Algorithm
Overview

In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, an algorithm is an effective method
Effective method
In computability theory, an effective method is a procedure that reduces the solution of some class of problems to a series of rote steps which, if followed to the letter, and as far as may be necessary, is bound to:...

expressed as a finite
Finite
Finite is the opposite of infinite. It may refer to:*Finite set, having a number of elements given by some natural number*Finite verb, being inflected for person and for tense...

list of well-defined instructions for calculating a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. Algorithms are used for calculation
Calculation
A calculation is a deliberate process for transforming one or more inputs into one or more results, with variable change.The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm to the vague heuristics of calculating a strategy in a competition...

, data processing
Data processing
Computer data processing is any process that a computer program does to enter data and summarise, analyse or otherwise convert data into usable information. The process may be automated and run on a computer. It involves recording, analysing, sorting, summarising, calculating, disseminating and...

, and automated reasoning
Automated reasoning
Automated reasoning is an area of computer science dedicated to understand different aspects of reasoning. The study in automated reasoning helps produce software which allows computers to reason completely, or nearly completely, automatically...

. In simple words an algorithm is a step-by-step procedure for calculations.

Starting from an initial state and initial input (perhaps empty), the instructions describe a computation
Computation
Computation is defined as any type of calculation. Also defined as use of computer technology in Information processing.Computation is a process following a well-defined model understood and expressed in an algorithm, protocol, network topology, etc...

that, when executed, will proceed through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state.
Discussions
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, an algorithm is an effective method
Effective method
In computability theory, an effective method is a procedure that reduces the solution of some class of problems to a series of rote steps which, if followed to the letter, and as far as may be necessary, is bound to:...

expressed as a finite
Finite
Finite is the opposite of infinite. It may refer to:*Finite set, having a number of elements given by some natural number*Finite verb, being inflected for person and for tense...

list of well-defined instructions for calculating a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. Algorithms are used for calculation
Calculation
A calculation is a deliberate process for transforming one or more inputs into one or more results, with variable change.The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm to the vague heuristics of calculating a strategy in a competition...

, data processing
Data processing
Computer data processing is any process that a computer program does to enter data and summarise, analyse or otherwise convert data into usable information. The process may be automated and run on a computer. It involves recording, analysing, sorting, summarising, calculating, disseminating and...

, and automated reasoning
Automated reasoning
Automated reasoning is an area of computer science dedicated to understand different aspects of reasoning. The study in automated reasoning helps produce software which allows computers to reason completely, or nearly completely, automatically...

. In simple words an algorithm is a step-by-step procedure for calculations.

Starting from an initial state and initial input (perhaps empty), the instructions describe a computation
Computation
Computation is defined as any type of calculation. Also defined as use of computer technology in Information processing.Computation is a process following a well-defined model understood and expressed in an algorithm, protocol, network topology, etc...

that, when executed, will proceed through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.

A partial formalization of the concept began with attempts to solve the Entscheidungsproblem
Entscheidungsproblem
In mathematics, the is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false...

(the "decision problem") posed by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

in 1928. Subsequent formalizations were framed as attempts to define "effective calculability" or "effective method"; those formalizations included the Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...

Herbrand
Jacques Herbrand
Jacques Herbrand was a French mathematician who was born in Paris, France and died in La Bérarde, Isère, France. Although he died at only 23 years of age, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse, and Richard Courant.He...

Kleene
Stephen Cole Kleene
Stephen Cole Kleene was an American mathematician who helped lay the foundations for theoretical computer science...

recursive function
Recursion (computer science)
Recursion in computer science is a method where the solution to a problem depends on solutions to smaller instances of the same problem. The approach can be applied to many types of problems, and is one of the central ideas of computer science....

s of 1930, 1934 and 1935, Alonzo Church
Alonzo Church
Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...

's lambda calculus
Lambda calculus
In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...

of 1936, Emil Post's "Formulation 1" of 1936, and Alan Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...

's Turing machines of 1936–7 and 1939. Giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem.

## Why algorithms are necessary: an informal definition

For a detailed presentation of the various points of view around the definition of "algorithm" see Algorithm characterizations
Algorithm characterizations
The word algorithm does not have a generally accepted definition. Researchers are actively working in formalizing this term. This article will present some of the "characterizations" of the notion of "algorithm" in more detail....

. For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations
Algorithm characterizations
The word algorithm does not have a generally accepted definition. Researchers are actively working in formalizing this term. This article will present some of the "characterizations" of the notion of "algorithm" in more detail....

, see Algorithm examples
Algorithm examples
This article Algorithm examples supplements Algorithm and Algorithm characterizations.- An example: Algorithm specification of addition m+n :Choice of machine model:...

.

While there is no generally accepted formal definition of "algorithm," an informal definition could be "a set of rules that precisely defines a sequence of operations." For some people, a program is only an algorithm if it stops eventually; for others, a program is only an algorithm if it stops before a given number of calculation steps.

A prototypical example of an algorithm is Euclid's algorithm to determine the maximum common divisor of two integers; an example (there are others) is described by the flow chart above and as an example in a later section.

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 ...you'd 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.

The term "enumerably infinite" means "countable using integers perhaps extending to infinity." 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 chosen from 0 to infinity. 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 (see more at Algorithm characterizations
Algorithm characterizations
The word algorithm does not have a generally accepted definition. Researchers are actively working in formalizing this term. This article will present some of the "characterizations" of the notion of "algorithm" in more detail....

) 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") for a fast, efficient, "good" process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities) to find, decode, and then process arbitrary input integers/symbols m and n, symbols + and = ... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format.

The concept of algorithm is also used to define the notion of decidability
Decidability (logic)
In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas. Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas can be effectively...

. That notion is central for explaining how formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

s come into being starting from a small set of axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

s and rules. In logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

, 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 program
Computer program
A computer program is a sequence of instructions written to perform a specified task with a computer. A computer requires programs to function, typically executing the program's instructions in a central processor. The program has an executable form that the computer can use directly to execute...

s 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 Turing-complete
Turing completeness
In computability theory, a system of data-manipulation rules is said to be Turing complete or computationally universal if and only if it can be used to simulate any single-taped Turing machine and thus in principle any computer. A classic example is the lambda calculus...

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: "...Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage [1987], an algorithm is a computational process defined by a Turing machine".

Typically, when an algorithm is associated with processing information, data is read from an input source, written to an output device, and/or stored for further processing. Stored data is regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structure
Data structure
In computer science, a data structure is a particular way of storing and organizing data in a computer so that it can be used efficiently.Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks...

s.

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, case-by-case; the criteria for each case must be clear (and computable).

Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom", an idea that is described more formally by flow of control
Control flow
In computer science, control flow refers to the order in which the individual statements, instructions, or function calls of an imperative or a declarative program are executed or evaluated....

.

So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming
Imperative programming
In computer science, imperative programming is a programming paradigm that describes computation in terms of statements that change a program state...

. 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
Memory
In psychology, memory is an organism's ability to store, retain, and recall information and experiences. Traditional studies of memory began in the fields of philosophy, including techniques of artificially enhancing 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
Functional programming
In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state...

and logic programming
Logic programming
Logic programming is, in its broadest sense, the use of mathematical logic for computer programming. In this view of logic programming, which can be traced at least as far back as John McCarthy's [1958] advice-taker proposal, logic is used as a purely declarative representation language, and a...

.

### Expressing algorithms

Algorithms can be expressed in many kinds of notation, including natural language
Natural language
In the philosophy of language, a natural language is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written...

s, pseudocode
Pseudocode
In computer science and numerical computation, pseudocode is a compact and informal high-level description of the operating principle of a computer program or other algorithm. It uses the structural conventions of a programming language, but is intended for human reading rather than machine reading...

, flowchart
Flowchart
A flowchart is a type of diagram that represents an algorithm or process, showing the steps as boxes of various kinds, and their order by connecting these with arrows. This diagrammatic representation can give a step-by-step solution to a given problem. Process operations are represented in these...

s, programming language
Programming language
A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....

s or control table
Control table
Control tables are tables that control the program flow or play a major part in program control. There are no rigid rules concerning the structure or content of a control table - its only qualifying attribute is its ability to direct program flow in some way through its 'execution' by an associated...

s (processed by interpreters
Interpreter (computing)
In computer science, an interpreter normally means a computer program that executes, i.e. performs, instructions written in a programming language...

). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts 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
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...

program as a sequence of machine tables (see more at finite state machine
Finite state machine
A finite-state machine or finite-state automaton , or simply a state machine, is a mathematical model used to design computer programs and digital logic circuits. It is conceived as an abstract machine that can be in one of a finite number of states...

and state transition table
State transition table
In automata theory and sequential logic, a state transition table is a table showing what state a finite semiautomaton or finite state machine will move to, based on the current state and other inputs...

), as flowcharts (see more at state diagram
State diagram
A state diagram is a type of diagram used in computer science and related fields to describe the behavior of systems. State diagrams require that the system described is composed of a finite number of states; sometimes, this is indeed the case, while at other times this is a reasonable abstraction...

), or as a form of rudimentary machine code
Machine code
Machine code or machine language is a system of impartible instructions executed directly by a computer's central processing unit. Each instruction performs a very specific task, typically either an operation on a unit of data Machine code or machine language is a system of impartible instructions...

or assembly code called "sets of quadruples" (see more at Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...

).

Representations of algorithms can be classed into three accepted levels of Turing machine description:
• 1 High-level 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 machine's "state table".

For an example of the simple algorithm "Add m+n" described in all three levels see Algorithm examples
Algorithm examples
This article Algorithm examples supplements Algorithm and Algorithm characterizations.- An example: Algorithm specification of addition m+n :Choice of machine model:...

.

## Implementation

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
Neural network
The term neural network was traditionally used to refer to a network or circuit of biological neurons. The modern usage of the term often refers to artificial neural networks, which are composed of artificial neurons or nodes...

(for example, the human brain
Human brain
The human brain has the same general structure as the brains of other mammals, but is over three times larger than the brain of a typical mammal with an equivalent body size. Estimates for the number of neurons in the human brain range from 80 to 120 billion...

implementing arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

or an insect looking for food), in an electrical circuit, or in a mechanical device.

## Computer algorithms

In computer systems, an algorithm is basically an instance of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

written in software by software developers to be effective for the intended "target" computer(s), in order for the target machines to produce output from given input (perhaps null).

"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 it's the smallest possible program for producing the output that it does"

Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant'"—such a proof would solve the Halting problem
Halting problem
In computability theory, the halting problem can be stated as follows: Given a description of a computer program, decide whether the program finishes running or continues to run forever...

(ibid).

Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This will be 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, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms".

Unfortunately there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example of using Euclid's algorithm will be shown below.

Computers (and computors), models of computation: A computer (or human "computor") is a restricted type of machine, a "discrete deterministic mechanical device" that blindly follows its instructions. Melzak's and Lambek's primitive models reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable counters (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent.

Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability". Minsky's machine proceeds sequentially through its five (or six depending on how one counts) instructions unless either a conditional IF–THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment (replacement, substitution) operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1). Rarely will a programmer have to write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT.

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". But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.

This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor).

But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...

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 Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" (division) instruction available rather than just subtraction (or worse: just Minsky's "decrement").

Structured programming, canonical structures: Per the Church-Turing thesis any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code
Spaghetti code
Spaghetti code is a pejorative term for source code that has a complex and tangled control structure, especially one using many GOTOs, exceptions, threads, or other "unstructured" branching constructs. It is named such because program flow tends to look like a bowl of spaghetti, i.e. twisted and...

" a programmer can write structured programs using these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language". Tausworthe augments the three Böhm-Jacopini canonical structures
Structured program theorem
The structured program theorem is a result in programming language theory. It states that every computable function can be implemented in a programming language that combines subprograms in only three specific ways...

: SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program will be one that lends itself to proofs of correctness using mathematical induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

.

Canonical flowchart symbols: The graphical aide called a flowchart
Flowchart
A flowchart is a type of diagram that represents an algorithm or process, showing the steps as boxes of various kinds, and their order by connecting these with arrows. This diagrammatic representation can give a step-by-step solution to a given problem. Process operations are represented in these...

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 4: the directed arrow showing program flow, the rectangle (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.

### Sorting example

One of the simplest algorithms is to find the largest number in an (unsorted) list of numbers. The solution necessarily requires looking at every number in the list, but only once at each. From this follows a simple algorithm, which can be stated in a high-level description English prose, as:

High-level description:
1. Assume the first item is largest.
2. Look at each of the remaining items in the list and if it is larger than the largest item so far, make a note of it.
3. The last noted item is the largest in the list when the process is complete.

(Quasi-)formal description:
Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in pseudocode
Pseudocode
In computer science and numerical computation, pseudocode is a compact and informal high-level description of the operating principle of a computer program or other algorithm. It uses the structural conventions of a programming language, but is intended for human reading rather than machine reading...

or pidgin code:
Input: A non-empty list of numbers L.
Output: The largest number in the list L.

largestL0
for each item in the list (Length(L)≥1), do
if the item > largest, then
largest ← the item
return largest

### Euclid’s algorithm

Euclid’s algorithm appears as Proposition II in Book VII ("Elementary Number Theory") of his Elements. Euclid poses the problem: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number [to be] a multitude composed of units": a counting number, a positive integer not including 0. And to "measure" is to place a shorter measuring length s successively (q times) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q*s, q being the quotient, or remainder r is the "modulus", the integer-fractional part left over after the division.

For Euclid’s method to succeed, the starting lengths must satisfy two requirements: (i) the lengths must not be 0, AND (ii) the subtraction must be “proper”, 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 0).

Euclid's original proof adds a third: the two lengths are not prime to one another. Euclid stipulated this so that he could construct a reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

proof that the two numbers' common measure is in fact the greatest. While Nicomachus' algorithm is the same as Euclid's, 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.

#### Example

Example of 1599 and 650 :
 Step 1 1599 = 650*2 + 299 Step 2 650 = 299*2 + 52 Step 3 299 = 52*5 + 39 Step 4 52 = 39*1 + 13 Step 5 39 = 13*3 + 0

#### Computer (computor) language for Euclid's algorithm

Only a few instruction types are required to execute Euclid's algorithm—some logical tests (conditional GOTO), unconditional GOTO, assignment (replacement), and subtraction.
• A location is symbolized by upper case letter(s), e.g. S, A, etc.
• The varying quantity (number) in a location will be written in lower case letter(s) and (usually) associated with the location's name. For example, location L at the start might contain the number l = 3009.

#### An inelegant program for Euclid's algorithm

The following algorithm is framed as Knuth's 4-step version of Euclid's and Nichomachus', 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 high-level description, shown in boldface, is adapted from Knuth 1973:2–4:

INPUT:
1 [Into two locations L and S put the numbers l and s that represent the two lengths]: INPUT L, S
2 [Initialize R: make the remaining length r equal to the starting/initial/input length l] R ← L

E0: [Insure rs.]
3 [Insure the smaller of the two numbers is in S and the larger in R]: IF R > S THEN the contents of L is the larger number so skip over the exchange-steps 4, 5 and 6: GOTO step 6 ELSE swap the contents of R and S.]
4 L ← R (this first step is redundant, but will be useful for later discussion).
5 R ← S
6 S ← L

E1:[Find remainder]: 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 [Remainder-loop]: GOTO 7.

E2: [Is the remainder 0?]: EITHER (i) the last measure was exact and the remainder in R is 0 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: [Interchange s and r ]: The nut of Euclid's 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 [Repeat the measuring process]: GOTO 7

OUTPUT:
15 [Done. S contains the greatest common divisor]: PRINT S

DONE:
16 HALT, END, STOP.

#### An elegant program for Euclid's algorithm

The following version of Euclid's algorithm requires only 6 core instructions to do what 13 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 Euclid's algorithm for greatest common divisor
6 PRINT "Type two integers greater than 0"
10 INPUT A,B
20 IF B=0 THEN GOTO 80
30 IF A > B THEN GOTO 60
40 LET B=B-A
50 GOTO 20
60 LET A=A-B
70 GOTO 20
80 PRINT A
90 END
```

How "Elegant" works: In place of an outer "Euclid loop", "Elegant" shifts back and forth between two "co-loops", 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, i.e. the domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...

of the function computed by the algorithm/program, is to include only positive integers including zero, then the failures at zero indicate that the algorithm (and the program that instantiates
Instance (computer science)
In object-oriented programming an instance is an occurrence or a copy of an object, whether currently executing or not. Instances of a class share the same set of attributes, yet will typically differ in what those attributes contain....

it) is a partial function
Partial function
In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...

rather than a total function. A notable failure due to exceptions is the Ariane V rocket failure.

Proof of program correctness by use of mathematical induction: Knuth demonstrates the application of mathematical induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

to an "extended" version of Euclid's algorithm, and he proposes "a general method applicable to proving the validity of any algorithm". Tausworthe proposes that a measure of the complexity of a program be the length of its correctness proof.

### Measuring and improving the Euclid algorithms

Elegance (compactness) versus goodness (speed) : With only 6 core instructions, "Elegant" is the clear winner compared to "Inelegant" at 13 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 loop-throughs, 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 "effective"—that is, it computes the function intended by its author—then the question becomes, can it be improved?

The compactness of "Inelegant" can be improved by the elimination of 5 steps. But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm; rather, it can only be done heuristically, i.e. by exhaustive search (examples to be found at Busy beaver
Busy beaver
In computability theory, a busy beaver is a Turing machine that attains the maximum "operational busyness" among all the Turing machines in a certain class...

), trial and error, cleverness, insight, application of inductive reasoning
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...

, 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 13 to 8, which makes it "more elegant" than "Elegant" at 9 steps.

The speed of "Elegant" can be improved by moving the B=0? test outside of the two subtraction loops. This change calls for the addition of 3 instructions (B=0?, A=0?, GOTO). Now "Elegant" computes the example-numbers faster; whether for any given A, B and R, S this is always the case would require a detailed analysis.

## Algorithmic analysis

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
Analysis of algorithms
To analyze an algorithm is to determine the amount of resources necessary to execute it. Most algorithms are designed to work with inputs of arbitrary length...

to obtain such quantitative answers (estimates); for example, the sorting algorithm above has a time requirement of O(n), using the big O notation
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...

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 O(1), if the space required to store the input numbers is not counted, or O(n) 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
Algorithmic efficiency
In computer science, efficiency is used to describe properties of an algorithm relating to how much of various types of resources it consumes. Algorithmic efficiency can be thought of as analogous to engineering productivity for a repeating or continuous process, where the goal is to reduce...

' than others. For example, a binary search algorithm will usually outperform a brute force
Brute-force search
In computer science, brute-force search or exhaustive search, also known as generate and test, is a trivial but very general problem-solving technique that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem's...

sequential search when used for table lookup
Lookup table
In computer science, a lookup table is a data structure, usually an array or associative array, often used to replace a runtime computation with a simpler array indexing operation. The savings in terms of processing time can be significant, since retrieving a value from memory is often faster than...

s on sorted lists.

### Formal versus empirical

The analysis and study of algorithms
Analysis of algorithms
To analyze an algorithm is to determine the amount of resources necessary to execute it. Most algorithms are designed to work with inputs of arbitrary length...

is a discipline of computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, and is often practiced abstractly without the use of a specific programming language
Programming language
A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....

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
Pseudocode
In computer science and numerical computation, pseudocode is a compact and informal high-level description of the operating principle of a computer program or other algorithm. It uses the structural conventions of a programming language, but is intended for human reading rather than machine reading...

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
Algorithmic efficiency
In computer science, efficiency is used to describe properties of an algorithm relating to how much of various types of resources it consumes. Algorithmic efficiency can be thought of as analogous to engineering productivity for a repeating or continuous process, where the goal is to reduce...

is eventually put to the test using real code.

Empirical testing is useful because it may uncover unexpected interactions that affect performance. Benchmark
Benchmark (computing)
In computing, a benchmark is the act of running a computer program, a set of programs, or other operations, in order to assess the relative performance of an object, normally by running a number of standard tests and trials against it...

s may be used to compare before/after potential improvements to an algorithm after program optimization.

### By implementation

One way to classify algorithms is by implementation means.
• Recursion or iteration: A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain condition matches, which is a method common to functional programming
Functional programming
In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state...

. Iterative
Iteration
Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.-Mathematics:Iteration in...

algorithms use repetitive constructs like loops and sometimes additional data structures like stacks
Stack (data structure)
In computer science, a stack is a last in, first out abstract data type and linear data structure. A stack can have any abstract data type as an element, but is characterized by only three fundamental operations: push, pop and stack top. The push operation adds a new item to the top of the stack,...

to solve the given problems. Some problems are naturally suited for one implementation or the other. For example, towers of Hanoi is a well understood in 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
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...

. 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
Logic programming
Logic programming is, in its broadest sense, the use of mathematical logic for computer programming. In this view of logic programming, which can be traced at least as far back as John McCarthy's [1958] advice-taker proposal, logic is used as a purely declarative representation language, and a...

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
Formal semantics of programming languages
In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages and models of computation...

: a change in the axioms has a well-defined change in the algorithm.
• Serial or 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 algorithm
Parallel algorithm
In computer science, a parallel algorithm or concurrent algorithm, as opposed to a traditional sequential algorithm, is an algorithm which can be executed a piece at a time on many different processing devices, and then put back together again at the end to get the correct result.Some algorithms...

s or distributed algorithms
Distributed algorithms
A distributed algorithm is an algorithm designed to run on computer hardware constructed from interconnected processors. Distributed algorithms are used in many varied application areas of distributed computing, such as telecommunications, scientific computing, distributed information processing,...

. 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 network
Computer network
A computer network, often simply referred to as a network, is a collection of hardware components and computers interconnected by communication channels that allow sharing of resources and information....

. 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. 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 non-deterministic: Deterministic algorithm
Deterministic algorithm
In computer science, a deterministic algorithm is an algorithm which, in informal terms, behaves predictably. Given a particular input, it will always produce the same output, and the underlying machine will always pass through the same sequence of states...

s solve the problem with exact decision at every step of the algorithm whereas non-deterministic 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 algorithm
Approximation algorithm
In computer science and operations research, approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP-hard problems; since it is unlikely that there can ever be efficient polynomial time exact...

s seek an approximation that is close to the true solution. Approximation may use either a deterministic or a random strategy. Such algorithms have practical value for many hard problems.
• Quantum algorithm: Quantum algorithm
Quantum algorithm
In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a...

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
Quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...

or quantum entanglement
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

.

Another way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories will include many different types of algorithms. Some commonly found paradigms include:
• Brute-force or exhaustive search. This is the naïve method of trying every possible solution to see which is best.
• Divide and conquer. A divide and conquer algorithm
Divide and conquer algorithm
In computer science, divide and conquer is an important algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same type, until these become simple enough to be solved directly...

repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually recursively
Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...

) 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 will be more complex than decrease and conquer algorithms. An example of decrease and conquer algorithm is the binary search algorithm
Binary search algorithm
In computer science, a binary search or half-interval search algorithm finds the position of a specified value within a sorted array. At each stage, the algorithm compares the input key value with the key value of the middle element of the array. If the keys match, then a matching element has been...

.
• Dynamic programming
Dynamic programming
In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller and optimal substructure...

. When a problem shows optimal substructure
Optimal substructure
In computer science, a problem is said to have optimal substructure if an optimal solution can be constructed efficiently from optimal solutions to its subproblems...

, 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, Floyd–Warshall algorithm, the shortest path to a goal from a vertex in a weighted graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and memoization
Memoization
In computing, memoization is an optimization technique used primarily to speed up computer programs by having function calls avoid repeating the calculation of results for previously processed inputs...

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
Mathematical table
Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments— to simplify and drastically speed up computation...

of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity.
• The greedy method. A greedy algorithm
Greedy algorithm
A greedy algorithm is any algorithm that follows the problem solving heuristic of making the locally optimal choice at each stagewith the hope of finding the global optimum....

is similar to a dynamic programming algorithm
Dynamic programming
In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller and optimal substructure...

, but the difference is that solutions to the subproblems do not have to be known at each stage; instead a "greedy" choice can be made of what looks best for the moment. The greedy method extends the solution with the best possible decision (not all feasible decisions) at an algorithmic stage based on the current local optimum and the best decision (not all possible decisions) made in a previous stage. It is not exhaustive, and does not give an accurate answer to many problems. But when it works, it will be the fastest method. The most popular greedy algorithm is finding the minimal spanning tree as given by Huffman Tree
Huffman coding
In computer science and information theory, Huffman coding is an entropy encoding algorithm used for lossless data compression. The term refers to the use of a variable-length code table for encoding a source symbol where the variable-length code table has been derived in a particular way based on...

, Kruskal
Kruskal's algorithm
Kruskal's algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized...

, Prim
Prim's algorithm
In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a connected weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized...

, Sollin.
• Linear programming. When solving a problem using linear programming
Linear programming
Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...

, specific inequalities involving the inputs are found and then an attempt is made to maximize (or minimize) some linear function of the inputs. Many problems (such as the maximum flow
Maximum flow problem
In optimization theory, the maximum flow problem is to find a feasible flow through a single-source, single-sink flow network that is maximum....

for directed graphs) can be stated in a linear programming way, and then be solved by a 'generic' algorithm such as the simplex algorithm
Simplex algorithm
In mathematical optimization, Dantzig's simplex algorithm is a popular algorithm for linear programming. The journal Computing in Science and Engineering listed it as one of the top 10 algorithms of the twentieth century....

. A more complex variant of linear programming is called integer programming, where the solution space is restricted to the integers.
• Reduction
Reduction (complexity)
In computability theory and computational complexity theory, a reduction is a transformation of one problem into another problem. Depending on the transformation used this can be used to define complexity classes on a set of problems....

. This technique involves solving a difficult problem by transforming it into a better known problem for which we have (hopefully) asymptotically optimal
Asymptotically optimal
In computer science, an algorithm is said to be asymptotically optimal if, roughly speaking, for large inputs it performs at worst a constant factor worse than the best possible algorithm...

algorithms. The goal is to find a reducing algorithm whose complexity
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...

is not dominated by the resulting reduced algorithm's. For example, one selection algorithm
Selection algorithm
In computer science, a selection algorithm is an algorithm for finding the kth smallest number in a list . This includes the cases of finding the minimum, maximum, and median elements. There are O, worst-case linear time, selection algorithms...

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.
• Search and enumeration. Many problems (such as playing chess
Chess
Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...

) can be modeled as problems on graphs
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

. A graph exploration algorithm specifies rules for moving around a graph and is useful for such problems. This category also includes search algorithm
Search algorithm
In computer science, a search algorithm is an algorithm for finding an item with specified properties among a collection of items. The items may be stored individually as records in a database; or may be elements of a search space defined by a mathematical formula or procedure, such as the roots...

s, branch and bound
Branch and bound
Branch and bound is a general algorithm for finding optimal solutions of various optimization problems, especially in discrete and combinatorial optimization...

enumeration and backtracking
Backtracking
Backtracking is a general algorithm for finding all solutions to some computational problem, that incrementally builds candidates to the solutions, and abandons each partial candidate c as soon as it determines that c cannot possibly be completed to a valid solution.The classic textbook example...

.
1. Randomized algorithm
Randomized algorithm
A randomized algorithm is an algorithm which employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits...

s are those that make some choices randomly (or pseudo-randomly); for some problems, it can in fact be proven that the fastest solutions must involve some randomness
Randomness
Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....

. There are two large classes of such algorithms:
1. Monte Carlo algorithm
Monte Carlo algorithm
In computing, a Monte Carlo algorithm is a randomized algorithm whose running time is deterministic, but whose output may be incorrect with a certain probability....

s return a correct answer with high-probability. E.g. RP
RP (complexity)
In complexity theory, RP is the complexity class of problems for which a probabilistic Turing machine exists with these properties:* It always runs in polynomial time in the input size...

is the subclass of these that run in polynomial time)
2. Las Vegas algorithm
Las Vegas algorithm
In computing, a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure. In other words, a Las Vegas algorithm does not gamble with the verity of the result; it gambles only with the resources...

s always return the correct answer, but their running time is only probabilistically bound, e.g. ZPP.
2. In optimization problem
Optimization problem
In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. An optimization problem with discrete...

s, heuristic
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...

algorithms do not try to find an optimal solution, but an approximate solution where the time or resources are limited. They are not practical to find perfect solutions. An example of this would be local search
Local search (optimization)
In computer science, local search is a metaheuristic method for solving computationally hard optimization problems. Local search can be used on problems that can be formulated as finding a solution maximizing a criterion among a number of candidate solutions...

, tabu search
Tabu search
Tabu search is a mathematical optimization method, belonging to the class of trajectory based techniques. Tabu search enhances the performance of a local search method by using memory structures that describe the visited solutions: once a potential solution has been determined, it is marked as...

, or simulated annealing
Simulated annealing
Simulated annealing is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete...

algorithms, a class of heuristic probabilistic algorithms that vary the solution of a problem by a random amount. The name "simulated annealing" alludes to the metallurgic term meaning the heating and cooling of metal to achieve freedom from defects. The purpose of the random variance is to find close to globally optimal solutions rather than simply locally optimal ones, the idea being that the random element will be decreased as the algorithm settles down to a solution. Approximation algorithms are those heuristic algorithms that additionally provide some bounds on the error. Genetic algorithm
Genetic algorithm
A genetic algorithm is a search heuristic that mimics the process of natural evolution. This heuristic is routinely used to generate useful solutions to optimization and search problems...

s attempt to find solutions to problems by mimicking biological evolution
Evolution
Evolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...

ary processes, with a cycle of random mutations yielding successive generations of "solutions". Thus, they emulate reproduction and "survival of the fittest". In genetic programming
Genetic programming
In artificial intelligence, genetic programming is an evolutionary algorithm-based methodology inspired by biological evolution to find computer programs that perform a user-defined task. It is a specialization of genetic algorithms where each individual is a computer program...

, this approach is extended to algorithms, by regarding the algorithm itself as a "solution" to a problem.

### By field of study

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 algorithm
Search algorithm
In computer science, a search algorithm is an algorithm for finding an item with specified properties among a collection of items. The items may be stored individually as records in a database; or may be elements of a search space defined by a mathematical formula or procedure, such as the roots...

s, sorting algorithm
Sorting algorithm
In computer science, a sorting algorithm is an algorithm that puts elements of a list in a certain order. The most-used orders are numerical order and lexicographical order...

s, merge algorithm
Merge algorithm
Merge algorithms are a family of algorithms that run sequentially over multiple sorted lists, typically producing more sorted lists as output. This is well-suited for machines with tape drives...

s, numerical algorithms
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

, graph algorithms
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

, string algorithms, computational geometric algorithms
Computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...

, combinatorial algorithms, medical algorithm
Medical algorithm
A medical algorithm is any computation, formula, statistical survey, nomogram, or look-up table, useful in healthcare. Medical algorithms include decision tree approaches to healthcare treatment and also less clear-cut tools aimed at reducing or defining uncertainty.-Scope:Medical algorithms are...

s, machine learning
Machine learning
Machine learning, a branch of artificial intelligence, is a scientific discipline concerned with the design and development of algorithms that allow computers to evolve behaviors based on empirical data, such as from sensor data or databases...

, cryptography
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...

, data compression
Data compression
In computer science and information theory, data compression, source coding or bit-rate reduction is the process of encoding information using fewer bits than the original representation would use....

algorithms and parsing techniques
Parsing
In computer science and linguistics, parsing, or, more formally, syntactic analysis, is the process of analyzing a text, made of a sequence of tokens , to determine its grammatical structure with respect to a given formal grammar...

.

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

Algorithms can be classified by the amount of time they need to complete compared to their input size. There is a wide variety: some algorithms complete in linear time relative to input size, some do so in an exponential amount of time or even worse, and some never halt. Additionally, 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.

Burgin (2005, p. 24) uses a generalized definition of algorithms that relaxes the common requirement that the output of the algorithm that computes a function must be determined after a finite number of steps. He defines a super-recursive class of algorithms as "a class of algorithms in which it is possible to compute functions not computable by any Turing machine" (Burgin 2005, p. 107). This is closely related to the study of methods of hypercomputation
Hypercomputation
Hypercomputation or super-Turing computation refers to models of computation that are more powerful than, or are incomparable with, Turing computability. This includes various hypothetical methods for the computation of non-Turing-computable functions, following super-recursive algorithms...

.

## Continuous algorithms

The adjective "continuous" when applied to the word "algorithm" can mean:
1. An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximations—such algorithms are studied in numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

; or
2. An algorithm in the form of a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

that operates continuously on the data, running on an analog computer
Analog computer
An analog computer is a form of computer that uses the continuously-changeable aspects of physical phenomena such as electrical, mechanical, or hydraulic quantities to model the problem being solved...

.

## Legal issues

See also: Software patents for a general overview of the patentability of software, including computer-implemented algorithms.

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
Gottschalk v. Benson
Gottschalk v. Benson, was a United States Supreme Court case in which the Court ruled that a process claim directed to a numerical algorithm, as such, was not patentable because "the patent would wholly pre-empt the mathematical formula and in practical effect would be a patent on the algorithm...

). However, practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr
Diamond v. Diehr
Diamond v. Diehr, , was a 1981 U.S. Supreme Court decision which held that the execution of a physical process, controlled by running a computer program was patentable...

, the application of a simple feedback
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...

algorithm to aid in the curing of synthetic rubber
Synthetic rubber
Synthetic rubber is is any type of artificial elastomer, invariably a polymer. An elastomer is a material with the mechanical property that it can undergo much more elastic deformation under stress than most materials and still return to its previous size without permanent deformation...

was deemed patentable. The patenting of software
Software patent debate
The software patent debate is the argument dealing with the extent to which it should be possible to patent software and computer-implemented inventions as a matter of public policy. Policy debate on software patents has been active for years. The opponents to software patents have gained more...

is highly controversial, and there are highly criticized patents involving algorithms, especially data compression
Data compression
In computer science and information theory, data compression, source coding or bit-rate reduction is the process of encoding information using fewer bits than the original representation would use....

algorithms, such as Unisys
Unisys
Unisys Corporation , headquartered in Blue Bell, Pennsylvania, United States, and incorporated in Delaware, is a long established business whose core products now involves computing and networking.-History:...

' LZW patent.

Additionally, some cryptographic algorithms have export restrictions (see export of cryptography
Export of cryptography
The export of cryptography in the United States is the transfer from the United States to another country of devices and technology related to cryptography....

).

## Etymology

The word "Algorithm" or "Algorism" in some other writing versions, comes from the name Al-Khwārizmī (c. 780-850), a Persian
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...

mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

, geographer
Geographer
A geographer is a scholar whose area of study is geography, the study of Earth's natural environment and human society.Although geographers are historically known as people who make maps, map making is actually the field of study of cartography, a subset of geography...

and a scholar in the House of Wisdom
House of Wisdom
The House of Wisdom was a library and translation institute established in Abbassid-era Baghdad, Iraq. It was a key institution in the Translation Movement and considered to have been a major intellectual centre during the Islamic Golden Age...

Baghdad is the capital of Iraq, as well as the coterminous Baghdad Governorate. The population of Baghdad in 2011 is approximately 7,216,040...

, whose name means "the native of Khwarezm
Khwarezm
Khwarezm, or Chorasmia, is a large oasis region on the Amu Darya river delta in western Central Asia, which borders to the north the Aral Sea, to the east the Kyzylkum desert, to the south the Karakum desert and to the west the Ustyurt Plateau...

"
, a city that was part of the Greater Iran
Greater Iran
Greater Iran refers to the regions that have significant Iranian cultural influence. It roughly corresponds to the territory on the Iranian plateau and its bordering plains, stretching from Iraq, the Caucasus, and Turkey in the west to the Indus River in the east...

during his era and now is in modern day Uzbekistan
Uzbekistan
Uzbekistan , officially the Republic of Uzbekistan is a doubly landlocked country in Central Asia and one of the six independent Turkic states. It shares borders with Kazakhstan to the west and to the north, Kyrgyzstan and Tajikistan to the east, and Afghanistan and Turkmenistan to the south....

He wrote a treatise in the Arabic language during the 9th century, which was translated into Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

in the 12th century under the title Algoritmi de numero Indorum. This title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name. Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through his other book, the Algebra. In late medieval Latin, algorismus, the corruption of his name, simply meant the "decimal number system" that is still the meaning of modern English algorism
Algorism
Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist...

. In 17th century French the word's form, but not its meaning, changed to algorithme. English adopted the French very soon afterwards, but it wasn't until the late 19th century that "Algorithm" took on the meaning that it has in modern English.

### Discrete and distinguishable symbols

Tally-marks: 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
Roman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...

and the abacus
Abacus
The abacus, also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes. Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of...

evolved (Dilson, p. 16–41). Tally marks appear prominently in unary numeral system
Unary numeral system
The unary numeral system is the bijective base-1 numeral system. It is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol representing 1 is repeated N times. For example, using the symbol | , the number 6 is represented as ||||||...

arithmetic used in Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...

and Post–Turing machine computations.

### Manipulation of symbols as "place holders" for numbers: algebra

The work of the ancient Greek geometers
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

(Euclidean algorithm
Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...

), Persian mathematician
Islamic mathematics
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...

Al-Khwarizmi
'There is some confusion in the literature on whether al-Khwārizmī's full name is ' or '. Ibn Khaldun notes in his encyclopedic work: "The first who wrote upon this branch was Abu ʿAbdallah al-Khowarizmi, after whom came Abu Kamil Shojaʿ ibn Aslam." . 'There is some confusion in the literature on...

(from whose name the terms "algorism
Algorism
Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist...

" and "algorithm" are derived), and Western European mathematicians culminated in Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

's notion of the calculus ratiocinator
Calculus ratiocinator
The Calculus Ratiocinator is a theoretical universal logical calculation framework, a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently mentioned characteristica universalis, a universal conceptual language....

(ca 1680):

### Mechanical contrivances with discrete states

The clock: Bolter credits the invention of the weight-driven clock
Clock
A clock is an instrument used to indicate, keep, and co-ordinate time. The word clock is derived ultimately from the Celtic words clagan and clocca meaning "bell". A silent instrument missing such a mechanism has traditionally been known as a timepiece...

as "The key invention [of Europe in the Middle Ages]", in particular the verge escapement
Verge escapement
The verge escapement is the earliest known type of mechanical escapement, the mechanism in a mechanical clock that controls its rate by advancing the gear train at regular intervals or 'ticks'. Its origin is unknown. Verge escapements were used from the 14th century until about 1800 in clocks...

that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata
Automata theory
In theoretical computer science, automata theory is the study of abstract machines and the computational problems that can be solved using these machines. These abstract machines are called automata...

" beginning in the 13th century and finally to "computational machines"—the difference engine
Difference engine
A difference engine is an automatic, mechanical calculator designed to tabulate polynomial functions. Both logarithmic and trigonometric functions can be approximated by polynomials, so a difference engine can compute many useful sets of numbers.-History:...

and analytical engine
Analytical engine
The Analytical Engine was a proposed mechanical general-purpose computer designed by English mathematician Charles Babbage. It was first described in 1837 as the successor to Babbage's difference engine, a design for a mechanical calculator...

s of Charles Babbage
Charles Babbage
Charles Babbage, FRS was an English mathematician, philosopher, inventor and mechanical engineer who originated the concept of a programmable computer...

Augusta Ada King, Countess of Lovelace , born Augusta Ada Byron, was an English writer chiefly known for her work on Charles Babbage's early mechanical general-purpose computer, the analytical engine...

.

Logical machines 1870—Stanley 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 map
Karnaugh map
The Karnaugh map , Maurice Karnaugh's 1953 refinement of Edward Veitch's 1952 Veitch diagram, is a method to simplify Boolean algebra expressions...

s. Jevons (1880) describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part or class of the [logical] combinations can be picked out mechanically . . . More recently however I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a Logical Machine" His machine came equipped with "certain moveable wooden rods" and "at the foot are 21 keys like those of a piano [etc] . . .". With this machine he could analyze a "syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...

or any other simple logical argument".

This machine he displayed in 1870 before the Fellows of the Royal Society. Another logician John Venn
John Venn
Donald A. Venn FRS , was a British logician and philosopher. He is famous for introducing the Venn diagram, which is used in many fields, including set theory, probability, logic, statistics, and computer science....

, however, in his 1881 Symbolic Logic, turned a jaundiced eye to this effort: "I have no high estimate myself of the interest or importance of what are sometimes called logical machines ... it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines"; see more at Algorithm characterizations
Algorithm characterizations
The word algorithm does not have a generally accepted definition. Researchers are actively working in formalizing this term. This article will present some of the "characterizations" of the notion of "algorithm" in more detail....

. But not to be outdone he too presented "a plan somewhat analogous, I apprehend, to Prof. Jevon's abacus ... [And] [a]gain, corresponding to Prof. Jevons's logical machine, the following contrivance may be described. I prefer to call it merely a logical-diagram machine ... but I suppose that it could do very completely all that can be rationally expected of any logical machine".

Jacquard loom, Hollerith punch cards, telegraphy and telephony—the electromechanical relay: Bell and Newell (1971) indicate that the Jacquard loom
Jacquard loom
The Jacquard loom is a mechanical loom, invented by Joseph Marie Jacquard in 1801, that simplifies the process of manufacturing textiles with complex patterns such as brocade, damask and matelasse. The loom is controlled by punched cards with punched holes, each row of which corresponds to one row...

(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 mid-19th 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
Ticker tape
Ticker tape was the earliest digital electronic communications medium, transmitting stock price information over telegraph lines, in use between around 1870 through 1970...

(ca 1870s) was in use, as was the use of Hollerith cards in the 1890 U.S. census. Then came the teleprinter
Teleprinter
A teleprinter is a electromechanical typewriter that can be used to communicate typed messages from point to point and point to multipoint over a variety of communication channels that range from a simple electrical connection, such as a pair of wires, to the use of radio and microwave as the...

(ca. 1910) with its punched-paper use of Baudot code
Baudot code
The Baudot code, invented by Émile Baudot, is a character set predating EBCDIC and ASCII. It was the predecessor to the International Telegraph Alphabet No 2 , the teleprinter code in use until the advent of ASCII. Each character in the alphabet is represented by a series of bits, sent over a...

on tape.

Telephone-switching networks of electromechanical relay
Relay
A relay is an electrically operated switch. Many relays use an electromagnet to operate a switching mechanism mechanically, but other operating principles are also used. Relays are used where it is necessary to control a circuit by a low-power signal , or where several circuits must be controlled...

s (invented 1835) was behind the work of George Stibitz
George Stibitz
George Robert Stibitz is internationally recognized as one of the fathers of the modern digital computer...

(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 mid-20th century

Symbols and rules: In rapid succession the mathematics of George Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...

(1847, 1854), Gottlob Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...

(1879), and Giuseppe Peano
Giuseppe Peano
Giuseppe Peano was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in...

(1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's 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: Frege's 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
Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...

and Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...

in their Principia Mathematica
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...

(1910–1913).

The paradoxes: At the same time a number of disturbing paradoxes appeared in the literature, in particular the Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction...

(1897), the Russell paradox (1902–03), and the Richard Paradox. The resultant considerations led to Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...

's paper (1931)—he specifically cites the paradox of the liar—that completely reduces rules of recursion
Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...

to numbers.

Effective calculability: In an effort to solve the Entscheidungsproblem
Entscheidungsproblem
In mathematics, the is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false...

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" (i.e., a calculation that would succeed). In rapid succession the following appeared: Alonzo Church
Alonzo Church
Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...

, Stephen Kleene and J.B. Rosser's λ-calculus a finely honed definition of "general recursion" from the work of Gödel acting on suggestions of Jacques Herbrand
Jacques Herbrand
Jacques Herbrand was a French mathematician who was born in Paris, France and died in La Bérarde, Isère, France. Although he died at only 23 years of age, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse, and Richard Courant.He...

(cf. Gödel's Princeton lectures of 1934) and subsequent simplifications by Kleene. Church's proof that the Entscheidungsproblem was unsolvable, Emil Post's definition of effective calculability as a worker mindlessly following a list of instructions to move left or right through a sequence of rooms and while there either mark or erase a paper or observe the paper and make a yes-no decision about the next instruction. Alan Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...

's proof of that the Entscheidungsproblem was unsolvable by use of his "a- [automatic-] machine"—in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine". S. C. Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I", and a few years later Kleene's renaming his Thesis "Church's Thesis" and proposing "Turing's Thesis".

### Emil Post (1936) and Alan Turing (1936–7, 1939)

Here is a remarkable coincidence of two men not knowing each other but describing a process of men-as-computers working on computations—and 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, i.e., 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 [i.e., INPUT] being marked with a stroke. Likewise the answer [i.e., OUTPUT] 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 will terminate only when it comes to the direction of type (C ) [i.e., STOP]". See more at Post–Turing machine

Alan Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...

's work preceded that of Stibitz (1937); it is unknown whether Stibitz knew of the work of Turing. Turing's biographer believed that Turing's use of a typewriter-like model derived from a youthful interest: "Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had a typewriter; and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'". Given the prevalence of Morse code and telegraphy, ticker tape machines, and teletypewriters we might conjecture that all were influences.

Turing—his model of computation is now called a Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...

—begins, 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 child's arithmetic book....I assume then that the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite....

"The behavior 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".

Turing's 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. Although it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have some more definite, mathematical expressible definition . . . [he discusses the history of the definition pretty much as presented above with respect to Gödel, Herbrand, Kleene, Church, Turing and Post] . . . 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 author's 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 boldly defined an 'effective [mathematical] method' in the following manner (boldface 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. [his footnote #5; see discussion immediately below]. 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 doesn't 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 1939:225–6)

Rosser's footnote #5 references the work of (1) Church and Kleene and their definition of λ-definability, in particular Church's use of it in his An Unsolvable Problem of Elementary Number Theory (1936); (2) Herbrand and Gödel and their use of recursion in particular Gödel's use in his famous paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931); and (3) Post (1936) and Turing (1936–7) in their mechanism-models of computation.

Stephen C. Kleene defined as his now-famous "Thesis I" known as the Church–Turing thesis
Church–Turing thesis
In computability theory, the Church–Turing thesis is a combined hypothesis about the nature of functions whose values are effectively calculable; in more modern terms, algorithmically computable...

. 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 1943:273)

### History after 1950

A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is on-going because of issues surrounding, in particular, foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...

(especially the Church–Turing thesis
Church–Turing thesis
In computability theory, the Church–Turing thesis is a combined hypothesis about the nature of functions whose values are effectively calculable; in more modern terms, algorithmically computable...

) and philosophy of mind
Philosophy of mind
Philosophy of mind is a branch of philosophy that studies the nature of the mind, mental events, mental functions, mental properties, consciousness and their relationship to the physical body, particularly the brain. The mind-body problem, i.e...

(especially arguments around artificial intelligence
Artificial intelligence
Artificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...

). For more, see Algorithm characterizations
Algorithm characterizations
The word algorithm does not have a generally accepted definition. Researchers are actively working in formalizing this term. This article will present some of the "characterizations" of the notion of "algorithm" in more detail....

.

### Secondary references

, ISBN 0-8078-4108-0 pbk., ISBN 0-312-10409-X (pbk.), 3rd edition 1976[?], ISBN 0-674-32449-8 (pbk.), ISBN 0-671-49207-1. Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.

• Jean-Luc Chabert, Évelyne Barbin, A history of algorithms: from the pebble to the microchip, Springer, 1999, ISBN 3-540-63369-3
• David Harel, Yishai A. Feldman, Algorithmics: the spirit of computing, Edition 3, Pearson Education, 2004, ISBN 0-321-11784-0
• Knuth, Donald E.
Donald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...

(2000). Selected Papers on Analysis of Algorithms. Stanford, California: Center for the Study of Language and Information.
• Knuth, Donald E.
Donald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...

(2010). Selected Papers on Design of Algorithms. Stanford, California: Center for the Study of Language and Information.

Algorithm repositories
• The Stony Brook Algorithm RepositoryState University of New York at Stony Brook
State University of New York at Stony Brook
The State University of New York at Stony Brook, also known as Stony Brook University, is a public research university located in Stony Brook, New York, on the North Shore of Long Island, about east of Manhattan....

• Library of Efficient Datastructures and Algorimths (LEDA)—previously from Max-Planck-Institut für Informatik
• Netlib ReposityUniversity of Tennessee
University of Tennessee
The University of Tennessee is a public land-grant university headquartered at Knoxville, Tennessee, United States...

and Oak Ridge National Laboratory
Oak Ridge National Laboratory
Oak Ridge National Laboratory is a multiprogram science and technology national laboratory managed for the United States Department of Energy by UT-Battelle. ORNL is the DOE's largest science and energy laboratory. ORNL is located in Oak Ridge, Tennessee, near Knoxville...

• Collected Algorithms of the ACMAssociation for Computing Machinery
Association for Computing Machinery
The Association for Computing Machinery is a learned society for computing. It was founded in 1947 as the world's first scientific and educational computing society. Its membership is more than 92,000 as of 2009...

• The Stanford GraphBaseStanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...

• CombinatoricaUniversity of Iowa
University of Iowa
The University of Iowa is a public state-supported research university located in Iowa City, Iowa, United States. It is the oldest public university in the state. The university is organized into eleven colleges granting undergraduate, graduate, and professional degrees...

and State University of New York at Stony Brook
State University of New York at Stony Brook
The State University of New York at Stony Brook, also known as Stony Brook University, is a public research university located in Stony Brook, New York, on the North Shore of Long Island, about east of Manhattan....

Lecture notes