching Robots Right from Wrong |isbn=978-0-19-537404-9 |publisher=Oxford University Press |location=US }} | |||
* ] | |||
* {{cite book |author=Bleakley, Chris |title=Poems that Solve Puzzles: The History and Science of Algorithms |year=2020 |publisher=Oxford University Press |isbn=978-0-19-885373-2 |url=https://books.google.com/books?id=3pr5DwAAQBAJ }} | |||
* ] | |||
{{refend}} | |||
==References== | |||
* ] | |||
==External links== | ==External links== | ||
{{wiktionary}} | |||
* {{DADS|algorithm|algorithm}} | |||
{{wikibooks|Algorithms}} | |||
* Algana.co.uk Free source code for algorithm Java applets and C++ modules | |||
{{Wikiversity department}} | |||
* Gaston H. Gonnet and Ricardo Baeza-Yates: Example programs from Free source code for many important algorithms. | |||
{{Commons category|Algorithms}} | |||
* . "This is a dictionary of algorithms, algorithmic techniques, data structures, archetypical problems, and related definitions." | |||
* {{springer|title=Algorithm|id=p/a011780|mode=cs1}} | |||
* | |||
* {{MathWorld | urlname=Algorithm | title=Algorithm}} | |||
* | |||
* – ] | |||
* - Discuss ideas, algorithms, challenges related to programming. Also announcements about Online Programming Contests will be posted in this group. | |||
; Algorithm repositories | |||
* An interesting way of using algorithms to make music. | |||
* – ] | |||
* is a web site dedicated to algorithms. | |||
* – ] | |||
* {{Webarchive|url=https://web.archive.org/web/20151206222112/http://www-cs-staff.stanford.edu/%7Eknuth/sgb.html |date=December 6, 2015 }} – ] | |||
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Latest revision as of 03:47, 9 December 2024
Sequence of operations for a task "Algorithms" redirects here. For the subfield of computer science, see Analysis of algorithms. For other uses, see Algorithm (disambiguation).
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages)
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In mathematics and computer science, an algorithm (/ˈælɡərɪðəm/ ) is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning).
In contrast, a heuristic is an approach to solving problems that do not have well-defined correct or optimal results. For example, although social media recommender systems 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 and in a well-defined formal language for calculating a function. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.
Etymology
Around 825 AD, Persian scientist and polymath Muḥammad ibn Mūsā al-Khwārizmī wrote kitāb al-ḥisāb al-hindī ("Book of Indian computation") and kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ("Addition and subtraction in Indian arithmetic"). In the early 12th century, Latin translations of said al-Khwarizmi 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. Hereby, alghoarismi or algorismi is the Latinization of Al-Khwarizmi's name; the text starts with the phrase Dixit Algorismi, or "Thus spoke Al-Khwarizmi". Around 1230, the English word algorism is attested and then by Chaucer in 1391, English adopted the French term. In the 15th century, under the influence of the Greek word ἀριθμός (arithmos, "number"; cf. "arithmetic"), the Latin word was altered to algorithmus.
Definition
For a detailed presentation of the various points of view on the definition of "algorithm", see Algorithm characterizations.One informal definition is "a set of rules that precisely defines a sequence of operations", which would include all computer programs (including programs that do not perform numeric calculations), and any prescribed bureaucratic procedure or cook-book recipe. In general, a program is an algorithm only if it stops eventually—even though infinite loops may sometimes prove desirable. Boolos, Jeffrey & 1974, 1999 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.
Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain performing arithmetic or an insect looking for food), in an electrical circuit, or a mechanical device.
History
This section is missing information about 20th and 21st century development of computer algorithms. Please expand the section to include this information. Further details may exist on the talk page. (October 2023) |
Ancient algorithms
Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in Babylonian mathematics (around 2500 BC), Egyptian mathematics (around 1550 BC), Indian mathematics (around 800 BC and later), the Ifa Oracle (around 500 BC), Greek mathematics (around 240 BC), Chinese mathematics (around 200 BC and later), and Arabic mathematics (around 800 AD).
The earliest evidence of algorithms is found in ancient Mesopotamian mathematics. A Sumerian clay tablet found in Shuruppak near Baghdad and dated to c. 2500 BC describes the earliest division algorithm. During the Hammurabi dynasty c. 1800 – c. 1600 BC, Babylonian clay tablets described algorithms for computing formulas. 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.
Algorithms for arithmetic are also found in ancient Egyptian mathematics, dating back to the Rhind Mathematical Papyrus c. 1550 BC. Algorithms were later used in ancient Hellenistic mathematics. Two examples are the Sieve of Eratosthenes, which was described in the Introduction to Arithmetic by Nicomachus, and the Euclidean algorithm, which was first described in Euclid's Elements (c. 300 BC).Examples of ancient Indian mathematics included the Shulba Sutras, the Kerala School, and the Brāhmasphuṭasiddhānta.
The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm.
Computers
Weight-driven clocks
Bolter credits the invention of the weight-driven clock as "the key invention ," specifically the verge escapement mechanism producing the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata" in the 13th century and "computational machines"—the difference and analytical engines of Charles Babbage and Ada Lovelace in the mid-19th century. 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 a 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
Bell and Newell (1971) write that the Jacquard loom, a precursor to Hollerith cards (punch cards), and "telephone switching technologies" led to the development of the first computers. By the mid-19th century, the telegraph, the precursor of the telephone, was in use throughout the world. By the late 19th century, the ticker tape (c. 1870s) was in use, as were Hollerith cards (c. 1890). Then came the teleprinter (c. 1910) with its punched-paper use of Baudot code on tape.
Telephone-switching networks of electromechanical relays were invented in 1835. These led to the invention of the digital adding device by George Stibitz 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".
Formalization
In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the Entscheidungsproblem (decision problem) posed by David Hilbert. Later formalizations were framed as attempts to define "effective calculability" or "effective method". Those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, and Alan Turing's Turing machines of 1936–37 and 1939.
Representations
Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon-charts, programming languages or control tables (processed by interpreters). 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. A high-level description describes qualities of the algorithm itself, ignoring how it is implemented on the Turing machine. 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. In the most detail, a formal description gives the exact state table and list of transitions of the Turing machine.
Flowchart representation
The graphical aid called a flowchart 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.
Algorithmic analysis
Main article: Analysis of algorithmsIt 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 , using big O notation. 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 , otherwise is required.
Different algorithms may complete the same task with a different set of instructions in less or more time, space, or 'effort' than others. For example, a binary search algorithm (with cost ) outperforms a sequential search (cost ) when used for table lookups on sorted lists or arrays.
Formal versus empirical
Main articles: Empirical algorithmics, Profiling (computer programming), and Program optimizationThe analysis, and study of algorithms 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. Benchmarks 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.
Execution efficiency
Main article: Algorithmic efficiencyTo illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging. In general, speed improvements depend on special properties of the problem, which are very common in practical applications. Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.
Design
See also: Algorithm § By design paradigmAlgorithm 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 divide-and-conquer or dynamic programming within operation research. Techniques for designing and implementing algorithm designs are also called algorithm design patterns, 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 big O notation is used to describe e.g., an algorithm's run-time growth as the size of its input increases.
Structured programming
Per the Church–Turing thesis, any algorithm can be computed by any Turing complete 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 "spaghetti code", 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". Tausworthe augments the three Böhm-Jacopini canonical structures: SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction.
Legal status
See also: Software patentBy 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 Gottschalk v. Benson). However practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr, the application of a simple feedback algorithm to aid in the curing of synthetic rubber was deemed patentable. The patenting of software is controversial, and there are criticized patents involving algorithms, especially data compression algorithms, such as Unisys's LZW patent. Additionally, some cryptographic algorithms have export restrictions (see export of cryptography).
Classification
By implementation
- Recursion
- A recursive algorithm invokes itself repeatedly until meeting a termination condition, and is a common functional programming method. Iterative algorithms use repetitions such as loops or data structures like stacks 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 or 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 algorithms solve the problem with exact decision at every step; whereas non-deterministic algorithms 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 algorithms 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 weights of items as well as their value.
- Quantum algorithm
- Quantum algorithms run on a realistic model of quantum computation. The term is usually used for those algorithms which seem inherently quantum or use some essential feature of Quantum computing such as quantum superposition or quantum entanglement.
By design paradigm
Another way of classifying algorithms is by their design methodology or paradigm. Some common paradigms are:
- Brute-force or exhaustive search
- 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 recursively) until the instances are small enough to solve easily. Merge sorting is an example of divide and conquer, where an unordered list can be divided into segments containing one item and sorting of entire list can be obtained by merging the segments. A simpler variant of divide and conquer is called a decrease-and-conquer algorithm, which solves one smaller instance of itself, and uses the solution to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithms. An example of a decrease and conquer algorithm is the binary search algorithm.
- Search and enumeration
- Many problems (such as playing chess) can be modelled as problems on graphs. A graph exploration algorithm specifies rules for moving around a graph and is useful for such problems. This category also includes search algorithms, 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. Whether randomized algorithms with polynomial time complexity can be the fastest algorithm for some problems is an open question known as the P versus NP problem. There are two large classes of such algorithms:
- Monte Carlo algorithms return a correct answer with high probability. E.g. RP is the subclass of these that run in polynomial time.
- Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. ZPP.
- 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 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), then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as 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.
Optimization problems
For optimization problems 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:
- Linear programming
- 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 simplex algorithm. Problems that can be solved with linear programming include the maximum flow problem for directed graphs. If a problem also requires that any of the unknowns be integers, then it is classified in integer programming. 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.
- Dynamic programming
- When a problem shows optimal substructures—meaning the optimal solution can be constructed from optimal solutions to subproblems—and overlapping subproblems, 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 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 algorithms, 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 optima. The most popular use of greedy algorithms is finding minimal spanning trees of graphs without negative cycles. Huffman Tree, Kruskal, Prim, Sollin are greedy algorithms that can solve this optimization problem.
- The heuristic method
- In optimization problems, heuristic algorithms 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, tabu search, simulated annealing, and genetic algorithms. 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.
Examples
Further information: List of algorithmsOne 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:
- If a set of numbers is empty, then there is no highest number.
- Assume the first number in the set is the largest.
- For each remaining number in the set: if this number is greater than the current largest, it becomes the new largest.
- When there are no unchecked numbers left in the set, consider the current largest number to be the largest in the set.
(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 pseudocode or pidgin code:
Algorithm LargestNumber Input: A list of numbers L. Output: The largest number in the list L.
if L.size = 0 return null largest ← L for each item in L, do if item > largest, then largest ← item return largest
- "←" denotes assignment. For instance, "largest ← item" means that the value of largest changes to the value of item.
- "return" terminates the algorithm and outputs the following value.
See also
- Abstract machine
- ALGOL
- Algorithm aversion
- Algorithm engineering
- Algorithm characterizations
- Algorithmic bias
- Algorithmic composition
- Algorithmic entities
- Algorithmic synthesis
- Algorithmic technique
- Algorithmic topology
- Computational mathematics
- Garbage in, garbage out
- Introduction to Algorithms (textbook)
- Government by algorithm
- List of algorithms
- List of algorithm general topics
- Medium is the message
- Regulation of algorithms
- Theory of computation
Notes
- ^ "Definition of ALGORITHM". Merriam-Webster Online Dictionary. Archived from the original on February 14, 2020. Retrieved November 14, 2019.
- ^ David A. Grossman, Ophir Frieder, Information Retrieval: Algorithms and Heuristics, 2nd edition, 2004, ISBN 1402030045
- ^ "Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2).
- ^ 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).
- "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).
- "An algorithm has zero or more inputs, i.e., quantities which are given to it initially before the algorithm begins" (Knuth 1973:5).
- "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).
- "An algorithm has one or more outputs, i.e., quantities which have a specified relation to the inputs" (Knuth 1973:5).
- 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).
- Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247
- Stone 1973:4
-
Simanowski, Roberto (2018). The Death Algorithm and Other Digital Dilemmas. Untimely Meditations. Vol. 14. Translated by Chase, Jefferson. Cambridge, Massachusetts: MIT Press. p. 147. ISBN 9780262536370. Archived from the original on December 22, 2019. Retrieved May 27, 2019.
the next level of abstraction of central bureaucracy: globally operating algorithms.
-
Dietrich, Eric (1999). "Algorithm". In Wilson, Robert Andrew; Keil, Frank C. (eds.). The MIT Encyclopedia of the Cognitive Sciences. MIT Cognet library. Cambridge, Massachusetts: MIT Press (published 2001). p. 11. ISBN 9780262731447. Retrieved July 22, 2020.
An algorithm is a recipe, method, or technique for doing something.
- Stone requires that "it must terminate in a finite number of steps" (Stone 1973:7–8).
- Boolos and Jeffrey 1974,1999:19
- ^ Chabert, Jean-Luc (2012). A History of Algorithms: From the Pebble to the Microchip. Springer Science & Business Media. pp. 7–8. ISBN 9783642181924.
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- Hayashi, T. (2023, January 1). Brahmagupta. Encyclopedia Britannica.
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- ^ Cooke, Roger L. (2005). The History of Mathematics: A Brief Course. John Wiley & Sons. ISBN 978-1-118-46029-0.
- Chabert, Jean-Luc, ed. (1999). "A History of Algorithms". SpringerLink. doi:10.1007/978-3-642-18192-4. ISBN 978-3-540-63369-3.
- ^ Dooley, John F. (2013). A Brief History of Cryptology and Cryptographic Algorithms. Springer Science & Business Media. pp. 12–3. ISBN 9783319016283.
- Knuth, Donald E. (1972). "Ancient Babylonian Algorithms" (PDF). Commun. ACM. 15 (7): 671–677. doi:10.1145/361454.361514. ISSN 0001-0782. S2CID 7829945. Archived from the original (PDF) on December 24, 2012.
- Aaboe, Asger (2001). Episodes from the Early History of Astronomy. New York: Springer. pp. 40–62. ISBN 978-0-387-95136-2.
- Ast, Courtney. "Eratosthenes". Wichita State University: Department of Mathematics and Statistics. Archived from the original on February 27, 2015. Retrieved February 27, 2015.
- Bolter 1984:24
- Bolter 1984:26
- Bolter 1984:33–34, 204–206.
- Bell and Newell diagram 1971:39, cf. Davis 2000
- Melina Hill, Valley News Correspondent, A Tinkerer Gets a Place in History, Valley News West Lebanon NH, Thursday, March 31, 1983, p. 13.
- Davis 2000:14
- Kleene 1943 in Davis 1965:274
- Rosser 1939 in Davis 1965:225
- ^ Sipser 2006:157
- Kriegel, Hans-Peter; Schubert, Erich; Zimek, Arthur (2016). "The (black) art of run-time evaluation: Are we comparing algorithms or implementations?". Knowledge and Information Systems. 52 (2): 341–378. doi:10.1007/s10115-016-1004-2. ISSN 0219-1377. S2CID 40772241.
- Gillian Conahan (January 2013). "Better Math Makes Faster Data Networks". discovermagazine.com. Archived from the original on May 13, 2014. Retrieved May 13, 2014.
- Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price, "ACM-SIAM Symposium On Discrete Algorithms (SODA) Archived July 4, 2013, at the Wayback Machine, Kyoto, January 2012. See also the sFFT Web Page Archived February 21, 2012, at the Wayback Machine.
- Goodrich, Michael T.; Tamassia, Roberto (2002). Algorithm Design: Foundations, Analysis, and Internet Examples. John Wiley & Sons, Inc. ISBN 978-0-471-38365-9. Archived from the original on April 28, 2015. Retrieved June 14, 2018.
- "Big-O notation (article) | Algorithms". Khan Academy. Retrieved June 3, 2024.
- John G. Kemeny and Thomas E. Kurtz 1985 Back to Basic: The History, Corruption, and Future of the Language, Addison-Wesley Publishing Company, Inc. Reading, MA, ISBN 0-201-13433-0.
- Tausworthe 1977:101
- Tausworthe 1977:142
- Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff and Chapter 9.1
- "The Experts: Does the Patent System Encourage Innovation?". The Wall Street Journal. May 16, 2013. ISSN 0099-9660. Retrieved March 29, 2017.
- Kellerer, Hans; Pferschy, Ulrich; Pisinger, David (2004). Knapsack Problems | Hans Kellerer | Springer. Springer. doi:10.1007/978-3-540-24777-7. ISBN 978-3-540-40286-2. S2CID 28836720. Archived from the original on October 18, 2017. Retrieved September 19, 2017.
- 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 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. CiteSeerX 10.1.1.145.4600. doi:10.1145/102782.102783. S2CID 13268711.
- George B. Dantzig and Mukund N. Thapa. 2003. Linear Programming 2: Theory and Extensions. Springer-Verlag.
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Further reading
- Bellah, Robert Neelly (1985). Habits of the Heart: Individualism and Commitment in American Life. Berkeley: University of California Press. ISBN 978-0-520-25419-0.
- Berlinski, David (2001). The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer. Harvest Books. ISBN 978-0-15-601391-8.
- Chabert, Jean-Luc (1999). A History of Algorithms: From the Pebble to the Microchip. Springer Verlag. ISBN 978-3-540-63369-3.
- Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest; Clifford Stein (2009). Introduction To Algorithms (3rd ed.). MIT Press. ISBN 978-0-262-03384-8.
- Harel, David; Feldman, Yishai (2004). Algorithmics: The Spirit of Computing. Addison-Wesley. ISBN 978-0-321-11784-7.
- 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: Praeger.
- Jon Kleinberg, Éva Tardos(2006): Algorithm Design, Pearson/Addison-Wesley, ISBN 978-0-32129535-4
- Knuth, Donald E. (2000). Selected Papers on Analysis of Algorithms Archived July 1, 2017, at the Wayback Machine. Stanford, California: Center for the Study of Language and Information.
- Knuth, Donald E. (2010). Selected Papers on Design of Algorithms Archived July 16, 2017, at the Wayback Machine. Stanford, California: Center for the Study of Language and Information.
- Wallach, Wendell; Allen, Colin (November 2008). Moral Machines: Teaching Robots Right from Wrong. US: Oxford University Press. ISBN 978-0-19-537404-9.
- Bleakley, Chris (2020). Poems that Solve Puzzles: The History and Science of Algorithms. Oxford University Press. ISBN 978-0-19-885373-2.
External links
- "Algorithm". Encyclopedia of Mathematics. EMS Press. 2001 .
- Weisstein, Eric W. "Algorithm". MathWorld.
- Dictionary of Algorithms and Data Structures – National Institute of Standards and Technology
- Algorithm repositories
- The Stony Brook Algorithm Repository – State University of New York at Stony Brook
- Collected Algorithms of the ACM – Associations for Computing Machinery
- The Stanford GraphBase Archived December 6, 2015, at the Wayback Machine – Stanford University
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