1598:. Parallel algorithms are algorithms that take advantage of computer architectures where multiple processors can work on a problem at the same time. Distributed algorithms are algorithms that use multiple machines connected to a computer network. Parallel and distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. For example, a CPU would be an example of a parallel algorithm. The resource consumption in such 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.
155:
817:
2133:
917:). 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 ways to express algorithms that avoid many of the ambiguities common in statements based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but they are also often used as a way to define or document algorithms.
2147:
4339:, 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,
511:
102:
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1858:. 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.
47:
971:(SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm–Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. The symbols and their use to build the canonical structures are shown in the diagram.
958:
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.
1701:, which 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 the conquer stage is more complex than decrease and conquer algorithms. An example of a decrease and conquer algorithm is the
957:
for more). Representations of algorithms 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
1901:
is similar to a dynamic programming algorithm in that it works by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution, which may be given or have been constructed in some way and improve it by making small modifications. For some
1743:
Such algorithms make some choices randomly (or pseudo-randomly). They can be very useful in finding approximate solutions for problems where finding exact solutions can be impractical (see heuristic method below). For some of these problems, it is known that the fastest approximations must involve
1678:
Brute force is a method of problem-solving that involves systematically trying every possible option until the optimal solution is found. This approach can be very time-consuming, as it requires going through every possible combination of variables. However, it is often used when other methods are
1886:
go together. The main difference between dynamic programming and divide and conquer is that subproblems are more or less independent in divide and conquer, whereas subproblems overlap in dynamic programming. The difference between dynamic programming and straightforward recursion is in caching or
1252:
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
1225:
is eventually put to the test using real code. For the solution of a "one-off" problem, the efficiency of a particular algorithm may not have significant consequences (unless n is extremely large) but for algorithms designed for fast interactive, commercial or long life scientific usage it may be
2435:
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
970:
offers a way to describe and document an algorithm (and a computer program corresponding to it). Like the program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only four: the directed arrow showing program flow, the rectangle
987:
It is frequently important to know how much of a particular resource (such as time or storage) is theoretically required for a given algorithm. Methods have been developed for the analysis of algorithms to obtain such quantitative answers (estimates); for example, an algorithm that adds up the
1936:
can be used to find a solution close to the optimal solution in cases where finding the optimal solution is impractical. These algorithms work by getting 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
1630:, where there is a set of given items. Its 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.
1279:. 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
1299:. In fact, it has been demonstrated that Turing completeness requires only 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 "
1841:
When searching for optimal solutions to a linear function bound to linear equality and inequality constraints, the constraints of the problem can be used directly in producing the optimal solutions. There are algorithms that can solve any problem in this category, such as the popular
4079:, 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
1972:
One of the simplest algorithms is to find 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 stated in a high-level description in
English prose, as:
4271:, 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
1625:
seek an approximation that is closer to the true solution. The approximation can be reached by either using a deterministic or a random strategy. Such algorithms have practical value for many hard problems. One of the examples of an approximate algorithm is the
807:
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".
1329:
Algorithms, by themselves, 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
4196:. 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.
1906:, that is, at solutions that cannot be improved by the algorithm but are not optimum. The most popular use of greedy algorithms is for finding the minimal spanning tree where finding the optimal solution is possible with this method.
1953:. Some of them, 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
283:
is an approach to problem-solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. For example, social media
1887:
memoization of recursive calls. When subproblems are independent and there is no repetition, memoization does not help; hence dynamic programming is not a solution for all complex problems. By using memoization or maintaining a
1303:", 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
1216:
or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation. Usually,
439:
define an algorithm to be a set of instructions for determining an output, given explicitly, in a form that can be followed by either a computing machine or a human who could only carry out specific elementary operations on
3793:
1697:. Sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in the conquer phase by merging the segments. A simpler variant of divide and conquer is called a
466:, the time an algorithm requires to complete cannot be measured, as it is unrelated to the customary physical dimension. Such uncertainties that characterize ongoing work stem from the unavailability of a definition of
5620:
1804:
for finding the median in an unsorted list involves first sorting the list (the expensive portion) and then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as
1233:
may be used to compare before/after potential improvements to an algorithm after program optimization. Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner.
226:
5665:
311:, 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
742:
in the mid-19th century. Lovelace is credited with the first creation of an algorithm intended for processing on a computer, Babbage's analytical engine, which is the first device considered a real
1029:. At all times the algorithm only needs to remember two values: the sum of all the elements so far, and its current position in the input list. Therefore, it is said to have a space requirement of
5670:
4911:
1832:
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:
2374:
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).
1221:
is used for analysis as it is the simplest and most general representation. However, ultimately, most algorithms are usually implemented on particular hardware/software platforms and their
5736:
750:. Lovelace is sometimes called "history's first programmer" as a result, though a full implementation of Babbage's second device would not be realized until decades after her lifetime.
1668:. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories includes many different types of algorithms. Some common paradigms are:
288:
rely on heuristics in such a way that, although widely characterized as "algorithms" in 21st-century popular media, cannot deliver correct results due to the nature of the problem.
5289:
1267:
Algorithm design refers to a method or a mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories, such as
1586:
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
5658:
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not available or too complex. Brute force can be used to solve a variety of problems, including finding the shortest path between two points and cracking passwords.
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is well understood using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.
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1058:
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is one that invokes (makes reference to) itself repeatedly until a certain condition (also known as termination condition) matches, which is a method common to
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5471:
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2327:
Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and
Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247
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4859:
4590:
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.
3285:(described using a membership oracle) can be approximated to high accuracy by a randomized polynomial time algorithm, but not by a deterministic one: see
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1988:
For each remaining number in the set: if this number is larger than the current largest number, consider this number to be the largest number in the set.
679:
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1307:: 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
5722:
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1866:
When a problem shows optimal substructures—meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems—and
1819:
In this approach, multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution.
2387:(concerning some chosen notation for integers) ... this limitation (to numerical functions) results in no loss of generality", (Rogers 1987:1).
1998:
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
1937:
solution. Their merit is that they can find a solution very close to the optimal solution in a relatively short time. Such algorithms include
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Dyer, Martin; Frieze, Alan; Kannan, Ravi (January 1991). "A Random
Polynomial-time Algorithm for Approximating the Volume of Convex Bodies".
3242:
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2806:
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2413:"A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method
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112:
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339:("Addition and subtraction in Indian arithmetic"). In the early 12th century, Latin translations of said al-Khwarizmi texts involving the
123:
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enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.
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622:. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.
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To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to
141:
83:
1991:
When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set.
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766:(punch cards), and "telephone switching technologies" led to the development of the first computers. By the mid-19th century, the
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This technique involves solving a difficult problem by transforming it into a better-known problem for which we have (hopefully)
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mechanism that provides the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical
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683:
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4358:. Intelligent Systems, Control and Automation: Science and Engineering. Vol. 74. Switzerland: Springer. pp. 111–127.
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1807:
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3567:) where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc.
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4743:
4727:
4707:
Hertzke, Allen D.; McRorie, Chris (1998). "The
Concept of Moral Ecology". In Lawler, Peter Augustine; McConkey, Dale (eds.).
4032:, p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper
2348:
3647:, p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes.
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3086:
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of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity.
57:
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2362:"Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2).
2090:
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3911:, ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources.
1875:
5914:
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2464:. Untimely Meditations. Vol. 14. Translated by Chase, Jefferson. Cambridge, Massachusetts: MIT Press. p. 147.
1879:
1686:
1268:
280:
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2426:"An algorithm has one or more outputs, i.e., quantities which have a specified relation to the inputs" (Knuth 1973:5).
2181:
2152:
1938:
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1608:
1363:
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Different algorithms may complete the same task with a different set of instructions in less or more time, space, or '
410:
3001:
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One informal definition is "a set of rules that precisely defines a sequence of operations", which would include all
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to solve the given problems. Some problems are naturally suited for one implementation or the other. For example,
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5563:
5496:
5491:
5454:
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4958:
4928:
4533:, p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.
4471:. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an
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4159:
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482:
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eventually. Using human characteristics as descriptors of machines in metaphorical ways was already practiced by
257:
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Since antiquity, step-by-step procedures for solving mathematical problems have been recorded. This includes in
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1643:. The term is usually used for those algorithms which seem inherently quantum or use some essential feature of
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critical. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign.
601:
389:
in 1391, English adopted the French term. In the 15th century, under the influence of the Greek word ἀριθμός (
1870:, meaning the same subproblems are used to solve many different problem instances, a quicker approach called
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5919:
5609:
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3193:
2234:
2221:
2191:
1954:
1702:
1622:
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914:
842:
640:
159:
3188:
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5400:
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5390:
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5343:
5306:
5252:
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1911:
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repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually
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1312:
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31:
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can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and
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644:
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237:
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that states a
Knowledge (XXG) editor's personal feelings or presents an original argument about a topic.
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3677:
The
Undecidable: Basic Papers On Undecidable Propositions, Unsolvable Problems and Computable Functions
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1919:
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specifies rules for moving around a graph and is useful for such problems. This category also includes
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In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the
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for directed graphs. If a problem additionally requires that one or more of the unknowns must be an
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Empirical testing is useful because it may uncover unexpected interactions that affect performance.
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2957:(2016). "The (black) art of run-time evaluation: Are we comparing algorithms or implementations?".
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1693:) until the instances are small enough to solve easily. One such example of divide and conquer is
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Bolter credits the invention of the weight-driven clock as "the key invention ," specifically the
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4572:, pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton.
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1967:
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770:, the precursor of the telephone, was in use throughout the world. By the late 19th century, the
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597:
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420:(including programs that do not perform numeric calculations), and (for example) any prescribed
409:
For a detailed presentation of the various points of view on the definition of "algorithm", see
4803:
3856:
3031:
1778:
always return the correct answer, but their running time is only probabilistically bound, e.g.
1611:
solve problems via guessing although typical guesses are made more accurate through the use of
1111:
295:, an algorithm can be expressed within a finite amount of space and time and in a well-defined
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2567:
2507:
2465:
2344:
1950:
1843:
1644:
1636:
890:
731:
727:
563:
3979:. Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
1283:
is used to describe e.g., an algorithm's run-time growth as the size of its input increases.
5904:
5889:
5439:
5230:
5218:
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5179:
5128:
5080:
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Elements of
Mathematical Logic and its Application to the theory of Subrecursive Algorithms
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5267:
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4731:
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3136:
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3038:
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1336:). However practical applications of algorithms are sometimes patentable. For example, in
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997:
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862:
743:
735:
723:
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249:
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Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1
3411:
2506:. MIT Cognet library. Cambridge, Massachusetts: MIT Press (published 2001). p. 11.
524:
Please expand the section to include this information. Further details may exist on the
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4205:
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3875:
3849:
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3563:, p. 89ff. The first expression of "Church's Thesis". See in particular page 100 (
3496:
where they discuss "certain enumerable sets not effectively (mechanically) enumerable".
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1907:
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clay tablets described algorithms for computing formulas. Algorithms were also used in
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3522:, and Costa, J.F. (2000) An analog characterization of the subrecursive functions. In
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5894:
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5461:
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4562:
4501:(1936–37). "On Computable Numbers, With An Application to the Entscheidungsproblem".
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Stone requires that "it must terminate in a finite number of steps" (Stone 1973:7–8).
2500:
Dietrich, Eric (1999). "Algorithm". In Wilson, Robert Andrew; Keil, Frank C. (eds.).
1903:
1814:
1569:
1103:
950:
942:
938:
910:
870:
858:
846:
838:
759:
698:
675:
455:
432:
312:
4330:
4301:(1939). "An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem".
4262:
3603:
3318:
3252:
3178:
Knuth 1973 section 1.2.1, expanded by
Tausworthe 1977 at pages 100ff and Chapter 9.1
2986:
2488:
the next level of abstraction of central bureaucracy: globally operating algorithms.
1354:
is controversial, and there are criticized patents involving algorithms, especially
5844:
5808:
5778:
5526:
5405:
5184:
5095:
4990:
4873:
4719:
4537:
4498:
4298:
4182:
4131:
4109:
3799:
3781:
3773:
3753:
3745:
3638:
2954:
2773:
2619:
1732:
1716:
1179:
946:
820:
739:
421:
328:
304:
4740:
4724:
803:
were invented in 1835. These led to the invention of the digital adding device by
689:
The first cryptographic algorithm for deciphering encrypted code was developed by
5714:
4775:
4669:
Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest; Clifford Stein (2009).
4609:
4524:
4036:
that proved the "decision problem" to be "undecidable" (i.e., a negative result).
3224:
3070:
2501:
2459:
5798:
5637:
5553:
4363:
4136:
Volume 2/Seminumerical Algorithms, The Art of Computer Programming First Edition
3702:
3027:
2794:
2146:
2003:
1942:
1883:
1752:
can be the fastest algorithm for some problems is an open question known as the
925:
There is a wide variety of representations possible and one can express a given
874:
793:
782:
771:
574:
486:
431:. In general, a program is an algorithm only if it stops eventually—even though
273:
245:
241:
174:
4825:
4553:
4516:
5788:
5573:
4632:
The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer
3519:
3234:
2970:
2128:
1999:
1745:
1612:
1218:
894:
747:
654:
490:
344:
269:
4028:
Presented to the American Mathematical Society, September 1935. Reprinted in
3835:
3818:
3202:
2978:
2765:
2657:
2634:"Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria"
1607:
solve the problem with exact decision at every step of the algorithm whereas
4830:
4230:
3714:
2404:
which are given to it initially before the algorithm begins" (Knuth 1973:5).
1694:
1690:
1565:
1366:. Additionally, some cryptographic algorithms have export restrictions (see
967:
898:
866:
767:
726:" beginning in the 13th century and finally to "computational machines"—the
615:
582:
261:
17:
3609:
Church, Alonzo (1936). "Correction to a Note on the Entscheidungsproblem".
3042:
1874:
avoids recomputing solutions that have already been computed. For example,
573:
The earliest evidence of algorithms is found in the mathematics of ancient
510:
4584:, Manual of Patent Examining Procedure (MPEP). Latest revision August 2006
4173:
4154:
3717:
are included; those cited in the article are listed here by author's name.
3442:(1984 ed.). Chapel Hill, NC: The University of North Carolina Press.
3310:
2757:
1902:
problems they can find the optimal solution while for others they stop at
4751:. Stanford, California: Center for the Study of Language and Information.
4735:. Stanford, California: Center for the Study of Language and Information.
3694:
2401:
1378:
There are various ways to classify algorithms, each with its own merits.
1343:
690:
470:
that suits both concrete (in some sense) and abstract usage of the term.
425:
381:
2557:
2555:
2553:
2551:
1664:
Another way of classifying algorithms is by their design methodology or
481:. However, algorithms are also implemented by other means, such as in a
5899:
4348:
4322:
4254:
4070:
4003:
3769:
3630:
3595:
3554:
3380:
2665:
2633:
1851:
1800:
is not dominated by the resulting reduced algorithms. For example, one
586:
30:"Algorithms" redirects here. For the subfield of computer science, see
4855:
4225:
Computability, Effective Procedures and Algorithms. Infinite machines.
2826:. Wichita State University: Department of Mathematics and Statistics.
1062:, if the space required to store the input numbers is not counted, or
5621:
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
5240:
5058:
4087:
3710:
3278:
1359:
902:
824:
800:
428:
233:
4314:
4246:
4061:
4044:
3622:
3587:
3546:
3371:
3354:
2649:
1985:
Assume the first number in the set is the largest number in the set.
1982:
If there are no numbers in the set, then there is no highest number.
1590:
for such an environment is called a serial algorithm, as opposed to
256:
to divert the code execution through various routes (referred to as
3920:
From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931
3823:
Baltic International Yearbook of Cognition, Logic and Communication
2525:
An algorithm is a recipe, method, or technique for doing something.
4611:
Habits of the Heart: Individualism and Commitment in American Life
4591:
4529:. Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in
3141:
Back to Basic: The History, Corruption, and Future of the Language
2166:
1846:. Problems that can be solved with linear programming include the
1712:
1212:, and is often practiced abstractly without the use of a specific
815:
578:
463:
459:
153:
3419:
Bulletin of European Association for Theoretical Computer Science
2436:
resort to random methods or devices, e.g., dice" (Rogers 1987:2).
889:
Algorithms can be expressed in many kinds of notation, including
5828:
3908:
Sequential Abstract State Machines Capture Sequential Algorithms
3809:
2397:
5718:
5676:
European Community on Computational Methods in Applied Sciences
4893:
4777:
Poems that Solve Puzzles: The History and Science of Algorithms
4432:
Unto Others: The Evolution and Psychology of Unselfish Behavior
3726:
Engines of Logic: Mathematicians and the Origin of the Computer
2888:, Valley News West Lebanon NH, Thursday, March 31, 1983, p. 13.
1922:
are greedy algorithms that can solve this optimization problem.
781:) was in use, as were Hollerith cards (c. 1890). Then came the
522:
about 20th and 21st century development of computer algorithms.
3072:
Algorithm Design: Foundations, Analysis, and Internet Examples
504:
379:, or "Thus spoke Al-Khwarizmi". Around 1230, the English word
95:
40:
4839:
3533:(1936). "An Unsolvable Problem of Elementary Number Theory".
3355:"On a Subrecursive Hierarchy and Primitive Recursive Degrees"
2594:. In Emch, Gerard G.; Sridharan, R.; Srinivas, M. D. (eds.).
215:
5671:
International Council for Industrial and Applied Mathematics
4481:
Standardized Development of Computer Software Part 1 Methods
303:. Starting from an initial state and initial input (perhaps
3189:"The Experts: Does the Patent System Encourage Innovation?"
2118:" terminates the algorithm and outputs the following value.
1796:
algorithms. The goal is to find a reducing algorithm whose
1386:
One way to classify algorithms is by implementation means.
845:" or "effective method". Those formalizations included the
562:(around 800 BC and later), the Ifa Oracle (around 500 BC),
212:
203:
113:
personal reflection, personal essay, or argumentative essay
3223:
Kellerer, Hans; Pferschy, Ulrich; Pisinger, David (2004).
2714:
A Brief History of Cryptology and Cryptographic Algorithms
1878:, the shortest path to a goal from a vertex in a weighted
841:. Later formalizations were framed as attempts to define "
236:
instructions, typically used to solve a class of specific
4652:
A History of Algorithms: From the Pebble to the Microchip
4581:
2106.02 **>Mathematical Algorithms: 2100 Patentability
4453:
Introduction to Computer Organization and Data Structures
4281:
Theory of Recursive Functions and Effective Computability
3524:
Proc. of the 4th Conference on Real Numbers and Computers
2564:
A History of Algorithms: From the Pebble to the Microchip
209:
200:
194:
4864:
3922:((1967) ed.). Harvard University Press, Cambridge.
2716:. Springer Science & Business Media. pp. 12–3.
375:
of Al-Khwarizmi's name; the text starts with the phrase
4233:(1936). "Finite Combinatory Processes, Formulation I".
4211:(First ed.). Prentice-Hall, Englewood Cliffs, NJ.
3143:, Addison-Wesley Publishing Company, Inc. Reading, MA,
2566:. Springer Science & Business Media. pp. 7–8.
1295:, any algorithm can be computed by a model known to be
244:. Algorithms are used as specifications for performing
158:
Flowchart of using successive subtractions to find the
119:
3855:. New York: Touchstone/Simon & Schuster. pp.
3693:
Davis gives commentary before each article. Papers of
674:).Examples of ancient Indian mathematics included the
307:), the instructions describe a computation that, when
276:
with terms such as "memory", "search" and "stimulus".
5290:
Numerical methods for ordinary differential equations
4392:(3rd ed.). Morgan Kaufmann Publishers/Elsevier.
1153:
1114:
1070:
1037:
1000:
5666:
Société de Mathématiques Appliquées et Industrielles
5659:
Japan Society for Industrial and Applied Mathematics
5295:
Numerical methods for partial differential equations
4094:(Tenth ed.). North-Holland Publishing Company.
3480:(4th ed.). Cambridge University Press, London.
1764:
return a correct answer with high probability. E.g.
625:
Algorithms for arithmetic are also found in ancient
206:
197:
5837:
5756:
5646:
5630:
5592:
5544:
5482:
5357:
5315:
5109:
5071:
5009:
4927:
4354:. In van Rysewyk, Simon; Pontier, Matthijs (eds.).
191:
4429:
4204:
3848:
3674:
3473:
2596:Contributions to the History of Indian Mathematics
1168:
1135:
1085:
1052:
1015:
695:A Manuscript On Deciphering Cryptographic Messages
4049:Transactions of the American Mathematical Society
4034:An Unsolvable Problem of Elementary Number Theory
3880:((1968, 1994) ed.). St. Martin's Press, NY.
3764:as the show-stealing villain. Very brief bios of
3440:Turing's Man: Western Culture in the Computer Age
3359:Transactions of the American Mathematical Society
1756:. There are two large classes of such algorithms:
4757:Moral Machines: Teaching Robots Right from Wrong
4755:Wallach, Wendell; Allen, Colin (November 2008).
3988:"General Recursive Functions of Natural Numbers"
3032:ACM-SIAM Symposium On Discrete Algorithms (SODA)
2341:Information Retrieval: Algorithms and Heuristics
941:for more), as flowcharts and drakon-charts (see
2875:Bell and Newell diagram 1971:39, cf. Davis 2000
1621:While many algorithms reach an exact solution,
60:for grammar, style, cohesion, tone, or spelling
5654:Society for Industrial and Applied Mathematics
4844:National Institute of Standards and Technology
4542:Proceedings of the London Mathematical Society
4540:(1939). "Systems of Logic Based on Ordinals".
4504:Proceedings of the London Mathematical Society
3574:(1936). "A Note on the Entscheidungsproblem".
3412:"Algorithms: A Quest for Absolute Definitions"
2503:The MIT Encyclopedia of the Cognitive Sciences
2461:The Death Algorithm and Other Digital Dilemmas
1572:and sometimes additional data structures like
1542:implementation of Euclid's algorithm from the
327:Around 825 AD, Persian scientist and polymath
5730:
4905:
2383:"an algorithm is a procedure for computing a
929:program as a sequence of machine tables (see
458:come into being starting from a small set of
436:
397:"arithmetic"), the Latin word was altered to
8:
5472:Supersymmetric theory of stochastic dynamics
4840:Dictionary of Algorithms and Data Structures
4614:. Berkeley: University of California Press.
4467:Cf. in particular the first chapter titled:
4428:Sober, Elliott; Wilson, David Sloan (1998).
4349:"Moral Ecology Approaches to Machine Ethics"
3805:Dictionary of Algorithms and Data Structures
3226:Knapsack Problems | Hans Kellerer | Springer
2799:Episodes from the Early History of Astronomy
2679:
2677:
2675:
857:recursive functions of 1930, 1934 and 1935,
454:—an idea that is central for explaining how
4860:State University of New York at Stony Brook
4483:. Englewood Cliffs NJ: Prentice–Hall, Inc.
3336:Linear Programming 2: Theory and Extensions
337:kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī
5737:
5723:
5715:
5479:
5006:
4912:
4898:
4890:
4116:. Reading, Massachusetts: Addison–Wesley.
3389:Computer Structures: Readings and Examples
3387:Bell, C. Gordon and Newell, Allen (1971),
2686:The History of Mathematics: A Brief Course
1568:algorithms use repetitive constructs like
4725:Selected Papers on Analysis of Algorithms
4576:United States Patent and Trademark Office
4561:
4469:Algorithms, Turing Machines, and Programs
4410:Introduction to the Theory of Computation
4207:Computation: Finite and Infinite Machines
4172:
4138:. Reading, Massachusetts: Addison–Wesley.
4060:
3834:
3434:Includes a bibliography of 56 references.
3370:
3300:
2707:
2705:
2017:. Output: The largest number in the list
1152:
1113:
1069:
1036:
999:
992:numbers would have a time requirement of
827:", the first published computer algorithm
349:Liber Alghoarismi de practica arismetrice
142:Learn how and when to remove this message
84:Learn how and when to remove this message
4455:(1972 ed.). McGraw-Hill, New York.
3002:"Better Math Makes Faster Data Networks"
2884:Melina Hill, Valley News Correspondent,
2638:The Two-Year College Mathematics Journal
1145:) outperforms a sequential search (cost
639:. Algorithms were later used in ancient
4741:Selected Papers on Design of Algorithms
4436:. Cambridge: Harvard University Press.
3109:"Big-O notation (article) | Algorithms"
2284:
2013:LargestNumber Input: A list of numbers
1262:
945:for more), or as a form of rudimentary
705:, the earliest codebreaking algorithm.
693:, a 9th-century Arab mathematician, in
4688:Harel, David; Feldman, Yishai (2004).
4045:"Recursive Predicates and Quantifiers"
3653:The Muslim contribution to mathematics
3391:, McGraw–Hill Book Company, New York.
2830:from the original on February 27, 2015
2801:. New York: Springer. pp. 40–62.
2478:from the original on December 22, 2019
2306:from the original on February 14, 2020
758:Bell and Newell (1971) write that the
493:or an insect looking for food), in an
4709:Community and Political Thought Today
4690:Algorithmics: The Spirit of Computing
4383:from the original on October 9, 2022.
4114:Fundamental Algorithms, Third Edition
3432:from the original on October 9, 2022.
3259:from the original on October 18, 2017
2927:
2925:
2923:
2921:
2585:
2583:
1768:is the subclass of these that run in
1748:. Whether randomized algorithms with
450:is also used to define the notion of
7:
4869:Associations for Computing Machinery
4856:The Stony Brook Algorithm Repository
3740:Davis offers concise biographies of
3526:, Odense University, pp. 91–109
3041:, Kyoto, January 2012. See also the
2370:
2368:
2358:
2356:
2335:
2333:
2323:
2321:
2290:
2288:
3089:from the original on April 28, 2015
1263:Algorithm § By design paradigm
697:. He gave the first description of
473:Most algorithms are intended to be
335:("Book of Indian computation") and
252:. More advanced algorithms can use
4921:Industrial and applied mathematics
2886:A Tinkerer Gets a Place in History
2592:"Algorithms in Indian Mathematics"
2339:David A. Grossman, Ophir Frieder,
1601:Deterministic or non-deterministic
1346:algorithm to aid in the curing of
1305:Böhm-Jacopini canonical structures
25:
5151:Stochastic differential equations
4347:Santos-Lang, Christopher (2015).
3651:Daffa', Ali Abdullah al- (1977).
3008:from the original on May 13, 2014
2959:Knowledge and Information Systems
2300:Merriam-Webster Online Dictionary
1206:analysis, and study of algorithms
953:called "sets of quadruples" (see
357:Liber Algorismi de numero Indorum
27:Sequence of operations for a task
5467:Supersymmetric quantum mechanics
3792: This article incorporates
3787:
3030:, Dina Katabi, and Eric Price, "
3000:Gillian Conahan (January 2013).
2245:List of algorithm general topics
2145:
2131:
1715:) can be modeled as problems on
1196:Profiling (computer programming)
799:Telephone-switching networks of
792:) with its punched-paper use of
509:
437:Boolos, Jeffrey & 1974, 1999
187:
100:
45:
5349:Stochastic variational calculus
5141:Ordinary differential equations
4865:Collected Algorithms of the ACM
4759:. US: Oxford University Press.
4592:https://doi.org/10.2307/3027363
4390:Programming Language Pragmatics
4092:Introduction to Metamathematics
3819:"Evolution and moral diversity"
3535:American Journal of Mathematics
2739:"Ancient Babylonian Algorithms"
2618:Hayashi, T. (2023, January 1).
2545:Boolos and Jeffrey 1974,1999:19
2270:Computational complexity theory
1711:Many problems (such as playing
1583:Serial, parallel or distributed
661:, which was first described in
435:may sometimes prove desirable.
5146:Partial differential equations
5019:Arbitrary-precision arithmetic
3075:. John Wiley & Sons, Inc.
1699:decrease-and-conquer algorithm
1342:, the application of a simple
1163:
1157:
1130:
1118:
1102:' than others. For example, a
1080:
1074:
1047:
1041:
1010:
1004:
329:Muḥammad ibn Mūsā al-Khwārizmī
1:
5034:Interactive geometry software
4479:Tausworthe, Robert C (1977).
4235:The Journal of Symbolic Logic
3611:The Journal of Symbolic Logic
3576:The Journal of Symbolic Logic
2915:Rosser 1939 in Davis 1965:225
2906:Kleene 1943 in Davis 1965:274
786:
775:
668:
647:, which was described in the
633:
604:
590:
5994:Theoretical computer science
4279:Rogers, Hartley Jr. (1987).
4146:, LSU Publ., Leningrad, 1981
1687:divide-and-conquer algorithm
1639:run on a realistic model of
1609:non-deterministic algorithms
837:(decision problem) posed by
319:, incorporate random input.
315:; some algorithms, known as
5086:Computational number theory
5049:Numerical-analysis software
4809:Encyclopedia of Mathematics
4780:. Oxford University Press.
4673:(3rd ed.). MIT Press.
4364:10.1007/978-3-319-08108-3_8
3334:and Mukund N. Thapa. 2003.
2866:Bolter 1984:33–34, 204–206.
2632:Zaslavsky, Claudia (1970).
2182:Algorithm characterizations
2153:Computer programming portal
1996:(Quasi-)formal description:
1721:graph exploration algorithm
1350:was deemed patentable. The
1182:on sorted lists or arrays.
966:The graphical aid called a
746:computer instead of just a
411:Algorithm characterizations
341:Hindu–Arabic numeral system
6010:
4671:Introduction To Algorithms
4650:Chabert, Jean-Luc (1999).
4413:. PWS Publishing Company.
4388:Scott, Michael L. (2009).
3728:. New York: W.W. Nortion.
3502:Super-Recursive Algorithms
3048:February 21, 2012, at the
2622:. Encyclopedia Britannica.
2562:Chabert, Jean-Luc (2012).
2228:Introduction to Algorithms
2101:" means that the value of
1965:
1750:polynomial time complexity
1322:
1260:
1241:
1189:
980:
650:Introduction to Arithmetic
631:Rhind Mathematical Papyrus
497:, or a mechanical device.
408:
232:) is a finite sequence of
36:Algorithm (disambiguation)
29:
5953:
5684:
5492:Algebra of physical space
4959:Automated theorem proving
4879:December 6, 2015, at the
4738:Knuth, Donald E. (2010).
4629:Berlinski, David (2001).
4563:21.11116/0000-0001-91CE-3
4451:Stone, Harold S. (1972).
4303:Journal of Symbolic Logic
4160:Communications of the ACM
4155:"Algorithm=Logic+Control"
3681:. New York: Raven Press.
3438:Bolter, David J. (1984).
3235:10.1007/978-3-540-24777-7
2971:10.1007/s10115-016-1004-2
2737:Knuth, Donald E. (1972).
2688:. John Wiley & Sons.
2598:. Springer. p. 153.
2296:"Definition of ALGORITHM"
2217:Computational mathematics
1854:then it is classified in
1543:
1136:{\displaystyle O(\log n)}
483:biological neural network
258:automated decision-making
5285:Numerical linear algebra
4774:Bleakley, Chris (2020).
4554:10.1112/plms/s2-45.1.161
4517:10.1112/plms/s2-42.1.230
4407:Sipser, Michael (2006).
3836:10.4148/biyclc.v7i0.1775
3760:, Gödel and Turing with
3004:. discovermagazine.com.
2712:Dooley, John F. (2013).
2684:Cooke, Roger L. (2005).
2255:Regulation of algorithms
2105:changes to the value of
1876:Floyd–Warshall algorithm
1623:approximation algorithms
1605:Deterministic algorithms
1392:
962:Flowchart representation
801:electromechanical relays
385:is attested and then by
5958:List of data structures
5024:Finite element analysis
4974:Constraint satisfaction
3952:Alan Turing: The Enigma
3851:Darwin's Dangerous Idea
3476:Computability and Logic
3194:The Wall Street Journal
2235:Government by algorithm
2222:Garbage in, garbage out
2192:Algorithmic composition
1977:High-level description:
1955:approximation algorithm
1868:overlapping subproblems
1788:Reduction of complexity
1703:binary search algorithm
1186:Formal versus empirical
843:effective calculability
754:Electromechanical relay
643:. Two examples are the
641:Hellenistic mathematics
596:describes the earliest
333:kitāb al-ḥisāb al-hindī
234:mathematically rigorous
160:greatest common divisor
5579:Mathematical economics
5253:Multivariable calculus
5136:Differential equations
4979:Constraint programming
4969:Computational geometry
4874:The Stanford GraphBase
4850:Algorithm repositories
4746:July 16, 2017, at the
4356:Machine Medical Ethics
3874:Dilson, Jesse (2007).
3794:public domain material
3655:. London: Croom Helm.
2590:Sriram, M. S. (2005).
2400:or more inputs, i.e.,
1794:asymptotically optimal
1762:Monte Carlo algorithms
1708:Search and enumeration
1596:distributed algorithms
1562:functional programming
1368:export of cryptography
1313:mathematical induction
1287:Structured programming
1244:Algorithmic efficiency
1223:algorithmic efficiency
1192:Empirical algorithmics
1170:
1137:
1087:
1054:
1017:
988:elements of a list of
983:Analysis of algorithms
935:state-transition table
828:
552:Babylonian mathematics
520:is missing information
347:appeared, for example
170:
122:by rewriting it in an
34:. For other uses, see
32:Analysis of algorithms
5532:Supersymmetry algebra
5517:Representation theory
5512:Renormalization group
5158:Differential geometry
5039:Optimization software
5011:Mathematical software
4730:July 1, 2017, at the
4606:Bellah, Robert Neelly
4174:10.1145/359131.359136
3992:Mathematische Annalen
3766:Joseph-Marie Jacquard
3500:Burgin, Mark (2004).
3311:10.1145/102782.102783
3037:July 4, 2013, at the
2783:on December 24, 2012.
2758:10.1145/361454.361514
2343:, 2nd edition, 2004,
2260:Theory of computation
2250:Medium is the message
2207:Algorithmic technique
2202:Algorithmic synthesis
2177:Algorithm engineering
1966:Further information:
1930:optimization problems
1830:optimization problems
1824:Optimization problems
1808:transform and conquer
1649:quantum superposition
1352:patenting of software
1309:proofs of correctness
1171:
1138:
1106:algorithm (with cost
1088:
1055:
1018:
907:programming languages
881:of 1936–37 and 1939.
819:
684:Brāhmasphuṭasiddhānta
645:Sieve of Eratosthenes
629:, dating back to the
581:clay tablet found in
566:(around 240 BC), and
317:randomized algorithms
157:
5855:Breadth-first search
5584:Mathematical finance
5569:Social choice theory
5484:Algebraic structures
5433:in quantum mechanics
5369:Analytical mechanics
5335:Stochastic processes
5307:Variational calculus
5119:Approximation theory
5044:Statistical software
4273:Church–Turing thesis
4194:Theory of algorithms
4018:on September 3, 2014
3934:, 3rd edition 1976,
3916:van Heijenoort, Jean
3063:Goodrich, Michael T.
2265:Computability theory
2212:Algorithmic topology
2197:Algorithmic entities
1934:heuristic algorithms
1925:The heuristic method
1848:maximum flow problem
1776:Las Vegas algorithms
1739:Randomized algorithm
1675:or exhaustive search
1653:quantum entanglement
1618:Exact or approximate
1358:algorithms, such as
1333:Gottschalk v. Benson
1293:Church–Turing thesis
1238:Execution efficiency
1214:programming language
1200:Program optimization
1169:{\displaystyle O(n)}
1151:
1112:
1086:{\displaystyle O(n)}
1068:
1053:{\displaystyle O(1)}
1035:
1016:{\displaystyle O(n)}
998:
977:Algorithmic analysis
931:finite-state machine
834:Entscheidungsproblem
714:Weight-driven clocks
627:Egyptian mathematics
620:Babylonian astronomy
556:Egyptian mathematics
5945:Topological sorting
5875:Dynamic programming
5559:Operations research
5428:Perturbation theory
5226:Multilinear algebra
5197:Functional analysis
5054:Numerical libraries
4986:Computational logic
4885:Stanford University
4654:. Springer Verlag.
3169:Tausworthe 1977:142
3160:Tausworthe 1977:101
3026:Haitham Hassanieh,
2953:; Schubert, Erich;
2951:Kriegel, Hans-Peter
2456:Simanowski, Roberto
1947:simulated annealing
1872:dynamic programming
1862:Dynamic programming
1856:integer programming
1802:selection algorithm
1754:P versus NP problem
1641:quantum computation
1592:parallel algorithms
1558:recursive algorithm
1273:dynamic programming
1208:is a discipline of
659:Euclidean algorithm
286:recommender systems
266:automated reasoning
260:) and deduce valid
5989:Mathematical logic
5963:List of algorithms
5870:Divide and conquer
5865:Depth-first search
5860:Brute-force search
5774:Binary search tree
5696:Mathematics portal
5593:Other applications
5317:Probability theory
5300:Validated numerics
5280:Numerical analysis
5173:Geometric analysis
5163:Differential forms
4996:Information theory
4823:Weisstein, Eric W.
4692:. Addison-Wesley.
4088:Kleene, Stephen C.
4041:Kleene, Stephen C.
4004:10.1007/BF01565439
3984:Kleene, Stephen C.
3957:Simon and Schuster
3817:Dean, Tim (2012).
3518:Campagnolo, M.L.,
3338:. Springer-Verlag.
3277:For instance, the
2940:cf Tausworthe 1977
2396:"An algorithm has
2240:List of algorithms
2172:Algorithm aversion
2139:Mathematics portal
2093:. For instance, "
2089:"←" denotes
1968:List of algorithms
1951:genetic algorithms
1837:Linear programming
1682:Divide and conquer
1660:By design paradigm
1637:Quantum algorithms
1588:algorithm designed
1277:operation research
1269:divide-and-conquer
1166:
1133:
1095:if it is counted.
1083:
1050:
1013:
829:
732:analytical engines
703:frequency analysis
598:division algorithm
568:Arabic mathematics
560:Indian mathematics
558:(around 1550 BC),
554:(around 2500 BC),
546:Ancient algorithms
495:electrical circuit
485:(for example, the
299:for calculating a
171:
124:encyclopedic style
111:is written like a
64:You can assist by
5971:
5970:
5769:Associative array
5712:
5711:
5546:Decision sciences
5540:
5539:
5522:Spacetime algebra
5214:Harmonic analysis
5180:Dynamical systems
5124:Clifford analysis
5101:Discrete geometry
5067:
5066:
4787:978-0-19-885373-2
4766:978-0-19-537404-9
4699:978-0-321-11784-7
4680:978-0-262-03384-8
4661:978-3-540-63369-3
4642:978-0-15-601391-8
4635:. Harvest Books.
4621:978-0-520-25419-0
4490:978-0-13-842195-3
4462:978-0-07-061726-1
4420:978-0-534-94728-6
4399:978-0-12-374514-9
4373:978-3-319-08107-6
4290:978-0-262-68052-3
4283:. The MIT Press.
4218:978-0-13-165449-5
4123:978-0-201-89683-1
4101:978-0-7204-2103-3
3966:978-0-671-49207-6
3929:978-0-674-32449-7
3887:978-0-312-10409-2
3866:978-0-684-80290-9
3735:978-0-393-32229-3
3688:978-0-486-43228-1
3662:978-0-85664-464-1
3511:978-0-387-95569-8
3487:978-0-521-20402-6
3449:978-0-8078-1564-9
3332:George B. Dantzig
3244:978-3-540-40286-2
3082:978-0-471-38365-9
3067:Tamassia, Roberto
2808:978-0-387-95136-2
2695:978-1-118-46029-0
2605:978-93-86279-25-5
2417:" (Knuth 1973:5).
2119:
2110:
1894:The greedy method
1844:simplex algorithm
1731:enumeration, and
1725:search algorithms
1645:Quantum computing
1633:Quantum algorithm
1550:
1549:
1382:By implementation
891:natural languages
823:'s diagram from "
762:, a precursor to
664:Euclid's Elements
602:Hammurabi dynasty
577:(modern Iraq). A
570:(around 800 AD).
564:Greek mathematics
543:
542:
479:computer programs
418:computer programs
152:
151:
144:
94:
93:
86:
16:(Redirected from
6001:
5940:String-searching
5739:
5732:
5725:
5716:
5497:Feynman integral
5480:
5440:Potential theory
5329:random variables
5219:Fourier analysis
5202:Operator algebra
5129:Clifford algebra
5081:Computer algebra
5007:
4914:
4907:
4900:
4891:
4836:
4835:
4817:
4791:
4770:
4720:Knuth, Donald E.
4716:
4711:. Westport, CT:
4703:
4684:
4665:
4646:
4625:
4567:
4565:
4528:
4494:
4466:
4447:
4435:
4424:
4403:
4384:
4382:
4353:
4334:
4294:
4266:
4222:
4210:
4186:
4176:
4151:Kowalski, Robert
4139:
4127:
4105:
4074:
4064:
4027:
4025:
4023:
4014:. Archived from
3970:
3933:
3891:
3870:
3854:
3840:
3838:
3813:
3791:
3790:
3739:
3692:
3680:
3666:
3642:
3607:
3558:
3515:
3492:: cf. Chapter 3
3491:
3479:
3470:Jeffrey, Richard
3453:
3433:
3431:
3416:
3384:
3374:
3339:
3329:
3323:
3322:
3304:
3275:
3269:
3268:
3266:
3264:
3220:
3214:
3213:
3211:
3209:
3197:. May 16, 2013.
3185:
3179:
3176:
3170:
3167:
3161:
3158:
3152:
3130:
3124:
3123:
3121:
3119:
3105:
3099:
3098:
3096:
3094:
3059:
3053:
3024:
3018:
3017:
3015:
3013:
2997:
2991:
2990:
2947:
2941:
2938:
2932:
2929:
2916:
2913:
2907:
2904:
2898:
2895:
2889:
2882:
2876:
2873:
2867:
2864:
2858:
2855:
2849:
2846:
2840:
2839:
2837:
2835:
2819:
2813:
2812:
2791:
2785:
2784:
2782:
2776:. Archived from
2743:
2734:
2728:
2727:
2709:
2700:
2699:
2681:
2670:
2669:
2629:
2623:
2616:
2610:
2609:
2587:
2578:
2577:
2559:
2546:
2543:
2537:
2534:
2528:
2527:
2522:
2520:
2497:
2491:
2490:
2485:
2483:
2452:
2446:
2443:
2437:
2433:
2427:
2424:
2418:
2416:
2411:
2405:
2394:
2388:
2381:
2375:
2372:
2363:
2360:
2351:
2337:
2328:
2325:
2316:
2315:
2313:
2311:
2292:
2187:Algorithmic bias
2162:Abstract machine
2155:
2150:
2149:
2141:
2136:
2135:
2113:
2088:
1899:greedy algorithm
1729:branch and bound
1628:Knapsack problem
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1389:
1388:
1356:data compression
1348:synthetic rubber
1339:Diamond v. Diehr
1210:computer science
1178:) when used for
1177:
1175:
1173:
1172:
1167:
1144:
1142:
1140:
1139:
1134:
1094:
1092:
1090:
1089:
1084:
1061:
1059:
1057:
1056:
1051:
1024:
1022:
1020:
1019:
1014:
791:
788:
780:
777:
720:verge escapement
673:
670:
638:
635:
613:
609:
606:
595:
592:
538:
535:
529:
513:
505:
359:, attributed to
351:, attributed to
293:effective method
264:(referred to as
240:or to perform a
231:
230:
229:
228:
221:
218:
217:
214:
211:
208:
205:
202:
199:
196:
193:
179:computer science
147:
140:
136:
133:
127:
104:
103:
96:
89:
82:
78:
75:
69:
49:
48:
41:
21:
6009:
6008:
6004:
6003:
6002:
6000:
5999:
5998:
5974:
5973:
5972:
5967:
5949:
5880:Graph traversal
5833:
5757:Data structures
5752:
5746:Data structures
5743:
5713:
5708:
5680:
5642:
5626:
5588:
5536:
5502:Poisson algebra
5478:
5360:
5353:
5311:
5207:Operator theory
5105:
5063:
5029:Tensor software
5005:
4954:Automata theory
4923:
4918:
4881:Wayback Machine
4821:
4820:
4802:
4799:
4794:
4788:
4773:
4767:
4754:
4748:Wayback Machine
4732:Wayback Machine
4706:
4700:
4687:
4681:
4668:
4662:
4649:
4643:
4628:
4622:
4604:
4600:
4598:Further reading
4587:
4570:The Undecidable
4538:Turing, Alan M.
4536:
4531:The Undecidable
4499:Turing, Alan M.
4497:
4491:
4478:
4463:
4450:
4444:
4427:
4421:
4406:
4400:
4387:
4380:
4374:
4351:
4346:
4341:The Undecidable
4337:The Undecidable
4315:10.2307/2269059
4297:
4291:
4278:
4269:The Undecidable
4247:10.2307/2269031
4229:
4219:
4199:
4149:
4142:Kosovsky, N.K.
4130:
4124:
4108:
4102:
4086:
4077:The Undecidable
4062:10.2307/1990131
4039:
4030:The Undecidable
4021:
4019:
3982:
3967:
3945:
3930:
3914:
3888:
3873:
3867:
3845:Dennett, Daniel
3843:
3816:
3798:Paul E. Black.
3797:
3788:
3736:
3720:
3689:
3669:
3663:
3650:
3645:The Undecidable
3623:10.2307/2269030
3608:
3588:10.2307/2269326
3570:
3565:The Undecidable
3561:The Undecidable
3547:10.2307/2371045
3529:
3512:
3499:
3494:Turing machines
3488:
3464:
3450:
3437:
3429:
3414:
3402:
3372:10.2307/1993169
3353:Axt, P (1959).
3352:
3348:
3343:
3342:
3330:
3326:
3302:10.1.1.145.4600
3286:
3283:convex polytope
3276:
3272:
3262:
3260:
3245:
3222:
3221:
3217:
3207:
3205:
3187:
3186:
3182:
3177:
3173:
3168:
3164:
3159:
3155:
3137:Thomas E. Kurtz
3131:
3127:
3117:
3115:
3107:
3106:
3102:
3092:
3090:
3083:
3061:
3060:
3056:
3050:Wayback Machine
3039:Wayback Machine
3025:
3021:
3011:
3009:
2999:
2998:
2994:
2949:
2948:
2944:
2939:
2935:
2931:Sipser 2006:157
2930:
2919:
2914:
2910:
2905:
2901:
2896:
2892:
2883:
2879:
2874:
2870:
2865:
2861:
2856:
2852:
2847:
2843:
2833:
2831:
2822:Ast, Courtney.
2821:
2820:
2816:
2809:
2793:
2792:
2788:
2780:
2741:
2736:
2735:
2731:
2724:
2711:
2710:
2703:
2696:
2683:
2682:
2673:
2650:10.2307/3027363
2631:
2630:
2626:
2617:
2613:
2606:
2589:
2588:
2581:
2574:
2561:
2560:
2549:
2544:
2540:
2535:
2531:
2518:
2516:
2514:
2499:
2498:
2494:
2481:
2479:
2472:
2454:
2453:
2449:
2444:
2440:
2434:
2430:
2425:
2421:
2414:
2412:
2408:
2395:
2391:
2382:
2378:
2373:
2366:
2361:
2354:
2338:
2331:
2326:
2319:
2309:
2307:
2294:
2293:
2286:
2281:
2276:
2151:
2144:
2137:
2130:
2127:
2122:
2085:
2022:
1970:
1964:
1826:
1770:polynomial time
1662:
1578:towers of Hanoi
1533:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1384:
1376:
1327:
1325:Software patent
1321:
1297:Turing complete
1289:
1265:
1259:
1246:
1240:
1202:
1190:Main articles:
1188:
1149:
1148:
1146:
1110:
1109:
1107:
1066:
1065:
1063:
1033:
1032:
1030:
996:
995:
993:
985:
979:
964:
923:
921:Turing machines
887:
885:Representations
879:Turing machines
863:lambda calculus
814:
789:
778:
764:Hollerith cards
756:
744:Turing-complete
736:Charles Babbage
716:
711:
671:
636:
611:
607:
593:
548:
539:
533:
530:
523:
514:
503:
446:The concept of
414:
407:
377:Dixit Algorismi
361:Adelard of Bath
353:John of Seville
325:
297:formal language
279:In contrast, a
250:data processing
225:
224:
223:
190:
186:
148:
137:
131:
128:
120:help improve it
117:
105:
101:
90:
79:
73:
70:
63:
50:
46:
39:
28:
23:
22:
15:
12:
11:
5:
6007:
6005:
5997:
5996:
5991:
5986:
5976:
5975:
5969:
5968:
5966:
5965:
5960:
5954:
5951:
5950:
5948:
5947:
5942:
5937:
5932:
5927:
5922:
5917:
5912:
5907:
5902:
5897:
5892:
5887:
5882:
5877:
5872:
5867:
5862:
5857:
5852:
5847:
5841:
5839:
5835:
5834:
5832:
5831:
5826:
5821:
5816:
5811:
5806:
5801:
5796:
5791:
5786:
5781:
5776:
5771:
5766:
5760:
5758:
5754:
5753:
5744:
5742:
5741:
5734:
5727:
5719:
5710:
5709:
5707:
5706:
5693:
5685:
5682:
5681:
5679:
5678:
5673:
5668:
5663:
5662:
5661:
5650:
5648:
5644:
5643:
5641:
5640:
5634:
5632:
5628:
5627:
5625:
5624:
5617:
5612:
5607:
5602:
5596:
5594:
5590:
5589:
5587:
5586:
5581:
5576:
5571:
5566:
5561:
5556:
5550:
5548:
5542:
5541:
5538:
5537:
5535:
5534:
5529:
5524:
5519:
5514:
5509:
5504:
5499:
5494:
5488:
5486:
5477:
5476:
5475:
5474:
5469:
5459:
5458:
5457:
5452:
5442:
5437:
5436:
5435:
5425:
5424:
5423:
5418:
5413:
5408:
5403:
5398:
5393:
5383:
5382:
5381:
5376:
5365:
5363:
5355:
5354:
5352:
5351:
5346:
5341:
5332:
5321:
5319:
5313:
5312:
5310:
5309:
5304:
5303:
5302:
5297:
5292:
5287:
5277:
5276:
5275:
5270:
5265:
5260:
5250:
5249:
5248:
5243:
5238:
5233:
5223:
5222:
5221:
5211:
5210:
5209:
5204:
5194:
5193:
5192:
5190:Control theory
5187:
5177:
5176:
5175:
5170:
5165:
5155:
5154:
5153:
5148:
5143:
5133:
5132:
5131:
5121:
5115:
5113:
5107:
5106:
5104:
5103:
5098:
5093:
5088:
5083:
5077:
5075:
5069:
5068:
5065:
5064:
5062:
5061:
5056:
5051:
5046:
5041:
5036:
5031:
5026:
5021:
5015:
5013:
5004:
5003:
4998:
4993:
4988:
4983:
4982:
4981:
4971:
4966:
4961:
4956:
4951:
4950:
4949:
4944:
4933:
4931:
4925:
4924:
4919:
4917:
4916:
4909:
4902:
4894:
4888:
4887:
4871:
4862:
4852:
4851:
4847:
4846:
4837:
4818:
4798:
4797:External links
4795:
4793:
4792:
4786:
4771:
4765:
4752:
4736:
4717:
4704:
4698:
4685:
4679:
4666:
4660:
4647:
4641:
4626:
4620:
4601:
4599:
4596:
4595:
4594:
4586:
4585:
4573:
4534:
4495:
4489:
4476:
4475:" (p. 4).
4461:
4448:
4442:
4425:
4419:
4404:
4398:
4385:
4372:
4344:
4295:
4289:
4276:
4241:(3): 103–105.
4227:
4217:
4201:Minsky, Marvin
4197:
4187:
4167:(7): 424–436.
4147:
4140:
4128:
4122:
4106:
4100:
4084:
4037:
3998:(5): 727–742.
3980:
3965:
3947:Hodges, Andrew
3943:
3928:
3912:
3900:
3886:
3871:
3865:
3841:
3814:
3785:
3778:Claude Shannon
3734:
3718:
3687:
3667:
3661:
3648:
3617:(3): 101–102.
3572:Church, Alonzo
3568:
3541:(2): 345–363.
3531:Church, Alonzo
3527:
3516:
3510:
3497:
3486:
3466:Boolos, George
3462:
3448:
3435:
3408:Gurevich, Yuri
3404:Blass, Andreas
3400:
3385:
3349:
3347:
3344:
3341:
3340:
3324:
3270:
3243:
3215:
3180:
3171:
3162:
3153:
3133:John G. Kemeny
3125:
3100:
3081:
3054:
3019:
2992:
2965:(2): 341–378.
2942:
2933:
2917:
2908:
2899:
2890:
2877:
2868:
2859:
2857:Bolter 1984:26
2850:
2848:Bolter 1984:24
2841:
2824:"Eratosthenes"
2814:
2807:
2786:
2752:(7): 671–677.
2729:
2722:
2701:
2694:
2671:
2624:
2611:
2604:
2579:
2572:
2547:
2538:
2529:
2512:
2492:
2470:
2447:
2438:
2428:
2419:
2406:
2389:
2376:
2364:
2352:
2329:
2317:
2283:
2282:
2280:
2277:
2275:
2274:
2273:
2272:
2267:
2257:
2252:
2247:
2242:
2237:
2232:
2224:
2219:
2214:
2209:
2204:
2199:
2194:
2189:
2184:
2179:
2174:
2169:
2164:
2158:
2157:
2156:
2142:
2126:
2123:
2121:
2120:
2111:
2023:
2009:
2008:
1993:
1992:
1989:
1986:
1983:
1963:
1960:
1959:
1958:
1926:
1923:
1895:
1892:
1864:
1859:
1839:
1825:
1822:
1821:
1820:
1817:
1812:
1790:
1784:
1783:
1773:
1758:
1757:
1741:
1736:
1709:
1706:
1683:
1680:
1676:
1661:
1658:
1657:
1656:
1634:
1631:
1619:
1616:
1602:
1599:
1584:
1581:
1554:
1548:
1547:
1535:
1534:
1393:
1383:
1380:
1375:
1374:Classification
1372:
1320:
1317:
1301:spaghetti code
1288:
1285:
1281:big O notation
1258:
1255:
1242:Main article:
1239:
1236:
1187:
1184:
1165:
1162:
1159:
1156:
1132:
1129:
1126:
1123:
1120:
1117:
1082:
1079:
1076:
1073:
1049:
1046:
1043:
1040:
1027:big O notation
1012:
1009:
1006:
1003:
981:Main article:
978:
975:
963:
960:
955:Turing machine
927:Turing machine
922:
919:
913:(processed by
911:control tables
886:
883:
813:
810:
805:George Stibitz
755:
752:
715:
712:
710:
707:
637: 1550 BC
612: 1600 BC
594: 2500 BC
547:
544:
541:
540:
517:
515:
508:
502:
499:
462:and rules. In
456:formal systems
433:infinite loops
406:
403:
324:
321:
150:
149:
108:
106:
99:
92:
91:
53:
51:
44:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6006:
5995:
5992:
5990:
5987:
5985:
5982:
5981:
5979:
5964:
5961:
5959:
5956:
5955:
5952:
5946:
5943:
5941:
5938:
5936:
5933:
5931:
5928:
5926:
5923:
5921:
5918:
5916:
5913:
5911:
5908:
5906:
5903:
5901:
5898:
5896:
5895:Hash function
5893:
5891:
5888:
5886:
5883:
5881:
5878:
5876:
5873:
5871:
5868:
5866:
5863:
5861:
5858:
5856:
5853:
5851:
5850:Binary search
5848:
5846:
5843:
5842:
5840:
5836:
5830:
5827:
5825:
5822:
5820:
5817:
5815:
5812:
5810:
5807:
5805:
5802:
5800:
5797:
5795:
5792:
5790:
5787:
5785:
5782:
5780:
5777:
5775:
5772:
5770:
5767:
5765:
5762:
5761:
5759:
5755:
5751:
5747:
5740:
5735:
5733:
5728:
5726:
5721:
5720:
5717:
5705:
5701:
5697:
5694:
5692:
5691:
5687:
5686:
5683:
5677:
5674:
5672:
5669:
5667:
5664:
5660:
5657:
5656:
5655:
5652:
5651:
5649:
5647:Organizations
5645:
5639:
5636:
5635:
5633:
5629:
5622:
5618:
5616:
5613:
5611:
5608:
5606:
5603:
5601:
5598:
5597:
5595:
5591:
5585:
5582:
5580:
5577:
5575:
5572:
5570:
5567:
5565:
5562:
5560:
5557:
5555:
5552:
5551:
5549:
5547:
5543:
5533:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5513:
5510:
5508:
5507:Quantum group
5505:
5503:
5500:
5498:
5495:
5493:
5490:
5489:
5487:
5485:
5481:
5473:
5470:
5468:
5465:
5464:
5463:
5462:Supersymmetry
5460:
5456:
5453:
5451:
5448:
5447:
5446:
5445:String theory
5443:
5441:
5438:
5434:
5431:
5430:
5429:
5426:
5422:
5419:
5417:
5414:
5412:
5409:
5407:
5404:
5402:
5399:
5397:
5394:
5392:
5389:
5388:
5387:
5384:
5380:
5377:
5375:
5372:
5371:
5370:
5367:
5366:
5364:
5362:
5356:
5350:
5347:
5345:
5344:Path integral
5342:
5340:
5336:
5333:
5330:
5326:
5325:Distributions
5323:
5322:
5320:
5318:
5314:
5308:
5305:
5301:
5298:
5296:
5293:
5291:
5288:
5286:
5283:
5282:
5281:
5278:
5274:
5271:
5269:
5266:
5264:
5261:
5259:
5256:
5255:
5254:
5251:
5247:
5244:
5242:
5239:
5237:
5234:
5232:
5229:
5228:
5227:
5224:
5220:
5217:
5216:
5215:
5212:
5208:
5205:
5203:
5200:
5199:
5198:
5195:
5191:
5188:
5186:
5183:
5182:
5181:
5178:
5174:
5171:
5169:
5166:
5164:
5161:
5160:
5159:
5156:
5152:
5149:
5147:
5144:
5142:
5139:
5138:
5137:
5134:
5130:
5127:
5126:
5125:
5122:
5120:
5117:
5116:
5114:
5112:
5108:
5102:
5099:
5097:
5094:
5092:
5091:Combinatorics
5089:
5087:
5084:
5082:
5079:
5078:
5076:
5074:
5070:
5060:
5057:
5055:
5052:
5050:
5047:
5045:
5042:
5040:
5037:
5035:
5032:
5030:
5027:
5025:
5022:
5020:
5017:
5016:
5014:
5012:
5008:
5002:
4999:
4997:
4994:
4992:
4989:
4987:
4984:
4980:
4977:
4976:
4975:
4972:
4970:
4967:
4965:
4964:Coding theory
4962:
4960:
4957:
4955:
4952:
4948:
4945:
4943:
4940:
4939:
4938:
4935:
4934:
4932:
4930:
4929:Computational
4926:
4922:
4915:
4910:
4908:
4903:
4901:
4896:
4895:
4892:
4886:
4882:
4878:
4875:
4872:
4870:
4866:
4863:
4861:
4857:
4854:
4853:
4849:
4848:
4845:
4841:
4838:
4833:
4832:
4827:
4824:
4819:
4815:
4811:
4810:
4805:
4801:
4800:
4796:
4789:
4783:
4779:
4778:
4772:
4768:
4762:
4758:
4753:
4750:
4749:
4745:
4742:
4737:
4734:
4733:
4729:
4726:
4721:
4718:
4714:
4710:
4705:
4701:
4695:
4691:
4686:
4682:
4676:
4672:
4667:
4663:
4657:
4653:
4648:
4644:
4638:
4634:
4633:
4627:
4623:
4617:
4613:
4612:
4607:
4603:
4602:
4597:
4593:
4589:
4588:
4583:
4582:
4577:
4574:
4571:
4568:Reprinted in
4564:
4559:
4555:
4551:
4547:
4543:
4539:
4535:
4532:
4526:
4522:
4518:
4514:
4510:
4506:
4505:
4500:
4496:
4492:
4486:
4482:
4477:
4474:
4470:
4464:
4458:
4454:
4449:
4445:
4443:9780674930469
4439:
4434:
4433:
4426:
4422:
4416:
4412:
4411:
4405:
4401:
4395:
4391:
4386:
4379:
4375:
4369:
4365:
4361:
4357:
4350:
4345:
4342:
4338:
4335:Reprinted in
4332:
4328:
4324:
4320:
4316:
4312:
4308:
4304:
4300:
4296:
4292:
4286:
4282:
4277:
4274:
4270:
4267:Reprinted in
4264:
4260:
4256:
4252:
4248:
4244:
4240:
4236:
4232:
4228:
4226:
4220:
4214:
4209:
4208:
4202:
4198:
4195:
4191:
4188:
4184:
4180:
4175:
4170:
4166:
4162:
4161:
4156:
4152:
4148:
4145:
4141:
4137:
4133:
4132:Knuth, Donald
4129:
4125:
4119:
4115:
4111:
4110:Knuth, Donald
4107:
4103:
4097:
4093:
4089:
4085:
4082:
4081:Church thesis
4078:
4075:Reprinted in
4072:
4068:
4063:
4058:
4054:
4050:
4046:
4042:
4038:
4035:
4031:
4022:September 30,
4017:
4013:
4009:
4005:
4001:
3997:
3993:
3989:
3985:
3981:
3978:
3977:0-671-49207-1
3974:
3968:
3962:
3958:
3954:
3953:
3948:
3944:
3941:
3940:0-674-32449-8
3937:
3931:
3925:
3921:
3917:
3913:
3910:
3909:
3904:
3903:Yuri Gurevich
3901:
3899:
3898:0-312-10409-X
3895:
3889:
3883:
3879:
3878:
3872:
3868:
3862:
3858:
3853:
3852:
3846:
3842:
3837:
3832:
3828:
3824:
3820:
3815:
3811:
3807:
3806:
3801:
3795:
3786:
3783:
3779:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3743:
3737:
3731:
3727:
3723:
3722:Davis, Martin
3719:
3716:
3712:
3708:
3704:
3700:
3699:Alonzo Church
3696:
3690:
3684:
3679:
3678:
3672:
3671:Davis, Martin
3668:
3664:
3658:
3654:
3649:
3646:
3643:Reprinted in
3640:
3636:
3632:
3628:
3624:
3620:
3616:
3612:
3605:
3601:
3597:
3593:
3589:
3585:
3581:
3577:
3573:
3569:
3566:
3562:
3559:Reprinted in
3556:
3552:
3548:
3544:
3540:
3536:
3532:
3528:
3525:
3521:
3517:
3513:
3507:
3503:
3498:
3495:
3489:
3483:
3478:
3477:
3471:
3467:
3463:
3461:
3460:0-8078-4108-0
3457:
3451:
3445:
3441:
3436:
3428:
3424:
3420:
3413:
3409:
3405:
3401:
3398:
3397:0-07-004357-4
3394:
3390:
3386:
3382:
3378:
3373:
3368:
3365:(1): 85–105.
3364:
3360:
3356:
3351:
3350:
3345:
3337:
3333:
3328:
3325:
3320:
3316:
3312:
3308:
3303:
3298:
3294:
3290:
3284:
3280:
3274:
3271:
3263:September 19,
3258:
3254:
3250:
3246:
3240:
3236:
3232:
3228:
3227:
3219:
3216:
3204:
3200:
3196:
3195:
3190:
3184:
3181:
3175:
3172:
3166:
3163:
3157:
3154:
3150:
3149:0-201-13433-0
3146:
3142:
3138:
3134:
3129:
3126:
3114:
3110:
3104:
3101:
3088:
3084:
3078:
3074:
3073:
3068:
3064:
3058:
3055:
3051:
3047:
3044:
3043:sFFT Web Page
3040:
3036:
3033:
3029:
3023:
3020:
3007:
3003:
2996:
2993:
2988:
2984:
2980:
2976:
2972:
2968:
2964:
2960:
2956:
2955:Zimek, Arthur
2952:
2946:
2943:
2937:
2934:
2928:
2926:
2924:
2922:
2918:
2912:
2909:
2903:
2900:
2897:Davis 2000:14
2894:
2891:
2887:
2881:
2878:
2872:
2869:
2863:
2860:
2854:
2851:
2845:
2842:
2829:
2825:
2818:
2815:
2810:
2804:
2800:
2796:
2790:
2787:
2779:
2775:
2771:
2767:
2763:
2759:
2755:
2751:
2747:
2740:
2733:
2730:
2725:
2723:9783319016283
2719:
2715:
2708:
2706:
2702:
2697:
2691:
2687:
2680:
2678:
2676:
2672:
2667:
2663:
2659:
2655:
2651:
2647:
2643:
2639:
2635:
2628:
2625:
2621:
2615:
2612:
2607:
2601:
2597:
2593:
2586:
2584:
2580:
2575:
2573:9783642181924
2569:
2565:
2558:
2556:
2554:
2552:
2548:
2542:
2539:
2533:
2530:
2526:
2515:
2513:9780262731447
2509:
2505:
2504:
2496:
2493:
2489:
2477:
2473:
2471:9780262536370
2467:
2463:
2462:
2457:
2451:
2448:
2442:
2439:
2432:
2429:
2423:
2420:
2410:
2407:
2403:
2399:
2393:
2390:
2386:
2380:
2377:
2371:
2369:
2365:
2359:
2357:
2353:
2350:
2346:
2342:
2336:
2334:
2330:
2324:
2322:
2318:
2305:
2301:
2297:
2291:
2289:
2285:
2278:
2271:
2268:
2266:
2263:
2262:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2236:
2233:
2230:
2229:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2159:
2154:
2148:
2143:
2140:
2134:
2129:
2124:
2117:
2112:
2108:
2104:
2100:
2096:
2092:
2087:
2086:
2084:
2081:
2078:
2074:
2071:
2067:
2063:
2060:
2057:
2053:
2050:
2047:
2044:
2041:
2037:
2033:
2029:
2026:
2020:
2016:
2012:
2007:
2005:
2001:
1997:
1990:
1987:
1984:
1981:
1980:
1979:
1978:
1974:
1969:
1961:
1956:
1952:
1948:
1944:
1940:
1935:
1931:
1927:
1924:
1921:
1917:
1913:
1909:
1905:
1900:
1896:
1893:
1890:
1885:
1881:
1877:
1873:
1869:
1865:
1863:
1860:
1857:
1853:
1849:
1845:
1840:
1838:
1835:
1834:
1833:
1831:
1823:
1818:
1816:
1815:Back tracking
1813:
1810:
1809:
1803:
1799:
1795:
1791:
1789:
1786:
1785:
1781:
1777:
1774:
1771:
1767:
1763:
1760:
1759:
1755:
1751:
1747:
1742:
1740:
1737:
1734:
1730:
1726:
1722:
1718:
1714:
1710:
1707:
1704:
1700:
1696:
1695:merge sorting
1692:
1688:
1684:
1681:
1677:
1674:
1671:
1670:
1669:
1667:
1659:
1654:
1650:
1646:
1642:
1638:
1635:
1632:
1629:
1624:
1620:
1617:
1614:
1610:
1606:
1603:
1600:
1597:
1593:
1589:
1585:
1582:
1579:
1575:
1571:
1567:
1563:
1559:
1555:
1552:
1551:
1545:
1541:
1537:
1536:
1391:
1390:
1387:
1381:
1379:
1373:
1371:
1369:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1340:
1335:
1334:
1326:
1318:
1316:
1314:
1310:
1306:
1302:
1298:
1294:
1286:
1284:
1282:
1278:
1274:
1270:
1264:
1256:
1254:
1251:
1245:
1237:
1235:
1232:
1227:
1224:
1220:
1215:
1211:
1207:
1201:
1197:
1193:
1185:
1183:
1181:
1180:table lookups
1160:
1154:
1127:
1124:
1121:
1115:
1105:
1104:binary search
1101:
1096:
1077:
1071:
1044:
1038:
1028:
1007:
1001:
991:
984:
976:
974:
973:
969:
961:
959:
956:
952:
951:assembly code
948:
944:
943:state diagram
940:
939:control table
936:
932:
928:
920:
918:
916:
912:
908:
904:
903:drakon-charts
900:
896:
892:
884:
882:
880:
876:
873:of 1936, and
872:
871:Formulation 1
868:
864:
860:
859:Alonzo Church
856:
852:
848:
844:
840:
839:David Hilbert
836:
835:
826:
822:
818:
812:Formalization
811:
809:
806:
802:
797:
795:
784:
773:
769:
765:
761:
760:Jacquard loom
753:
751:
749:
745:
741:
737:
733:
729:
725:
721:
713:
708:
706:
704:
700:
699:cryptanalysis
696:
692:
687:
685:
681:
680:Kerala School
677:
676:Shulba Sutras
672: 300 BC
666:
665:
660:
656:
652:
651:
646:
642:
632:
628:
623:
621:
617:
603:
600:. During the
599:
589:and dated to
588:
584:
580:
576:
571:
569:
565:
561:
557:
553:
545:
537:
527:
521:
518:This section
516:
512:
507:
506:
500:
498:
496:
492:
489:implementing
488:
484:
480:
476:
471:
469:
465:
461:
457:
453:
449:
444:
443:
438:
434:
430:
427:
424:procedure or
423:
419:
412:
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402:
400:
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392:
388:
384:
383:
378:
374:
370:
366:
362:
358:
354:
350:
346:
342:
338:
334:
330:
322:
320:
318:
314:
313:deterministic
310:
306:
302:
298:
294:
289:
287:
282:
277:
275:
271:
268:), achieving
267:
263:
259:
255:
251:
247:
243:
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235:
227:
220:
184:
180:
176:
169:
165:
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146:
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125:
121:
115:
114:
109:This article
107:
98:
97:
88:
85:
77:
67:
61:
59:
54:This article
52:
43:
42:
37:
33:
19:
5920:Root-finding
5845:Backtracking
5809:Segment tree
5779:Fenwick tree
5749:
5702: /
5698: /
5688:
5564:Optimization
5527:Superalgebra
5386:Field theory
5359:Mathematical
5337: /
5185:Chaos theory
5168:Gauge theory
5096:Graph theory
4991:Cryptography
4936:
4829:
4807:
4776:
4756:
4739:
4723:
4708:
4689:
4670:
4651:
4631:
4610:
4580:
4569:
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4508:
4507:. Series 2.
4502:
4480:
4472:
4468:
4452:
4431:
4409:
4389:
4355:
4340:
4336:
4309:(2): 53–60.
4306:
4302:
4299:Rosser, J.B.
4280:
4268:
4238:
4234:
4224:
4206:
4193:
4164:
4158:
4143:
4135:
4113:
4091:
4076:
4055:(1): 41–73.
4052:
4048:
4033:
4029:
4020:. Retrieved
4016:the original
3995:
3991:
3955:. New York:
3951:
3919:
3907:
3876:
3850:
3826:
3822:
3803:
3782:Howard Aiken
3774:Ada Lovelace
3725:
3676:
3652:
3644:
3614:
3610:
3582:(1): 40–41.
3579:
3575:
3564:
3560:
3538:
3534:
3523:
3504:. Springer.
3501:
3493:
3475:
3439:
3422:
3418:
3388:
3362:
3358:
3346:Bibliography
3335:
3327:
3292:
3288:
3273:
3261:. Retrieved
3229:. Springer.
3225:
3218:
3206:. Retrieved
3192:
3183:
3174:
3165:
3156:
3140:
3128:
3116:. Retrieved
3113:Khan Academy
3112:
3103:
3091:. Retrieved
3071:
3057:
3022:
3010:. Retrieved
2995:
2962:
2958:
2945:
2936:
2911:
2902:
2893:
2885:
2880:
2871:
2862:
2853:
2844:
2834:February 27,
2832:. Retrieved
2817:
2798:
2795:Aaboe, Asger
2789:
2778:the original
2749:
2745:
2732:
2713:
2685:
2644:(2): 76–99.
2641:
2637:
2627:
2614:
2595:
2563:
2541:
2532:
2524:
2517:. Retrieved
2502:
2495:
2487:
2480:. Retrieved
2460:
2450:
2445:Stone 1973:4
2441:
2431:
2422:
2409:
2392:
2384:
2379:
2340:
2310:November 14,
2308:. Retrieved
2299:
2226:
2115:
2106:
2102:
2098:
2094:
2082:
2079:
2076:
2072:
2069:
2065:
2061:
2058:
2055:
2051:
2048:
2045:
2042:
2039:
2035:
2031:
2027:
2024:
2018:
2014:
2010:
1995:
1994:
1976:
1975:
1971:
1939:local search
1908:Huffman Tree
1904:local optima
1871:
1827:
1806:
1733:backtracking
1698:
1663:
1385:
1377:
1337:
1331:
1328:
1319:Legal status
1290:
1266:
1247:
1228:
1203:
1097:
989:
986:
972:
965:
947:machine code
924:
915:interpreters
888:
832:
830:
821:Ada Lovelace
798:
779: 1870s
757:
740:Ada Lovelace
717:
694:
688:
662:
648:
624:
572:
549:
534:October 2023
531:
519:
472:
467:
452:decidability
447:
445:
441:
422:bureaucratic
415:
398:
394:
393:, "number";
390:
380:
376:
373:Latinization
368:
364:
356:
348:
336:
332:
326:
290:
278:
254:conditionals
246:calculations
182:
172:
167:
163:
138:
129:
110:
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71:
58:copy editing
56:may require
55:
18:Algorithmics
5799:Linked list
5704:topics list
5638:Mathematics
5554:Game theory
5455:Topological
5421:Topological
5416:Statistical
5379:Hamiltonian
4826:"Algorithm"
4804:"Algorithm"
4548:: 161–228.
4511:: 230–265.
4190:A.A. Markov
3800:"algorithm"
3762:von Neumann
3295:(1): 1–17.
3028:Piotr Indyk
2746:Commun. ACM
2620:Brahmagupta
2004:pidgin code
1943:tabu search
1884:memoization
1691:recursively
1673:Brute-force
875:Alan Turing
794:Baudot code
790: 1910
783:teleprinter
772:ticker tape
608: 1800
575:Mesopotamia
487:human brain
475:implemented
399:algorithmus
365:alghoarismi
274:Alan Turing
242:computation
175:mathematics
5984:Algorithms
5978:Categories
5935:Sweep line
5910:Randomized
5838:Algorithms
5789:Hash table
5750:algorithms
5610:Psychology
5574:Statistics
5374:Lagrangian
5001:Statistics
4937:Algorithms
4231:Post, Emil
3877:The Abacus
2402:quantities
2349:1402030045
2231:(textbook)
2091:assignment
2000:pseudocode
1798:complexity
1746:randomness
1613:heuristics
1546:flowchart
1538:Recursive
1364:LZW patent
1323:See also:
1261:See also:
1231:Benchmarks
1219:pseudocode
899:flowcharts
895:pseudocode
748:calculator
728:difference
682:, and the
657:, and the
655:Nicomachus
616:Babylonian
610: – c.
491:arithmetic
405:Definition
363:. Hereby,
345:arithmetic
270:automation
262:inferences
162:of number
132:April 2024
74:April 2024
66:editing it
5930:Streaming
5915:Recursion
5615:Sociology
5605:Chemistry
5401:Effective
5396:Conformal
5391:Classical
5263:Geometric
5236:Geometric
4831:MathWorld
4814:EMS Press
4473:algorithm
4090:(1991) .
4012:120517999
3715:Emil Post
3520:Moore, C.
3472:(1999) .
3297:CiteSeerX
3208:March 29,
3203:0099-9660
2979:0219-1377
2766:0001-0782
2658:0049-4925
2011:Algorithm
1566:Iterative
1553:Recursion
1125:
968:flowchart
867:Emil Post
865:of 1936,
796:on tape.
768:telegraph
709:Computers
583:Shuruppak
526:talk page
468:algorithm
448:algorithm
426:cook-book
369:algorismi
323:Etymology
281:heuristic
183:algorithm
5690:Category
5339:analysis
5258:Exterior
5231:Exterior
5111:Analysis
5073:Discrete
4947:analysis
4877:Archived
4816:. 2001 .
4744:Archived
4728:Archived
4722:(2000).
4608:(1985).
4578:(2006),
4378:Archived
4331:39499392
4263:40284503
4203:(1967).
4153:(1979).
4134:(1969).
4112:(1997).
4043:(1943).
3986:(1936).
3949:(1983).
3918:(2001).
3847:(1995).
3724:(2000).
3673:(1965).
3604:42323521
3427:Archived
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3319:13268711
3257:Archived
3253:28836720
3093:June 14,
3087:Archived
3069:(2002).
3046:Archived
3035:Archived
3006:Archived
2987:40772241
2828:Archived
2797:(2001).
2519:July 22,
2476:Archived
2458:(2018).
2385:function
2304:Archived
2125:See also
2097:←
2043:for each
1962:Examples
1666:paradigm
1647:such as
1344:feedback
1291:Per the
1025:, using
851:Herbrand
724:automata
691:Al-Kindi
579:Sumerian
391:arithmos
382:algorism
309:executed
301:function
238:problems
5925:Sorting
5900:Minimax
5700:outline
5631:Related
5600:Biology
5450:Bosonic
5411:Quantum
5361:physics
5327: (
5059:Solvers
4713:Praeger
4323:2269059
4255:2269031
4192:(1954)
4183:2509896
4071:1990131
3770:Babbage
3758:Hilbert
3742:Leibniz
3639:5557237
3631:2269030
3596:2269326
3555:2371045
3381:1993169
3118:June 3,
3012:May 13,
2774:7829945
2666:3027363
2482:May 27,
2103:largest
2095:largest
2083:largest
2073:largest
2066:largest
2036:largest
1912:Kruskal
1852:integer
1275:within
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587:Baghdad
501:History
440:symbols
387:Chaucer
371:is the
118:Please
5905:Online
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1360:Unisys
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1198:, and
1100:effort
937:, and
855:Kleene
825:Note G
678:, the
460:axioms
429:recipe
355:, and
331:wrote
291:As an
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4525:73712
4521:S2CID
4381:(PDF)
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4067:JSTOR
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3139:1985
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2770:S2CID
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2662:JSTOR
2279:Notes
2167:ALGOL
2064:>
2034:null
1889:table
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1713:chess
1570:loops
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847:Gödel
585:near
464:logic
305:empty
222:
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5885:Fold
5829:Trie
5824:Tree
5794:Heap
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3265:2017
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3199:ISSN
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3120:2024
3095:2018
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3014:2014
2975:ISSN
2836:2015
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2654:ISSN
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2521:2020
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38:.
20:)
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