Algorithm - Knowledge (XXG)

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: 3789: 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: 5294: 1141: 4904: 5018: 1679:

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.

1580:

is well understood using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.

1174: 1091: 1058: 1021: 5653: 4843: 4503: 3005: 1560:

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

3987: 5729: 5471: 3804: 2327:

Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247

4897: 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 5516: 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: 4575: 5675: 5023: 1779: 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: 5432: 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

5993: 5033: 4785: 4764: 4697: 4678: 4659: 4640: 4619: 4488: 4460: 4418: 4397: 4371: 4288: 4216: 4121: 4099: 3964: 3927: 3885: 3864: 3733: 3686: 3660: 3509: 3485: 3447: 3287:

Dyer, Martin; Frieze, Alan; Kannan, Ravi (January 1991). "A Random Polynomial-time Algorithm for Approximating the Volume of Convex Bodies".

3242: 3080: 2806: 2693: 2603: 2413:"A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method 5599: 112: 3034: 339:("Addition and subtraction in Indian arithmetic"). In the early 12th century, Latin translations of said al-Khwarizmi texts involving the 123: 3108: 1253:

enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.

5053: 5048: 4868: 2827: 622:. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events. 253: 2738: 4441: 3976: 3939: 3897: 3459: 3396: 3148: 2721: 2571: 2511: 2469: 1248:

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.

3426: 2303: 766:(punch cards), and "telephone switching technologies" led to the development of the first computers. By the mid-19th century, the 5466: 5150: 2244: 1792:

This technique involves solving a difficult problem by transforming it into a better-known problem for which we have (hopefully)

1195: 722:

mechanism that provides the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical

4876: 683: 340: 5213: 4973: 4358:. Intelligent Systems, Control and Automation: Science and Engineering. Vol. 74. Switzerland: Springer. pp. 111–127. 2269: 1807: 1797: 3906: 3567:) where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc. 5140: 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. 3256: 3086: 2475: 1891:

of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity.

57: 5145: 5038: 4813: 2362:"Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2). 2090: 816: 3911:, ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources. 1875: 5914: 5884: 5869: 5699: 5043: 3721: 3670: 2464:. Untimely Meditations. Vol. 14. Translated by Chase, Jefferson. Cambridge, Massachusetts: MIT Press. p. 147. 1879: 1686: 1268: 280: 5988: 5085: 4808: 3045: 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: 1720: 1608: 1363: 1098:

Different algorithms may complete the same task with a different set of instructions in less or more time, space, or '

410: 3001: 416:

One informal definition is "a set of rules that precisely defines a sequence of operations", which would include all

5939: 5813: 5803: 5783: 5703: 5167: 4015: 2227: 1304: 649: 630: 65: 35: 1576:

to solve the given problems. Some problems are naturally suited for one implementation or the other. For example,

5818: 5563: 5496: 5491: 5454: 4985: 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 4377: 4159: 2216: 482: 272:

eventually. Using human characteristics as descriptors of machines in metaphorical ways was already practiced by

257: 4272: 1292: 550:

Since antiquity, step-by-step procedures for solving mathematical problems have been recorded. This includes in

5420: 5415: 5324: 5284: 5000: 2254: 1643:. The term is usually used for those algorithms which seem inherently quantum or use some essential feature of 1539: 1226:

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 5957: 5919: 5609: 3950: 3193: 2234: 2221: 2191: 1954: 1702: 1622: 1604: 914: 842: 640: 159: 3188: 5763: 5614: 5604: 5578: 5400: 5395: 5390: 5385: 5343: 5306: 5252: 4978: 4968: 4946: 3296: 1911: 1867: 1793: 1787: 1689:

repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually

1573: 1561: 1367: 1351: 1312: 1249: 1243: 1222: 1205: 1191: 1099: 982: 934: 551: 300: 31: 5823: 5793: 5531: 5511: 5449: 5378: 5157: 5135: 5110: 5010: 3765: 2259: 2249: 2206: 2201: 2176: 1882:

can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and

1761: 1648: 1595: 1230: 906: 644: 372: 308: 237: 154: 116:

that states a Knowledge (XXG) editor's personal feelings or presents an original argument about a topic.

4579: 3677:

The Undecidable: Basic Papers On Undecidable Propositions, Unsolvable Problems and Computable Functions

2132: 1919: 1723:

specifies rules for moving around a graph and is useful for such problems. This category also includes

831:

In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the

5934: 5909: 5854: 5689: 5583: 5568: 5483: 5410: 5373: 5368: 5358: 5118: 5072: 2264: 2211: 2196: 1929: 1847: 1829: 1738: 1665: 1652: 1332: 1308: 1213: 1199: 930: 833: 663: 626: 619: 555: 451: 316: 3301: 1850:

for directed graphs. If a problem additionally requires that one or more of the unknowns must be an

1229:

Empirical testing is useful because it may uncover unexpected interactions that affect performance.

5983: 5944: 5929: 5874: 5558: 5427: 5338: 5225: 5196: 4920: 4884: 3915: 3062: 2957:(2016). "The (black) art of run-time evaluation: Are we comparing algorithms or implementations?". 1946: 1933: 1915: 1861: 1855: 1801: 1775: 1753: 1640: 1557: 1272: 854: 658: 265: 1693:) until the instances are small enough to solve easily. One such example of divide and conquer is 718:

Bolter credits the invention of the weight-driven clock as "the key invention ," specifically the

5962: 5864: 5859: 5773: 5695: 5348: 5334: 5316: 5299: 5279: 5262: 5172: 4995: 4712: 4572:, pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton. 4520: 4326: 4318: 4258: 4250: 4189: 4178: 4066: 4007: 3956: 3634: 3626: 3599: 3591: 3550: 3376: 3314: 3248: 2982: 2950: 2769: 2661: 2455: 2239: 2171: 2138: 1967: 1888: 1836: 1672: 1591: 1276: 770:, the precursor of the telephone, was in use throughout the world. By the late 19th century, the 702: 597: 567: 559: 494: 285: 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 5924: 5768: 5521: 5257: 5235: 5162: 5123: 5100: 4822: 4781: 4760: 4693: 4674: 4655: 4636: 4615: 4484: 4456: 4437: 4414: 4393: 4367: 4284: 4212: 4117: 4095: 3972: 3960: 3935: 3923: 3893: 3881: 3860: 3741: 3729: 3706: 3682: 3656: 3505: 3481: 3455: 3443: 3392: 3331: 3238: 3198: 3144: 3076: 2974: 2802: 2761: 2717: 2689: 2653: 2599: 2591: 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: 5201: 5179: 5128: 5080: 4941: 4605: 4557: 4549: 4512: 4359: 4310: 4242: 4168: 4056: 3999: 3830: 3761: 3618: 3583: 3542: 3474: 3366: 3306: 3230: 3066: 2966: 2823: 2753: 2645: 2186: 2161: 1898: 1728: 1724: 1627: 1587: 1355: 1347: 1338: 1209: 850: 719: 478: 417: 386: 292: 188: 178: 4144:

Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms

5879: 5545: 5501: 5328: 5272: 5267: 5245: 5206: 5028: 4953: 4880: 4747: 4731: 4150: 3469: 3282: 3136: 3049: 3038: 2777: 1769: 1765: 1577: 1336:). However practical applications of algorithms are sometimes patentable. For example, in 1324: 1296: 1150: 1067: 1034: 997: 878: 862: 743: 735: 723: 360: 352: 296: 249: 4223:

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

5745: 5189: 4630: 4205: 4040: 3983: 3875: 3849: 3844: 3777: 3675: 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".

3132: 2295: 1907: 1749: 1300: 1280: 1026: 954: 926: 804: 763: 618:

clay tablets described algorithms for computing formulas. Algorithms were also used in

525: 474: 3522:, and Costa, J.F. (2000) An analog characterization of the subrecursive functions. In 5977: 5894: 5849: 5506: 5461: 5444: 5090: 4963: 4889: 4562: 4501:(1936–37). "On Computable Numbers, With An Application to the Entscheidungsproblem". 4430: 4408: 4200: 4080: 4011: 3946: 3902: 3757: 3749: 3698: 3571: 3530: 3465: 3407: 3403: 2536:

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: 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: 17: 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: 404: 402: 400: 396: 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: 239: 235: 227: 220: 184: 180: 176: 169: 165: 161: 156: 146: 143: 135: 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: 4545: 4541: 4530: 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: 80: 71: 58:copy editing 56:may require 55: 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 18:Algorythm 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 3410:(2003). 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 1176:⁠ 1147:⁠ 1143:⁠ 1108:⁠ 1093:⁠ 1064:⁠ 1060:⁠ 1031:⁠ 1023:⁠ 994:⁠ 587:Baghdad 501:History 440:symbols 387:Chaucer 371:is the 118:Please 5905:Online 5890:Greedy 5819:String 5273:Vector 5268:Tensor 5246:Vector 5241:Tensor 4942:design 4784: 4763: 4696: 4677: 4658: 4639: 4618: 4523: 4487: 4459: 4440: 4417: 4396: 4370: 4329: 4321: 4287: 4261: 4253: 4215: 4181: 4120: 4098: 4069: 4010: 3975: 3963: 3942:(pbk.) 3938: 3926: 3896: 3884: 3863: 3784:, etc. 3754:Cantor 3732: 3713:, and 3711:Kleene 3707:Rosser 3703:Turing 3685: 3659: 3637: 3629: 3602: 3594: 3553: 3508: 3484: 3458: 3446: 3395: 3379: 3317: 3299: 3289:J. ACM 3279:volume 3251: 3241: 3201: 3147: 3079: 2985: 2977: 2805: 2772: 2764: 2720: 2692: 2664: 2656: 2602: 2570: 2510: 2468: 2347: 2116:return 2080:return 2032:return 2028:L.size 1949:, and 1920:Sollin 1717:graphs 1574:stacks 1503:return 1473:return 1443:return 1360:Unisys 1311:using 1257:Design 1198:, and 1100:effort 937:, and 855:Kleene 825:Note G 678:, the 460:axioms 429:recipe 355:, and 331:wrote 291:As an 5814:Stack 5804:Queue 5784:Graph 5764:Array 5406:Gauge 4525:73712 4521:S2CID 4381:(PDF) 4352:(PDF) 4327:S2CID 4319:JSTOR 4259:S2CID 4251:JSTOR 4179:S2CID 4067:JSTOR 4008:S2CID 3859:–36. 3796:from 3750:Frege 3746:Boole 3695:Gödel 3635:S2CID 3627:JSTOR 3600:S2CID 3592:JSTOR 3551:JSTOR 3430:(PDF) 3415:(PDF) 3377:JSTOR 3315:S2CID 3281:of a 3249:S2CID 3139:1985 2983:S2CID 2781:(PDF) 2770:S2CID 2742:(PDF) 2662:JSTOR 2279:Notes 2167:ALGOL 2064:> 2034:null 1889:table 1880:graph 1744:some 1713:chess 1570:loops 1544:above 847:Gödel 585:near 464:logic 305:empty 222: 181:, an 5885:Fold 5829:Trie 5824:Tree 5794:Heap 5748:and 4782:ISBN 4761:ISBN 4694:ISBN 4675:ISBN 4656:ISBN 4637:ISBN 4616:ISBN 4485:ISBN 4457:ISBN 4438:ISBN 4415:ISBN 4394:ISBN 4368:ISBN 4285:ISBN 4213:ISBN 4118:ISBN 4096:ISBN 4024:2013 3973:ISBN 3961:ISBN 3936:ISBN 3924:ISBN 3894:ISBN 3882:ISBN 3861:ISBN 3810:NIST 3730:ISBN 3683:ISBN 3657:ISBN 3506:ISBN 3482:ISBN 3456:ISBN 3444:ISBN 3393:ISBN 3265:2017 3239:ISBN 3210:2017 3199:ISSN 3145:ISBN 3135:and 3120:2024 3095:2018 3077:ISBN 3014:2014 2975:ISSN 2836:2015 2803:ISBN 2762:ISSN 2718:ISBN 2690:ISBN 2654:ISSN 2600:ISBN 2568:ISBN 2521:2020 2508:ISBN 2484:2019 2466:ISBN 2398:zero 2345:ISBN 2312:2019 2107:item 2099:item 2077:item 2070:then 2062:item 2046:item 2030:= 0 1916:Prim 1828:For 1719:. A 1500:else 1464:> 1452:else 1204:The 738:and 730:and 343:and 248:and 177:and 166:and 4558:hdl 4550:doi 4513:doi 4360:doi 4311:doi 4243:doi 4169:doi 4057:doi 4000:doi 3996:112 3831:doi 3619:doi 3584:doi 3543:doi 3367:doi 3307:doi 3231:doi 2967:doi 2754:doi 2646:doi 2002:or 1928:In 1780:ZPP 1651:or 1594:or 1506:gcd 1476:gcd 1413:int 1404:int 1398:gcd 1395:int 1370:). 1362:'s 1271:or 1250:FFT 1122:log 949:or 909:or 877:'s 869:'s 861:'s 734:of 701:by 686:. 653:by 477:as 395:cf. 367:or 173:In 5980:: 4883:– 4867:– 4858:– 4842:– 4828:. 4812:. 4806:. 4556:. 4546:45 4544:. 4519:. 4509:42 4376:. 4366:. 4325:. 4317:. 4305:. 4257:. 4249:. 4237:. 4177:. 4165:22 4163:. 4157:. 4083:). 4065:. 4053:53 4051:. 4047:. 4006:. 3994:. 3990:. 3971:, 3959:. 3905:, 3892:, 3857:32 3829:. 3825:. 3821:. 3808:. 3802:. 3780:, 3776:, 3772:, 3768:, 3756:, 3752:, 3748:, 3744:, 3709:, 3705:, 3701:, 3697:, 3633:. 3625:. 3613:. 3598:. 3590:. 3578:. 3549:. 3539:58 3537:. 3468:; 3454:, 3425:. 3423:81 3421:. 3417:. 3406:; 3375:. 3363:92 3361:. 3357:. 3313:. 3305:. 3293:38 3291:. 3255:. 3247:. 3237:. 3191:. 3111:. 3085:. 3065:; 2981:. 2973:. 2963:52 2961:. 2920:^ 2768:. 2760:. 2750:15 2748:. 2744:. 2704:^ 2674:^ 2660:. 2652:. 2640:. 2636:. 2582:^ 2550:^ 2523:. 2486:. 2474:. 2367:^ 2355:^ 2332:^ 2320:^ 2302:. 2298:. 2287:^ 2075:← 2068:, 2059:if 2056:do 2054:, 2049:in 2038:← 2025:if 2021:. 2006:: 1945:, 1941:, 1932:, 1918:, 1914:, 1910:, 1897:A 1766:RP 1727:, 1685:A 1564:. 1556:A 1527:); 1497:); 1455:if 1434:== 1425:if 1315:. 1194:, 933:, 905:, 901:, 897:, 893:, 787:c. 776:c. 669:c. 634:c. 614:, 605:c. 591:c. 401:. 216:əm 5738:e 5731:t 5724:v 5623:" 5619:" 5331:) 4913:e 4906:t 4899:v 4834:. 4790:. 4769:. 4715:. 4702:. 4683:. 4664:. 4645:. 4624:. 4566:. 4560:: 4552:: 4527:. 4515:: 4493:. 4465:. 4446:. 4423:. 4402:. 4362:: 4343:) 4333:. 4313:: 4307:4 4293:. 4275:. 4265:. 4245:: 4239:1 4221:. 4185:. 4171:: 4126:. 4104:. 4073:. 4059:: 4026:. 4002:: 3969:. 3932:. 3890:. 3869:. 3839:. 3833:: 3827:7 3812:. 3738:. 3691:. 3665:. 3641:. 3621:: 3615:1 3606:. 3586:: 3580:1 3557:. 3545:: 3514:. 3490:. 3452:. 3399:. 3383:. 3369:: 3321:. 3309:: 3267:. 3233:: 3212:. 3151:. 3122:. 3097:. 3052:. 3016:. 2989:. 2969:: 2838:. 2811:. 2756:: 2726:. 2698:. 2668:. 2648:: 2642:1 2608:. 2576:. 2415:' 2314:. 2114:" 2109:. 2052:L 2040:L 2019:L 2015:L 1957:. 1811:. 1782:. 1772:. 1735:. 1705:. 1655:. 1615:. 1540:C 1530:} 1524:A 1521:- 1518:B 1515:, 1512:A 1509:( 1494:B 1491:, 1488:B 1485:- 1482:A 1479:( 1470:) 1467:B 1461:A 1458:( 1449:; 1446:A 1440:) 1437:0 1431:B 1428:( 1422:{ 1419:) 1416:B 1410:, 1407:A 1401:( 1164:) 1161:n 1158:( 1155:O 1131:) 1128:n 1119:( 1116:O 1081:) 1078:n 1075:( 1072:O 1048:) 1045:1 1042:( 1039:O 1011:) 1008:n 1005:( 1002:O 990:n 853:– 849:– 785:( 774:( 667:( 536:) 532:( 528:. 442:. 413:. 219:/ 213:ð 210:ɪ 207:r 204:ə 201:ɡ 198:l 195:æ 192:ˈ 189:/ 185:( 168:s 164:r 145:) 139:( 134:) 130:( 126:. 87:) 81:( 76:) 72:( 68:. 62:. 38:. 20:)

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index