algorithm for other uses see algorithm disambiguation flowchart of an algorithm euclids algorithm for calculating the greatest common divisor gcd of two numbers a and b in locations named a and b the algorithm proceeds by successive subtractions in two loops if the test b a yields yes or true more accurately the number b in location b is greater than or equal to the number a in location a then the algorithm specifies b b a meaning the number b a replaces the old b similarly if a b then a a b the process terminates when the contents of b is 0 yielding the gcd in a algorithm derived from scott 200913 symbols and drawing style from tausworthe 1977 in mathematics and computer science an algorithm lrmlisten is an unambiguous specification of how to solve a class of problems algorithms can perform calculation data processing and automated reasoning tasks as an effective method an algorithm can be expressed within a finite amount of space and time the concept of algorithm has existed for centuries greek mathematicians used algorithms in for example the sieve of eratosthenes for finding prime numbers and the euclidean algorithm for finding the greatest common divisor of two numbers the word algorithm itself derives from the 9th century mathematician muammad ibn ms alkhwrizm latinized algoritmi a partial formalization of what would become the modern concept of algorithm began with attempts to solve the entscheidungsproblem decision problem posed by david hilbert in 1928 later formalizations were framed as attempts to define effective calculability those formalizations included the gdelherbrandkleene recursive functions of 1930 1934 and 1935 alonzo churchs lambda calculus of 1936 emil posts formulation 1 of 1936 and alan turings turing machines of 193637 and 1939 etymology the word algorithm has its roots in latinizing the name of muhammad ibn musa alkhwarizmi in a first step to algorismus about 825 alkhwarizmi wrote an arabic language treatise on the hinduarabic numeral system which was translated into latin during the 12th century under the title algoritmi de numero indorum this title means algoritmi on the numbers of the indians where algoritmi was the translators latinization of alkhwarizmis name in english it was first used in about 1230 and then by chaucer in 1391 english adopted the french term but it wasnt until the late 19th century that algorithm took on the meaning that it has in modern english another early use of the word is from 1240 in a manual titled carmen de algorismo composed by alexandre de villedieu it begins thus mwparseroutput templatequoteoverflowhiddenmargin1em 0padding0 40pxmwparseroutput templatequote templatequotecitelineheight15emtextalignleftpaddingleft16emmargintop0haec algorismus ars praesens dicitur in qua talibus indorum fruimur bis quinque figuris which translates as algorism is the art by which at present we use those indian figures which number two times five the poem is a few hundred lines long and summarizes the art of calculating with the new style of indian dice or talibus indorum or hindu numerals informal definition for a detailed presentation of the various points of view on the definition of algorithm see algorithm characterizations an informal definition could be a set of rules that precisely defines a sequence of operations a prototypical example of an algorithm is the euclidean algorithm to determine the maximum common divisor of two integers an example there are others is described by the flowchart above and as an example in a later section boolos jeffrey 1974 1999 offer an informal meaning of the word in the following quotation no human being can write fast enough or long enough or small enough smaller and smaller without limit youd be trying to write on molecules on atoms on electrons to list all members of an enumerably infinite set by writing out their names one after another in some notation but humans can do something equally useful in the case of certain enumerably infinite sets they can give explicit instructions for determining the nth member of the set for arbitrary finite n such instructions are to be given quite explicitly in a form in which they could be followed by a computing machine or by a human who is capable of carrying out only very elementary operations on symbols an enumerably infinite set is one whose elements can be put into onetoone correspondence with the integers thus boolos and jeffrey are saying that an algorithm implies instructions for a process that creates output integers from an arbitrary input integer or integers that in theory can be arbitrarily large thus an algorithm can be an algebraic equation such as y m n two arbitrary input variables m and n that produce an output y but various authors attempts to define the notion indicate that the word implies much more than this something on the order of for the addition example precise instructions in language understood by the computer outputinteger y at a specified place and in a specified format the concept of algorithm is also used to define the notion of decidability that notion is central for explaining how formal systems come into being starting from a small set of axioms and rules in logic the time that an algorithm requires to complete cannot be measured as it is not apparently related with our customary physical dimension from such uncertainties that characterize ongoing work stems the unavailability of a definition of algorithm that suits both concrete in some sense and abstract usage of the term formalization algorithms are essential to the way computers process data many computer programs contain algorithms that detail the specific instructions a computer should perform in a specific order to carry out a specified task such as calculating employees paychecks or printing students report cards thus an algorithm can be considered to be any sequence of operations that can be simulated by a turingcomplete system authors who assert this thesis include minsky 1967 savage 1987 and gurevich 2000 minsky but we will also maintain with turing that any procedure which could naturally be called effective can in fact be realized by a simple machine although this may seem extreme the arguments in its favor are hard to refute gurevich turings informal argument in favor of his thesis justifies a stronger thesis every algorithm can be simulated by a turing machine according to savage typically when an algorithm is associated with processing information data can be read from an input source written to an output device and stored for further processing stored data are regarded as part of the internal state of the entity performing the algorithm in practice the state is stored in one or more data structures for some such computational process the algorithm must be rigorously defined specified in the way it applies in all possible circumstances that could arise that is any conditional steps must be systematically dealt with casebycase the criteria for each case must be clear and computable because an algorithm is a precise list of precise steps the order of computation is always crucial to the functioning of the algorithm instructions are usually assumed to be listed explicitly and are described as starting from the top and going down to the bottom an idea that is described more formally by flow of control so far this discussion of the formalization of an algorithm has assumed the premises of imperative programming this is the most common conception and it attempts to describe a task in discrete mechanical means unique to this conception of formalized algorithms is the assignment operation setting the value of a variable it derives from the intuition of memory as a scratchpad there is an example below of such an assignment for some alternate conceptions of what constitutes an algorithm see functional programming and logic programming expressing algorithms algorithms can be expressed in many kinds of notation including natural languages pseudocode flowcharts drakoncharts programming languages or control tables processed by interpreters natural language expressions of algorithms tend to be verbose and ambiguous and are rarely used for complex or technical algorithms pseudocode flowcharts drakoncharts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer but are often used as a way to define or document algorithms there is a wide variety of representations possible and one can express a given turing machine program as a sequence of machine tables see more at finitestate machine state transition table and control table as flowcharts and drakoncharts see more at state diagram or as a form of rudimentary machine code or assembly code called sets of quadruples see more at turing machine representations of algorithms can be classed into three accepted levels of turing machine description 1 highlevel description prose to describe an algorithm ignoring the implementation details at this level we do not need to mention how the machine manages its tape or head 2 implementation description prose used to define the way the turing machine uses its head and the way that it stores data on its tape at this level we do not give details of states or transition function 3 formal description most detailed lowest level gives the turing machines state table for an example of the simple algorithm add mn described in all three levels see algorithmexamples design see also algorithm by design paradigm algorithm design refers to a method or mathematical process for problem solving and engineering algorithms the design of algorithms is part of many solution theories of operation research such as dynamic programming and divideandconquer techniques for designing and implementing algorithm designs are also called algorithm design patterns such as the template method pattern and decorator pattern one of the most important aspects of algorithm design is creating an algorithm that has an efficient runtime also known as its big o typical steps in development of algorithms problem definition development of a model specification of algorithm designing an algorithm checking the correctness of algorithm analysis of algorithm implementation of algorithm program testing documentation preparation implementation logical nand algorithm implemented electronically in 7400 chip most algorithms are intended to be implemented as computer programs however algorithms are also implemented by other means such as in a biological neural network for example the human brain implementing arithmetic or an insect looking for food in an electrical circuit or in a mechanical device computer algorithms flowchart examples of the canonical bhmjacopini structures the sequence rectangles descending the page the whiledo and the ifthenelse the three structures are made of the primitive conditional goto if testtrue then goto step xxx a diamond the unconditional goto rectangle various assignment operators rectangle and halt rectangle nesting of these structures inside assignmentblocks result in complex diagrams cf tausworthe 1977100114 in computer systems an algorithm is basically an instance of logic written in software by software developers to be effective for the intended target computers to produce output from given perhaps null input an optimal algorithm even running in old hardware would produce faster results than a nonoptimal higher time complexity algorithm for the same purpose running in more efficient hardware that is why algorithms like computer hardware are considered technology elegant compact programs good fast programs the notion of simplicity and elegance appears informally in knuth and precisely in chaitin knuth we want good algorithms in some loosely defined aesthetic sense one criterion is the length of time taken to perform the algorithm other criteria are adaptability of the algorithm to computers its simplicity and elegance etc chaitin a program is elegant by which i mean that its the smallest possible program for producing the output that it does chaitin prefaces his definition with ill show you cant prove that a program is elegantsuch a proof would solve the halting problem ibid algorithm versus function computable by an algorithm for a given function multiple algorithms may exist this is true even without expanding the available instruction set available to the programmer rogers observes that it is important to distinguish between the notion of algorithm ie procedure and the notion of function computable by algorithm ie mapping yielded by procedure the same function may have several different algorithms unfortunately there may be a tradeoff between goodness speed and elegance compactnessan elegant program may take more steps to complete a computation than one less elegant an example that uses euclids algorithm appears below computers and computors models of computation a computer or human computor minsky describes a more congenial variation of lambeks abacus model in his very simple bases for computability simulation of an algorithm computer computor language knuth advises the reader that the best way to learn an algorithm is to try it immediately take pen and paper and work through an example this means that the programmer must know a language that is effective relative to the target computing agent computercomputor but what model should be used for the simulation van emde boas observes even if we base complexity theory on abstract instead of concrete machines arbitrariness of the choice of a model remains it is at this point that the notion of simulation enters when speed is being measured the instruction set matters for example the subprogram in euclids algorithm to compute the remainder would execute much faster if the programmer had a modulus instruction available rather than just subtraction or worse just minskys decrement structured programming canonical structures per the churchturing thesis any algorithm can be computed by a model known to be turing complete and per minskys demonstrations turing completeness requires only four instruction typesconditional goto unconditional goto assignment halt kemeny and kurtz observe that while undisciplined use of unconditional gotos and conditional ifthen gotos can result in spaghetti code a programmer can write structured programs using only these instructions on the other hand it is also possible and not too hard to write badly structured programs in a structured language canonical flowchart symbols the graphical aide called a flowchart offers a way to describe and document an algorithm and a computer program of one like 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 sequence goto the diamond ifthenelse and the dot ortie the bhmjacopini canonical structures are made of these primitive shapes substructures 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 examples further information list of algorithms algorithm example an animation of the quicksort algorithm sorting an array of randomized values the red bars mark the pivot element at the start of the animation the element farthest to the right hand side is chosen as the pivot 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 highlevel description english prose as highlevel description if there are no numbers in the set then there is no highest number assume the first number in the set is the largest number in the set 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 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 quasiformal description written in prose but much closer to the highlevel language of a computer program the following is the more formal coding of the algorithm in pseudocode or pidgin code algorithm largestnumber input a list of numbers l output the largest number in the list l if lsize 0 return null largest l for each item in l do if item largest then largest item return largest denotes assignment for instance largest item means that the value of largest changes to the value of item return terminates the algorithm and outputs the following value euclids algorithm further information euclids algorithm the examplediagram of euclids algorithm from tl heath 1908 with more detail added euclid does not go beyond a third measuring and gives no numerical examples nicomachus gives the example of 49 and 21 i subtract the less from the greater 28 is left then again i subtract from this the same 21 for this is possible 7 is left i subtract this from 21 14 is left from which i again subtract 7 for this is possible 7 is left but 7 cannot be subtracted from 7 heath comments that the last phrase is curious but the meaning of it is obvious enough as also the meaning of the phrase about ending at one and the same numberheath 1908300 euclids algorithm to compute the greatest common divisor gcd to two numbers appears as proposition ii in book vii elementary number theory of his elements for euclids method to succeed the starting lengths must satisfy two requirements i the lengths must not be zero and ii the subtraction must be proper ie a test must guarantee that the smaller of the two numbers is subtracted from the larger alternately the two can be equal so their subtraction yields zero euclids original proof adds a third requirement the two lengths must not be prime to one another euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers common measure is in fact the greatest while nicomachus algorithm is the same as euclids when the numbers are prime to one another it yields the number 1 for their common measure so to be precise the following is really nicomachus algorithm a graphical expression of euclids algorithm to find the greatest common divisor for 1599 and 650 1599 6502 299 650 2992 52 299 525 39 52 391 13 39 133 0 computer language for euclids algorithm only a few instruction types are required to execute euclids algorithmsome logical tests conditional goto unconditional goto assignment replacement and subtraction a location is symbolized by upper case letters eg s a etc the varying quantity number in a location is written in lower case letters and usually associated with the locations name for example location l at the start might contain the number l 3009 an inelegant program for euclids algorithm inelegant is a translation of knuths version of the algorithm with a subtractionbased remainderloop replacing his use of division or a modulus instruction derived from knuth 197324 depending on the two numbers inelegant may compute the gcd in fewer steps than elegant the following algorithm is framed as knuths fourstep version of euclids and nicomachus but rather than using division to find the remainder it uses successive subtractions of the shorter length s from the remaining length r until r is less than s the highlevel description shown in boldface is adapted from knuth 197324 input 1 input l s 2 r l e0 3 if r s then the contents of l is the larger number so skip over the exchangesteps 4 5 and 6 goto step 6 else swap the contents of r and s 4 l r this first step is redundant but is useful for later discussion 5 r s 6 s l e1 until the remaining length r in r is less than the shorter length s in s repeatedly subtract the measuring number s in s from the remaining length r in r 7 if s r then done measuring so goto 10 else measure again 8 r r s 9 goto 7 e2 either i the last measure was exact the remainder in r is zero and the program can halt or ii the algorithm must continue the last measure left a remainder in r less than measuring number in s 10 if r 0 then done so goto step 15 else continue to step 11 e3 the nut of euclids algorithm use remainder r to measure what was previously smaller number s l serves as a temporary location 11 l r 12 r s 13 s l 14 goto 7 output 15 print s done 16 halt end stop an elegant program for euclids algorithm the following version of euclids algorithm requires only six core instructions to do what thirteen are required to do by inelegant worse inelegant requires more types of instructions the flowchart of elegant can be found at the top of this article in the unstructured basic language the steps are numbered and the instruction let is the assignment instruction symbolized by 5 rem euclids algorithm for greatest common divisor 6 print type two integers greater than 0 10 input ab 20 if b0 then goto 80 30 if a b then goto 60 40 let bba 50 goto 20 60 let aab 70 goto 20 80 print a 90 end the following version can be used with object oriented languages euclids algorithm for greatest common divisor integer euclidalgorithm int a int b amathabsa bmathabsb while b0 if ab aab else bba return a how elegant works in place of an outer euclid loop elegant shifts back and forth between two coloops an a b loop that computes a a b and a b a loop that computes b b a this works because when at last the minuend m is less than or equal to the subtrahend s difference minuend subtrahend the minuend can become s the new measuring length and the subtrahend can become the new r the length to be measured in other words the sense of the subtraction reverses testing the euclid algorithms does an algorithm do what its author wants it to do a few test cases usually suffice to confirm core functionality one source uses 3009 and 884 knuth suggested 40902 24140 another interesting case is the two relatively prime numbers 14157 and 5950 but exceptional cases must be identified and tested will inelegant perform properly when r s s r r s ditto for elegant b a a b a b yes to all what happens when one number is zero both numbers are zero inelegant computes forever in all cases elegant computes forever when a 0 what happens if negative numbers are entered fractional numbers if the input numbers ie the domain of the function computed by the algorithmprogram is to include only positive integers including zero then the failures at zero indicate that the algorithm and the program that instantiates it is a partial function rather than a total function a notable failure due to exceptions is the ariane 5 flight 501 rocket failure june 4 1996 proof of program correctness by use of mathematical induction knuth demonstrates the application of mathematical induction to an extended version of euclids algorithm and he proposes a general method applicable to proving the validity of any algorithm measuring and improving the euclid algorithms elegance compactness versus goodness speed with only six core instructions elegant is the clear winner compared to inelegant at thirteen instructions however inelegant is faster it arrives at halt in fewer steps algorithm analysis indicates why this is the case elegant does two conditional tests in every subtraction loop whereas inelegant only does one as the algorithm usually requires many loopthroughs on average much time is wasted doing a b 0 test that is needed only after the remainder is computed can the algorithms be improved once the programmer judges a program fit and effectivethat is it computes the function intended by its authorthen the question becomes can it be improved the compactness of inelegant can be improved by the elimination of five steps but chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm rather it can only be done heuristically ie by exhaustive search examples to be found at busy beaver trial and error cleverness insight application of inductive reasoning etc observe that steps 4 5 and 6 are repeated in steps 11 12 and 13 comparison with elegant provides a hint that these steps together with steps 2 and 3 can be eliminated this reduces the number of core instructions from thirteen to eight which makes it more elegant than elegant at nine steps the speed of elegant can be improved by moving the b0 test outside of the two subtraction loops this change calls for the addition of three instructions b 0 a 0 goto now elegant computes the examplenumbers faster whether this is always the case for any given a b and r s would require a detailed analysis algorithmic analysis analysis of algorithms 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 the sorting algorithm above has a time requirement of on using the big o notation with n as the length of the list at all times the algorithm only needs to remember two values the largest number found so far and its current position in the input list therefore it is said to have a space requirement of o1 if the space required to store the input numbers is not counted or on if it is counted different algorithms may complete the same task with a different set of instructions in less or more time space or effort than others for example a binary search algorithm with cost olog n outperforms a sequential search cost on when used for table lookups on sorted lists or arrays formal versus empirical empirical algorithmics profiling computer programming and program optimization the analysis and study of algorithms is a discipline of computer science and is often practiced abstractly without the use of a specific programming language 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 pseudocode 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 algorithmic efficiency 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 critical scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign empirical testing is useful because it may uncover unexpected interactions that affect performance benchmarks may be used to compare beforeafter 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 execution efficiency algorithmic efficiency to illustrate the potential improvements possible even in well established algorithms a recent significant innovation relating to fft algorithms used heavily in the field of image processing can decrease processing time up to 1000 times for applications like medical imaging speedups of this magnitude enable computing devices that make extensive use of image processing like digital cameras and medical equipment to consume less power classification there are various ways to classify algorithms each with its own merits by implementation one way to classify algorithms is by implementation means recursion a recursive algorithm 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 functional programming iterative algorithms use repetitive constructs like loops and sometimes additional data structures like stacks to solve the given problems some problems are naturally suited for one implementation or the other for example towers of hanoi is well understood using recursive implementation every recursive version has an equivalent but possibly more or less complex iterative version and vice versa logical an algorithm may be viewed as controlled logical deduction this notion may be expressed as algorithm logic control the logic component expresses the axioms that may be used in the computation and the control component determines the way in which deduction is applied to the axioms this is the basis for the logic programming paradigm in pure logic programming languages the control component is fixed and algorithms are specified by supplying only the logic component the appeal of this approach is the elegant semantics a change in the axioms produces a welldefined change in the algorithm serial parallel or distributed algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time those computers are sometimes called serial computers an algorithm designed for such an environment is called a serial algorithm as opposed to parallel algorithms or distributed algorithms parallel algorithms take advantage of computer architectures where several processors can work on a problem at the same time whereas distributed algorithms utilize multiple machines connected with a computer network parallel or distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together 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 some problems have no parallel algorithms and are called inherently serial problems deterministic or nondeterministic deterministic algorithms solve the problem with exact decision at every step of the algorithm whereas nondeterministic algorithms solve problems via guessing although typical guesses are made more accurate through the use of heuristics exact or approximate while many algorithms reach an exact solution approximation algorithms seek an approximation that is closer to the true solution 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 knapsack problem the knapsack problem is a problem where there is a set of given items the goal of the problem is to pack the knapsack to get the maximum total value each item has some weight and some value total weight that we can carry is no more than some fixed number x so we must consider weights of items as well as their value quantum algorithm they run on a realistic model of quantum computation the term is usually used for those algorithms which seem inherently quantum or use some essential feature of quantum computation such as quantum superposition or quantum entanglement by design paradigm another way of classifying algorithms is by their design methodology or paradigm there is a certain number of paradigms each different from the other furthermore each of these categories include many different types of algorithms some common paradigms are bruteforce or exhaustive search this is the naive method of trying every possible solution to see which is best divide and conquer a divide and conquer algorithm repeatedly reduces an instance of a problem to one or more smaller instances of the same problem usually recursively until the instances are small enough to solve easily one such example of divide and conquer is merge sorting sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in the conquer phase by merging the segments a simpler variant of divide and conquer is called a decrease and conquer algorithm that solves an identical subproblem and uses the solution of this subproblem to solve the bigger problem divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithms an example of decrease and conquer algorithm is the binary search algorithm search and enumeration many problems such as playing chess can be modeled as problems on graphs a graph exploration algorithm specifies rules for moving around a graph and is useful for such problems this category also includes search algorithms branch and bound enumeration and backtracking randomized algorithm such algorithms make some choices randomly or pseudorandomly 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 some randomness whether randomized algorithms with polynomial time complexity can be the fastest algorithms for some problems is an open question known as the p versus np problem there are two large classes of such algorithms monte carlo algorithms return a correct answer with highprobability eg rp is the subclass of these that run in polynomial time las vegas algorithms always return the correct answer but their running time is only probabilistically bound eg zpp reduction of complexity this technique involves solving a difficult problem by transforming it into a better known problem for which we have hopefully asymptotically optimal algorithms the goal is to find a reducing algorithm whose complexity is not dominated by the resulting reduced algorithms for example one selection algorithm 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 transform and conquer back tracking in this approach multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution optimization problems for optimization problems there is a more specific classification of algorithms an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following linear programming when searching for optimal solutions to a linear function bound 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 simplex algorithm problems that can be solved with linear programming include the maximum flow problem for directed graphs if a problem additionally requires that one or more of the unknowns must be an integer then it is classified in integer programming a linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial ie the solutions satisfy these restrictions anyway in the general case a specialized algorithm or an algorithm that finds approximate solutions is used depending on the difficulty of the problem dynamic programming when a problem shows optimal substructures meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems and overlapping subproblems meaning the same subproblems are used to solve many different problem instances a quicker approach called dynamic programming avoids recomputing solutions that have already been computed for example floydwarshall algorithm the shortest path to a goal from a vertex in a weighted graph can be found by using the shortest path to the goal from all adjacent vertices dynamic programming and memoization 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 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 table of subproblems already solved dynamic programming reduces the exponential nature of many problems to polynomial complexity the greedy method a greedy algorithm 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 problems they can find the optimal solution while for others they stop at local optima 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 huffman tree kruskal prim sollin are greedy algorithms that can solve this optimization problem the heuristic method in optimization problems heuristic algorithms 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 solution their merit is that they can find a solution very close to the optimal solution in a relatively short time such algorithms include local search tabu search simulated annealing and genetic algorithms some of them like simulated annealing are nondeterministic algorithms while others like tabu search are deterministic when a bound on the error of the nonoptimal solution is known the algorithm is further categorized as an approximation algorithm by field of study see also list of algorithms every field of science has its own problems and needs efficient algorithms related problems in one field are often studied together some example classes are search algorithms sorting algorithms merge algorithms numerical algorithms graph algorithms string algorithms computational geometric algorithms combinatorial algorithms medical algorithms machine learning cryptography data compression algorithms and parsing techniques fields tend to overlap with each other and algorithm advances in one field may improve those of other sometimes completely unrelated fields for example dynamic programming was invented for optimization of resource consumption in industry but is now used in solving a broad range of problems in many fields by complexity see also complexity class and parameterized complexity algorithms can be classified by the amount of time they need to complete compared to their input size constant time if the time needed by the algorithm is the same regardless of the input size eg an access to an array element linear time if the time is proportional to the input size eg the traverse of a list logarithmic time if the time is a logarithmic function of the input size eg binary search algorithm polynomial time if the time is a power of the input size eg the bubble sort algorithm has quadratic time complexity exponential time if the time is an exponential function of the input size eg bruteforce search some problems may have multiple algorithms of differing complexity while other problems might have no algorithms or no known efficient algorithms there are also mappings from some problems to other problems owing to this it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them continuous algorithms the adjective continuous when applied to the word algorithm can mean an algorithm operating on data that represents continuous quantities even though this data is represented by discrete approximationssuch algorithms are studied in numerical analysis or an algorithm in the form of a differential equation that operates continuously on the data running on an analog computer legal issues see also software patent 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 and hence algorithms are not patentable as in gottschalk v benson however practical applications of algorithms are sometimes patentable for example in diamond v diehr the application of a simple feedback algorithm to aid in the curing of synthetic rubber was deemed patentable the patenting of software is highly controversial and there are highly criticized patents involving algorithms especially data compression algorithms such as unisys lzw patent additionally some cryptographic algorithms have export restrictions see export of cryptography history development of the notion of algorithm ancient near east algorithms were used in ancient greece two examples are the sieve of eratosthenes which was described in introduction to arithmetic by nicomachus discrete and distinguishable symbols tallymarks to keep track of their flocks their sacks of grain and their money the ancients used tallying accumulating stones or marks scratched on sticks or making discrete symbols in clay through the babylonian and egyptian use of marks and symbols eventually roman numerals and the abacus evolved dilson p1641 tally marks appear prominently in unary numeral system arithmetic used in turing machine and postturing machine computations manipulation of symbols as place holders for numbers algebra the work of the ancient greek geometers euclidean algorithm the indian mathematician brahmagupta and the persian mathematician alkhwarizmi from whose name the terms algorism and algorithm are derived and western european mathematicians culminated in leibnizs notion of the calculus ratiocinator ca 1680 a good century and a half ahead of his time leibniz proposed an algebra of logic an algebra that would specify the rules for manipulating logical concepts in the manner that ordinary algebra specifies the rules for manipulating numbers mechanical contrivances with discrete states the clock bolter credits the invention of the weightdriven clock as the key invention lovelace is credited with the first creation of an algorithm intended for processing on a computer babbages analytical engine the first device considered a real turingcomplete computer instead of just a calculator and is sometimes called historys first programmer as a result though a full implementation of babbages second device would not be realized until decades after her lifetime logical machines 1870stanley jevons logical abacus and logical machine the technical problem was to reduce boolean equations when presented in a form similar to what are now known as karnaugh maps jevons 1880 describes first a simple abacus of slips of wood furnished with pins contrived so that any part or class of the this machine he displayed in 1870 before the fellows of the royal society jacquard loom hollerith punch cards telegraphy and telephonythe electromechanical relay bell and newell 1971 indicate that the jacquard loom 1801 precursor to hollerith cards punch cards 1887 and telephone switching technologies were the roots of a tree leading to the development of the first computers by the mid19th century the telegraph the precursor of the telephone was in use throughout the world its discrete and distinguishable encoding of letters as dots and dashes a common sound by the late 19th century the ticker tape ca 1870s was in use as was the use of hollerith cards in the 1890 us census then came the teleprinter ca 1910 with its punchedpaper use of baudot code on tape telephoneswitching networks of electromechanical relays invented 1835 was behind the work of george stibitz 1937 the inventor of the digital adding device as he worked 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 davis 2000 observes the particular importance of the electromechanical relay with its two binary states open and closed it was only with the development beginning in the 1930s of electromechanical calculators using electrical relays that machines were built having the scope babbage had envisioned mathematics during the 19th century up to the mid20th century symbols and rules in rapid succession the mathematics of george boole 1847 1854 gottlob frege 1879 and giuseppe peano 18881889 reduced arithmetic to a sequence of symbols manipulated by rules peanos the principles of arithmetic presented by a new method 1888 was the first attempt at an axiomatization of mathematics in a symbolic language but heijenoort gives frege 1879 this kudos freges is perhaps the most important single work ever written in logic in which we see a formula language that is a lingua characterica a language written with special symbols for pure thought that is free from rhetorical embellishments constructed from specific symbols that are manipulated according to definite rules the work of frege was further simplified and amplified by alfred north whitehead and bertrand russell in their principia mathematica 19101913 the paradoxes at the same time a number of disturbing paradoxes appeared in the literature in particular the buraliforti paradox 1897 the russell paradox 190203 and the richard paradox the resultant considerations led to kurt gdels paper 1931he specifically cites the paradox of the liarthat completely reduces rules of recursion to numbers effective calculability in an effort to solve the entscheidungsproblem defined precisely by hilbert in 1928 mathematicians first set about to define what was meant by an effective method or effective calculation or effective calculability ie a calculation that would succeed in rapid succession the following appeared alonzo church stephen kleene and j b rossers calculus emil post 1936 and alan turing 193637 1939 here is a remarkable coincidence of two men not knowing each other but describing a process of menascomputers working on computationsand they yield virtually identical definitions emil post 1936 described the actions of a computer human being as follows two concepts are involved that of a symbol space in which the work leading from problem to answer is to be carried out and a fixed unalterable set of directions his symbol space would be a two way infinite sequence of spaces or boxes the problem solver or worker is to move and work in this symbol space being capable of being in and operating in but one box at a time a box is to admit of but two possible conditions ie being empty or unmarked and having a single mark in it say a vertical stroke one box is to be singled out and called the starting point a specific problem is to be given in symbolic form by a finite number of boxes is to be given in symbolic form by such a configuration of marked boxes a set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem this process terminates only when it comes to the direction of type c see more at postturing machine alan turings statue at bletchley park alan turings work might conjecture that all were influences turinghis model of computation is now called a turing machinebegins as did post with an analysis of a human computer that he whittles down to a simple set of basic motions and states of mind but he continues a step further and creates a machine as a model of computation of numbers computing is normally done by writing certain symbols on paper we may suppose this paper is divided into squares like a childs arithmetic booki assume then that the computation is carried out on onedimensional paper ie on a tape divided into squares i shall also suppose that the number of symbols which may be printed is finite the behaviour of the computer at any moment is determined by the symbols which he is observing and his state of mind at that moment we may suppose that there is a bound b to the number of symbols or squares which the computer can observe at one moment if he wishes to observe more he must use successive observations we will also suppose that the number of states of mind which need be taken into account is finite let us imagine that the operations performed by the computer to be split up into simple operations which are so elementary that it is not easy to imagine them further divided turings reduction yields the following the simple operations must therefore include a changes of the symbol on one of the observed squares b changes of one of the squares observed to another square within l squares of one of the previously observed squares it may be that some of these change necessarily invoke a change of state of mind the most general single operation must therefore be taken to be one of the following a a possible change a of symbol together with a possible change of state of mind b a possible change b of observed squares together with a possible change of state of mind we may now construct a machine to do the work of this computer a few years later turing expanded his analysis thesis definition with this forceful expression of it a function is said to be effectively calculable if its values can be found by some purely mechanical process though it is fairly easy to get an intuitive grasp of this idea it is nevertheless desirable to have some more definite mathematical expressible definition we may take this statement literally understanding by a purely mechanical process one which could be carried out by a machine it is possible to give a mathematical description in a certain normal form of the structures of these machines the development of these ideas leads to the authors definition of a computable function and to an identification of computability with effective calculability we shall use the expression computable function to mean a function calculable by a machine and we let effectively calculable refer to the intuitive idea without particular identification with any one of these definitions j b rosser 1939 and s c kleene 1943 j barkley rosser defined an effective method in the following manner italicization added effective method is used here in the rather special sense of a method each step of which is precisely determined and which is certain to produce the answer in a finite number of steps with this special meaning three different precise definitions have been given to date the simplest of these to state due to post and turing says essentially that an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and later reading the answer all three definitions are equivalent so it doesnt matter which one is used moreover the fact that all three are equivalent is a very strong argument for the correctness of any one rosser 19392256 rossers footnote no 5 references the work of 1 church and kleene and their definition of definability in particular churchs use of it in his an unsolvable problem of elementary number theory 1936 2 herbrand and gdel and their use of recursion in particular gdels use in his famous paper on formally undecidable propositions of principia mathematica and related systems i 1931 and 3 post 1936 and turing 193637 in their mechanismmodels of computation stephen c kleene defined as his nowfamous thesis i known as the churchturing thesis but he did this in the following context boldface in original 12 algorithmic theories in setting up a complete algorithmic theory what we do is to describe a procedure performable for each set of values of the independent variables which procedure necessarily terminates and in such manner that from the outcome we can read a definite answer yes or no to the question is the predicate value true kleene 1943273 history after 1950 a number of efforts have been directed toward further refinement of the definition of algorithm and activity is ongoing because of issues surrounding in particular foundations of mathematics especially the churchturing thesis and philosophy of mind especially arguments about artificial intelligence for more see algorithm characterizations heuristic abstract machine algorithm engineering algorithm characterizations algorithmic composition algorithmic entities algorithmic synthesis algorithmic trading garbage in garbage out introduction to algorithms textbook list of algorithms list of algorithm general topics list of important publications in theoretical computer science algorithms theory of computation computability theory computational complexity theory notes bibliography mwparseroutput refbeginfontsize90marginbottom05emmwparseroutput refbeginhangingindentsulliststyletypenonemarginleft0mwparseroutput refbeginhangingindentsullimwparseroutput refbeginhangingindentsdlddmarginleft0paddingleft32emtextindent32emliststylenonemwparseroutput refbegin100fontsize100 axt p 1959 on a subrecursive hierarchy and primitive recursive degrees transactions of the american mathematical society 92 85105 doi1023071993169 jstor1993169 bell c gordon and newell allen 1971 computer structures readings and examples mcgrawhill book company new york isbn0070043574 blass andreas gurevich yuri 2003 algorithms a quest for absolute definitions pdf bulletin of european association for theoretical computer science 81 includes an excellent bibliography of 56 references bolter david j 1984 turings man western culture in the computer age 1984 ed the university of north carolina press chapel hill nc isbn0807815640 isbn0807841080 pbk boolos george jeffrey richard 1999 computability and logic 4th ed cambridge university press london isbn052120402x cf chapter 3 turing machines where they discuss certain enumerable sets not effectively mechanically enumerable burgin mark 2004 superrecursive algorithms springer isbn9780387955698 campagnolo ml moore c and costa jf 2000 an analog characterization of the subrecursive functions in proc of the 4th conference on real numbers and computers odense university pp91109 church alonzo 1936a an unsolvable problem of elementary number theory the american journal of mathematics 58 2 345363 doi1023072371045 jstor2371045 reprinted in the undecidable p89ff the first expression of churchs thesis see in particular page 100 the undecidable where he defines the notion of effective calculability in terms of an algorithm and he uses the word terminates etc church alonzo 1936b a note on the entscheidungsproblem the journal of symbolic logic 1 1 4041 doi1023072269326 jstor2269326 church alonzo 1936 correction to a note on the entscheidungsproblem the journal of symbolic logic 1 3 101102 doi1023072269030 jstor2269030 reprinted in the undecidable p110ff church shows that the entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes daffa ali abdullah al 1977 the muslim contribution to mathematics london croom helm isbn0856644641 davis martin 1965 the undecidable basic papers on undecidable propositions unsolvable problems and computable functions new york raven press isbn0486432289 davis gives commentary before each article papers of gdel alonzo church turing rosser kleene and emil post are included those cited in the article are listed here by authors name davis martin 2000 engines of logic mathematicians and the origin of the computer new york w w nortion isbn0393322297 davis offers concise biographies of leibniz boole frege cantor hilbert gdel and turing with von neumann as the showstealing villain very brief bios of josephmarie jacquard babbage ada lovelace claude shannon howard aiken etc this article incorporates public domain materialfrom thenist documentblack paul e algorithm dictionary of algorithms and data structures dean tim 2012 evolution and moral diversity baltic international yearbook of cognition logic and communication 7 dennett daniel 1995 darwins dangerous idea new york touchstonesimon schuster isbn0684802902 dilson jesse 2007 the abacus 19681994 ed st martins press ny isbn031210409x isbn031210409x pbk yuri gurevich sequential abstract state machines capture sequential algorithms acm transactions on computational logic vol 1 no 1 july 2000 pages 77111 includes bibliography of 33 sources van heijenoort jean 2001 from frege to gdel a source book in mathematical logic 18791931 1967 ed harvard university press cambridge ma isbn0674324498 3rd edition 1976 isbn0674324498 pbk hodges andrew 1983 alan turing the enigma new york simon and schuster isbn0671492071 isbn0671492071 cf chapter the spirit of truth for a history leading to and a discussion of his proof kleene stephen c 1936 general recursive functions of natural numbers mathematische annalen 112 5 727742 doi101007bf01565439 presented to the american mathematical society september 1935 reprinted in the undecidable p237ff kleenes definition of general recursion known now as murecursion was used by church in his 1935 paper an unsolvable problem of elementary number theory that proved the decision problem to be undecidable ie a negative result kleene stephen c 1943 recursive predicates and quantifiers american mathematical society transactions 54 1 4173 doi1023071990131 jstor1990131 reprinted in the undecidable p255ff kleene refined his definition of general recursion and proceeded in his chapter 12 algorithmic theories to posit thesis i p274 he would later repeat this thesis in kleene 1952300 and name it churchs thesiskleene 1952317 ie the church thesis kleene stephen c 1991 introduction to metamathematics tenth ed northholland publishing company isbn0720421039 excellentaccessible readablereference source for mathematical foundations knuth donald 1997 fundamental algorithms third edition reading massachusetts addisonwesley isbn0201896834 knuth donald 1969 volume 2seminumerical algorithms the art of computer programming first edition reading massachusetts addisonwesley kosovsky n k elements of mathematical logic and its application to the theory of subrecursive algorithms lsu publ leningrad 1981 kowalski robert 1979 algorithmlogiccontrol communications of the acm 22 7 424436 doi101145359131359136 a a markov 1954 theory of algorithms minsky marvin 1967 computation finite and infinite machines first ed prenticehall englewood cliffs nj isbn0131654497 minsky expands his idea of an algorithman effective procedure in chapter 51 computability effective procedures and algorithms infinite machines post emil 1936 finite combinatory processes formulation i the journal of symbolic logic 1 3 103105 doi1023072269031 jstor2269031 reprinted in the undecidable p289ff post defines a simple algorithmiclike 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 socalled churchturing thesis rogers jr hartley 1987 theory of recursive functions and effective computability the mit press isbn0262680521 rosser j b 1939 an informal exposition of proofs of godels theorem and churchs theorem journal of symbolic logic 4 2 5360 doi1023072269059 jstor2269059 reprinted in the undecidable p223ff herein is rossers 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 p225226 the undecidable santoslang christopher 2014 moral ecology approaches to machine ethics in van rysewyk simon pontier matthijs machine medical ethics pdf switzerland springer pp111127 doi10100797833190810838 scott michael l 2009 programming language pragmatics 3rd ed morgan kaufmann publisherselsevier isbn9780123745149 sipser michael 2006 introduction to the theory of computation pws publishing company isbn053494728x sober elliott wilson david sloan 1998 unto others the evolution and psychology of unselfish behavior cambridge harvard university press stone harold s 1972 introduction to computer organization and data structures 1972 ed mcgrawhill new york isbn0070617260 cf in particular the first chapter titled algorithms turing machines and programs his succinct informal definition any sequence of instructions that can be obeyed by a robot is called an algorithm p4 tausworthe robert c 1977 standardized development of computer software part 1 methods englewood cliffs nj prenticehall inc isbn0138421951 turing alan m 193637 on computable numbers with an application to the entscheidungsproblem proceedings of the london mathematical society series 2 42 230265 doi101112plmss2421230 corrections ibid vol 431937 pp544546 reprinted in the undecidable p116ff turings famous paper completed as a masters dissertation while at kings college cambridge uk turing alan m 1939 systems of logic based on ordinals proceedings of the london mathematical society 45 161228 doi101112plmss2451161 reprinted in the undecidable p155ff turings paper that defined the oracle was his phd thesis while at princeton usa united states patent and trademark office 2006 210602 mathematical algorithms 2100 patentability manual of patent examining procedure mpep latest revision august 2006 bellah robert neelly 1985 habits of the heart individualism and commitment in american life berkeley university of california press isbn9780520254190 berlinski david 2001 the advent of the algorithm the 300year journey from an idea to the computer harvest books isbn9780156013918 chabert jeanluc 1999 a history of algorithms from the pebble to the microchip springer verlag isbn9783540633693 thomas h cormen charles e leiserson ronald l rivest clifford stein 2009 introduction to algorithms third edition mit press isbn9780262033848 harel david feldman yishai 2004 algorithmics the spirit of computing addisonwesley isbn9780321117847cs1 maint uses authors parameter link hertzke allen d mcrorie chris 1998 the concept of moral ecology in lawler peter augustine mcconkey dale community and political thought today westport ct praeger knuth donald e 2000 selected papers on analysis of algorithms stanford california center for the study of language and information knuth donald e 2010 selected papers on design of algorithms stanford california center for the study of language and information wallach wendell allen colin november 2008 moral machines teaching robots right from wrong usa oxford university press isbn9780195374049 look up algorithm in wiktionary the free dictionary wikibooks has a book on the topic of algorithms at wikiversity you can learn more and teach others about algorithm at the department of algorithm wikimedia commons has media related to algorithms hazewinkel michiel ed 2001 algorithm encyclopedia of mathematics springer sciencebusiness media bv kluwer academic publishers isbn9781556080104 algorithms at curlie weisstein eric w algorithm mathworld dictionary of algorithms and data structuresnational institute of standards and technology algorithms and data structures by dr nikolai bezroukov algorithm repositories opengenus cosmos largest crowdsourced algorithm repositoryopengenus foundation the stony brook algorithm repositorystate university of new york at stony brook netlib repositoryuniversity of tennessee and oak ridge national laboratory collected algorithms of the acmassociation for computing machinery the stanford graphbasestanford university combinatoricauniversity of iowa and state university of new york at stony brook library of efficient datastructures and algorithms ledapreviously from maxplanckinstitut fr informatik lecture notes algorithms course materials jeff erickson university of illinois authority control bne xx527980 bnf cb119358199 data gnd 40011835 lccn sh85003487 ndl 00560337 