In mathematics and computer science, an algorithm
(/ˈælɡərɪðəm/ ( listen) AL-gə-ridh-əm) is an
unambiguous specification of how to solve a class of problems.
Algorithms can perform calculation, data processing and automated
An algorithm is an effective method that can be expressed within a
finite amount of space and time and in a well-defined formal
language for calculating a function. Starting from an initial
state and initial input (perhaps empty), the instructions describe
a computation that, when executed, proceeds through a finite number
of well-defined successive states, eventually producing "output"
and terminating at a final ending state. The transition from one state
to the next is not necessarily deterministic; some algorithms, known
as randomized algorithms, incorporate random input.
The concept of algorithm has existed for centuries and the use of the
concept can be ascribed to Greek mathematicians, e.g. the sieve of
Eratosthenes and Euclid's algorithm; the term algorithm itself
derives from the 9th Century mathematician Muḥammad ibn Mūsā
al'Khwārizmī, latinized 'Algoritmi'. A partial formalization of what
would become the modern notion of algorithm began with attempts to
Entscheidungsproblem (the "decision problem") posed by David
Hilbert in 1928. Subsequent formalizations were framed as attempts to
define "effective calculability" or "effective method"; those
formalizations included the Gödel–Herbrand–
functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of
1936, Emil Post's "Formulation 1" of 1936, and Alan Turing's Turing
machines of 1936–7 and 1939.
2 Informal definition
3.1 Expressing algorithms
5 Computer algorithms
6.2 Euclid's algorithm
6.2.1 Computer language for Euclid's algorithm
6.2.2 An inelegant program for Euclid's algorithm
6.2.3 An elegant program for Euclid's algorithm
6.3 Testing the
6.4 Measuring and improving the
7 Algorithmic analysis
7.1 Formal versus empirical
7.2 Execution efficiency
8.1 By implementation
8.2 By design paradigm
8.3 Optimization problems
8.4 By field of study
8.5 By complexity
9 Continuous algorithms
10 Legal issues
11 History: Development of the notion of "algorithm"
11.1 Ancient Near East
11.2 Discrete and distinguishable symbols
11.3 Manipulation of symbols as "place holders" for numbers: algebra
11.4 Mechanical contrivances with discrete states
Mathematics during the 19th century up to the mid-20th century
Emil Post (1936) and
Alan Turing (1936–37, 1939)
11.7 J. B. Rosser (1939) and
S. C. Kleene
S. C. Kleene (1943)
11.8 History after 1950
12 See also
16 Secondary references
17 Further reading
18 External links
The word 'algorithm' probably has its roots in latinizing the name of
Muhammad ibn Musa al-Khwarizmi
Muhammad ibn Musa al-Khwarizmi in a first step to algorismus.
Al-Khwārizmī (Persian: خوارزمی, c. 780–850) was a
Persian mathematician, astronomer, geographer, and scholar in the
House of Wisdom
House of Wisdom in Baghdad, whose name means 'the native of Khwarezm',
a region that was part of
Greater Iran and is now in
About 825, al-Khwarizmi wrote an
Arabic language treatise on the
Hindu–Arabic numeral system, which was translated into
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 another of his
books, the Algebra. In late medieval Latin, algorismus, English
'algorism', the corruption of his name, simply meant the "decimal
number system". In the 15th century, under the influence of the Greek
word ἀριθμός 'number' (cf. 'arithmetic'), the
Latin word was
altered to algorithmus, and the corresponding English term 'algorithm'
is first attested in the 17th century; the modern sense was introduced
in the 19th century.
In English, it was first used in about 1230 and then by Chaucer in
1391. English adopted the French term, but it wasn't until the late
19th century that "algorithm" took on the meaning that it has in
Another early use of the word is from 1240, in a manual titled Carmen
de Algorismo composed by Alexandre de Villedieu. It begins thus:
Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur
bis quinque figuris.
which translates as:
Algorism is the art by which at present we use those Indian figures,
which number two times five.
The poem is a few hundred lines long and summarizes the art of
calculating with the new style of Indian dice, or Talibus Indorum, or
For a detailed presentation of the various points of view on the
definition of "algorithm", see
An informal definition could be "a set of rules that precisely defines
a sequence of operations." which would include all computer
programs, including programs that do not perform numeric calculations.
Generally, a program is only an algorithm if it stops eventually.
A prototypical example of an algorithm is the
Euclidean algorithm to
determine the maximum common divisor of two integers; an example
(there are others) is described by the flow chart above and as an
example in a later section.
Boolos, Jeffrey & 1974, 1999 offer an informal meaning of the word
in the following quotation:
No human being can write fast enough, or long enough, or small
enough† ( †"smaller and smaller without limit ...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.
An "enumerably infinite set" is one whose elements can be put into
one-to-one correspondence with the integers. Thus, Boolos and Jeffrey
are saying that an algorithm implies instructions for a process that
"creates" output integers from an arbitrary "input" integer or
integers that, in theory, can be arbitrarily large. Thus an algorithm
can be an algebraic equation such as y = m + n – two arbitrary
"input variables" m and n that produce an output y. But various
authors' attempts to define the notion indicate that the word implies
much more than this, something on the order of (for the addition
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
The concept of algorithm is also used to define the notion of
decidability. That notion is central for explaining how formal systems
come into being starting from a small set of axioms and rules. In
logic, the time that an algorithm requires to complete cannot be
measured, as it is not apparently related with our customary physical
dimension. From such uncertainties, that characterize ongoing work,
stems the unavailability of a definition of algorithm that suits both
concrete (in some sense) and abstract usage of the term.
Algorithms are essential to the way computers process data. Many
computer programs contain algorithms that detail the specific
instructions a computer should perform (in a specific order) to carry
out a specified task, such as calculating employees' paychecks or
printing students' report cards. Thus, an algorithm can be considered
to be any sequence of operations that can be simulated by a
Turing-complete 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 , an algorithm is a
computational process defined by a Turing machine".
Typically, when an algorithm is associated with processing
information, data can be read from an input source, written to an
output device and stored for further processing. Stored data are
regarded as part of the internal state of the entity performing the
algorithm. In practice, the state is stored in one or more data
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 is always crucial to the functioning of the algorithm.
Instructions are usually assumed to be listed explicitly, and are
described as starting "from the top" and going "down to the bottom",
an idea that is described more formally by flow of control.
So far, this discussion of the formalization of an algorithm has
assumed the premises of imperative programming. This is the most
common conception, and it attempts to describe a task in discrete,
"mechanical" means. Unique to this conception of formalized algorithms
is the assignment operation, setting the value of a variable. It
derives from the intuition of "memory" as a scratchpad. There is an
example below of such an assignment.
For some alternate conceptions of what constitutes an algorithm see
functional programming and logic programming.
Algorithms can be expressed in many kinds of notation, including
natural languages, pseudocode, flowcharts, drakon-charts, programming
languages or control tables (processed by interpreters). Natural
language expressions of algorithms tend to be verbose and ambiguous,
and are rarely used for complex or technical algorithms. Pseudocode,
flowcharts, drakon-charts and control tables are structured 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
There is a wide variety of representations possible and one can
express a given
Turing machine program as a sequence of machine tables
(see more at finite-state machine, state transition table and control
table), as flowcharts and drakon-charts (see more at state diagram),
or as a form of rudimentary machine code or assembly code called "sets
of quadruples" (see more at Turing machine).
Representations of algorithms can be classed into three accepted
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
For an example of the simple algorithm "Add m+n" described in all
three levels, see Algorithm#Examples.
Logical NAND algorithm implemented electronically in 7400 chip
Most algorithms are intended to be implemented as computer programs.
However, algorithms are also implemented by other means, such as in a
biological neural network (for example, the human brain implementing
arithmetic or an insect looking for food), in an electrical circuit,
or in a mechanical device.
Flowchart examples of the canonical Böhm-Jacopini structures: the
SEQUENCE (rectangles descending the page), the WHILE-DO and the
IF-THEN-ELSE. The three structures are made of the primitive
conditional GOTO (IF test=true THEN GOTO step xxx) (a diamond), the
unconditional GOTO (rectangle), various assignment operators
(rectangle), and HALT (rectangle). Nesting of these structures inside
assignment-blocks result in complex diagrams (cf Tausworthe
In computer systems, an algorithm is basically an instance of logic
written in software by software developers to be effective for the
intended "target" computer(s) to produce output from given (perhaps
null) input. An optimal algorithm, even running in old hardware, would
produce faster results than a non-optimal (higher time complexity)
algorithm for the same purpose, running in more efficient hardware;
that is why algorithms, like computer hardware, are considered
"Elegant" (compact) programs, "good" (fast) programs : The notion of
"simplicity and elegance" appears informally in Knuth and precisely in
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
Algorithm versus function computable by an algorithm: For a given
function multiple algorithms may exist. This is true, even without
expanding the available instruction set available to the programmer.
Rogers observes that "It is . . . important to distinguish between the
notion of algorithm, 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 that uses
Euclid's algorithm appears 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 must a programmer 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 the algorithm, to be effective, 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 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
Euclid's algorithm to compute the remainder would
execute much faster if the programmer had a "modulus" instruction
available rather than just subtraction (or worse: just Minsky's
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", a programmer can write
structured programs using only these instructions; on the other hand
"it is also possible, and not too hard, to write badly structured
programs in a structured language". Tausworthe augments the three
Böhm-Jacopini canonical structures: SEQUENCE, IF-THEN-ELSE, and
WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit
of a structured program is that it lends itself to proofs of
correctness using mathematical induction.
Canonical flowchart symbols: The graphical aide called a flowchart
offers a way to describe and document an algorithm (and a computer
program of one). Like program flow of a Minsky machine, a flowchart
always starts at the top of a page and proceeds down. Its primary
symbols are only four: the directed arrow showing program flow, the
rectangle (SEQUENCE, GOTO), the diamond (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.
Further information: List of algorithms
An animation of the quicksort algorithm sorting an array of randomized
values. The red bars mark the pivot element; at the start of the
animation, the element farthest to the right hand side is chosen as
One of the simplest algorithms is to find the largest number in a list
of numbers of random order. Finding the solution requires looking at
every number in the list. From this follows a simple algorithm, which
can be stated in a high-level description English prose, as:
If there are no numbers in the set then there is no highest number.
Assume the first number in the set is the largest number in the set.
For each remaining number in the set: if this number is larger than
the current largest number, consider this number to be the largest
number in the set.
When there are no numbers left in the set to iterate over, consider
the current largest number to be the largest number of the set.
(Quasi-)formal description: Written in prose but much closer to the
high-level language of a computer program, the following is the more
formal coding of the algorithm in pseudocode or pidgin code:
Input: A list of numbers L.
Output: The largest number in the list L.
if L.size = 0 return null
largest ← L
for each item in L, do
if item > largest, then
largest ← item
"←" denotes assignment. For instance, "largest ← item" means that
the value of largest changes to the value of item.
"return" terminates the algorithm and outputs the following value.
Further information: Euclid's algorithm
The example-diagram of
Euclid's algorithm from T.L. Heath (1908), with
more detail added.
Euclid does not go beyond a third measuring, and
gives no numerical examples.
Nicomachus gives the example of 49 and
21: "I subtract the less from the greater; 28 is left; then again I
subtract from this the same 21 (for this is possible); 7 is left; I
subtract this from 21, 14 is left; from which I again subtract 7 (for
this is possible); 7 is left, but 7 cannot be subtracted from 7."
Heath comments that, "The last phrase is curious, but the meaning of
it is obvious enough, as also the meaning of the phrase about ending
'at one and the same number'."(Heath 1908:300).
Euclid's algorithm to compute the greatest common divisor (GCD) to two
numbers appears as Proposition II in Book VII ("Elementary Number
Theory") of his Elements.
Euclid poses the problem thus: "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 zero. 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 zero, AND (ii) the
subtraction must be “proper”; i.e., a test must guarantee that the
smaller of the two numbers is subtracted from the larger (alternately,
the two can be equal so their subtraction yields zero).
Euclid's original proof adds a third requirement: the two lengths must
not be prime to one another.
Euclid stipulated this so that he could
construct a reductio ad absurdum proof that the two numbers' common
measure is in fact the greatest. While Nicomachus' algorithm is
the same as 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.
A graphical expression of
Euclid's algorithm to find the greatest
common divisor for 1599 and 650.
1599 = 650×2 + 299
650 = 299×2 + 52
299 = 52×5 + 39
52 = 39×1 + 13
39 = 13×3 + 0
Computer 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 is 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
"Inelegant" is a translation of Knuth's version of the algorithm with
a subtraction-based remainder-loop replacing his use of division (or a
"modulus" instruction). Derived from Knuth 1973:2–4. Depending on
the two numbers "Inelegant" may compute the g.c.d. in fewer steps than
The following algorithm is framed as Knuth's four-step version of
Euclid's and Nicomachus', but, rather than using division to find the
remainder, it uses successive subtractions of the shorter length s
from the remaining length r until r is less than s. The high-level
description, shown in boldface, is adapted from Knuth 1973:2–4:
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: [Ensure r ≥ s.]
3 [Ensure 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
swap the contents of R and S.
4 L ← R (this first step is redundant, but is useful for later
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
8 R ← R − S
E2: [Is the remainder zero?]: EITHER (i) the last measure was exact,
the remainder in R is zero, and the program can halt, OR (ii) the
algorithm must continue: the last measure left a remainder in R less
than measuring number in S.
10 IF R = 0 THEN
GOTO step 15
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]:
15 [Done. S contains the greatest common divisor]:
16 HALT, END, STOP.
An elegant program for Euclid's algorithm
The following version of
Euclid's algorithm requires only six core
instructions to do what thirteen are required to do by "Inelegant";
worse, "Inelegant" requires more types of instructions. The flowchart
of "Elegant" can be found at the top of this article. In the
(unstructured) Basic language, the steps are numbered, and the
instruction LET  =  is the assignment instruction symbolized by
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
The following version can be used with Object Oriented languages:
Euclid's algorithm for greatest common divisor
Algorithm (int A, int B)
if (A>B) A=A-B;
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.
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 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 it) is a partial function rather than a
total function. A notable failure due to exceptions is the Ariane 5
Flight 501 rocket failure (June 4, 1996).
Proof of program correctness by use of mathematical induction: Knuth
demonstrates the application of mathematical induction to an
"extended" version of 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
Elegance (compactness) versus goodness (speed): With only six core
instructions, "Elegant" is the clear winner, compared to "Inelegant"
at thirteen instructions. However, "Inelegant" is faster (it arrives
at HALT in fewer steps).
Algorithm analysis indicates why this is
the case: "Elegant" does two conditional tests in every subtraction
loop, whereas "Inelegant" only does one. As the algorithm (usually)
requires many 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
five 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), trial and error, cleverness, insight, application of
inductive reasoning, etc. Observe that steps 4, 5 and 6 are repeated
in steps 11, 12 and 13. Comparison with "Elegant" provides a hint that
these steps, together with steps 2 and 3, can be eliminated. This
reduces the number of core instructions from thirteen to eight, which
makes it "more elegant" than "Elegant", at nine steps.
The speed of "Elegant" can be improved by moving the "B=0?" test
outside of the two subtraction loops. This change calls for the
addition of three instructions (B = 0?, A = 0?, GOTO). Now "Elegant"
computes the example-numbers faster; whether this is always the case
for any given A, B and R, S would require a detailed analysis.
Main article: Analysis of algorithms
It is frequently important to know how much of a particular resource
(such as time or storage) is theoretically required for a given
algorithm. Methods have been developed for the analysis of algorithms
to obtain such quantitative answers (estimates); for example, the
sorting algorithm above has a time requirement of O(n), using the big
O notation with n as the length of the list. At all times the
algorithm only needs to remember two values: the largest number found
so far, and its current position in the input list. Therefore, it is
said to have a space requirement of 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' than others.
For example, a binary search algorithm (with cost O(log n) )
outperforms a sequential search (cost O(n) ) when used for table
lookups on sorted lists or arrays.
Formal versus empirical
Main articles: Empirical algorithmics, Profiling (computer
programming), and Program optimization
The analysis and study of algorithms is a discipline of computer
science, and is often practiced abstractly without the use of a
specific programming language or implementation. In this sense,
algorithm analysis resembles other mathematical disciplines in that it
focuses on the underlying properties of the algorithm and not on the
specifics of any particular implementation. Usually pseudocode is used
for analysis as it is the simplest and most general representation.
However, ultimately, most algorithms are usually implemented on
particular hardware / software platforms and their algorithmic
efficiency is eventually put to the test using real code. For the
solution of a "one off" problem, the efficiency of a particular
algorithm may not have significant consequences (unless n is extremely
large) but for algorithms designed for fast interactive, commercial or
long life scientific usage it may be critical. Scaling from small n to
large n frequently exposes inefficient algorithms that are otherwise
Empirical testing is useful because it may uncover unexpected
interactions that affect performance. Benchmarks may be used to
compare before/after potential improvements to an algorithm after
program optimization. Empirical tests cannot replace formal analysis,
though, and are not trivial to perform in a fair manner.
Main article: Algorithmic efficiency
To illustrate the potential improvements possible even in well
established algorithms, a recent significant innovation, relating to
FFT algorithms (used heavily in the field of image processing), can
decrease processing time up to 1,000 times for applications like
medical imaging. In general, speed improvements depend on special
properties of the problem, which are very common in practical
applications. Speedups of this magnitude enable computing devices
that make extensive use of image processing (like digital cameras and
medical equipment) to consume less power.
There are various ways to classify algorithms, each with its own
One way to classify algorithms is by implementation means.
A recursive algorithm is one that invokes (makes reference to) itself
repeatedly until a certain condition (also known as termination
condition) matches, which is a method common to functional
programming. Iterative algorithms use repetitive constructs like loops
and sometimes additional data structures like stacks to solve the
given problems. Some problems are naturally suited for one
implementation or the other. For example, towers of Hanoi is well
understood using recursive implementation. Every recursive version has
an equivalent (but possibly more or less complex) iterative version,
and vice versa.
An algorithm may be viewed as controlled logical deduction. This
notion may be expressed as:
Algorithm = logic + control. The logic
component expresses the axioms that may be used in the computation and
the control component determines the way in which deduction is applied
to the axioms. This is the basis for the logic programming paradigm.
In pure logic programming languages the control component is fixed and
algorithms are specified by supplying only the logic component. The
appeal of this approach is the elegant semantics: a change in the
axioms has a well-defined change in the algorithm.
Serial, parallel or distributed
Algorithms are usually discussed with the assumption that computers
execute one instruction of an algorithm at a time. Those computers are
sometimes called serial computers. An algorithm designed for such an
environment is called a serial algorithm, as opposed to parallel
algorithms or distributed algorithms. Parallel algorithms take
advantage of computer architectures where several processors can work
on a problem at the same time, whereas distributed algorithms utilize
multiple machines connected with a network. Parallel or distributed
algorithms divide the problem into more symmetrical or asymmetrical
subproblems and collect the results back together. The resource
consumption in such algorithms is not only processor cycles on each
processor but also the communication overhead between the processors.
Some sorting algorithms can be parallelized efficiently, but their
communication overhead is expensive. Iterative algorithms are
generally parallelizable. Some problems have no parallel algorithms,
and are called inherently serial problems.
Deterministic or non-deterministic
Deterministic algorithms 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
algorithms seek an approximation that is closer to the true solution.
Approximation can be reached by either using a deterministic or a
random strategy. Such algorithms have practical value for many hard
problems. One of the examples of an approximate algorithm is the
Knapsack problem. The Knapsack problem is a problem where there is a
set of given items. The goal of the problem is to pack the knapsack to
get the maximum total value. Each item has some weight and some value.
Total weight that we can carry is no more than some fixed number X.
So, we must consider weights of items as well as their value.
They run on a realistic model of quantum computation. The term is
usually used for those algorithms which seem inherently quantum, or
use some essential feature of quantum computation such as quantum
superposition or quantum entanglement.
By design paradigm
Another way of classifying algorithms is by their design methodology
or paradigm. There is a certain number of paradigms, each different
from the other. Furthermore, each of these categories include many
different types of algorithms. Some common paradigms are:
Brute-force or exhaustive search
This is the naive method of trying every possible solution to see
which is best.
Divide and conquer
A divide and conquer algorithm repeatedly reduces an instance of a
problem to one or more smaller instances of the same problem (usually
recursively) until the instances are small enough to solve easily. One
such example of divide and conquer is merge sorting. Sorting can be
done on each segment of data after dividing data into segments and
sorting of entire data can be obtained in the conquer phase by merging
the segments. A simpler variant of divide and conquer is called a
decrease and conquer algorithm, that solves an identical subproblem
and uses the solution of this subproblem to solve the bigger problem.
Divide and conquer divides the problem into multiple subproblems and
so the conquer stage is more complex than decrease and conquer
algorithms. An example of decrease and conquer algorithm is the binary
Search and enumeration
Many problems (such as playing chess) can be modeled as problems on
graphs. A graph exploration algorithm specifies rules for moving
around a graph and is useful for such problems. This category also
includes search algorithms, branch and bound enumeration and
Such algorithms make some choices randomly (or pseudo-randomly). They
can be very useful in finding approximate solutions for problems where
finding exact solutions can be impractical (see heuristic method
below). For some of these problems, it is known that the fastest
approximations must involve some randomness. Whether randomized
algorithms with polynomial time complexity can be the fastest
algorithms for some problems is an open question known as the P versus
NP problem. There are two large classes of such algorithms:
Monte Carlo algorithms return a correct answer with high-probability.
E.g. RP is the subclass of these that run in polynomial time.
Las Vegas algorithms always return the correct answer, but their
running time is only probabilistically bound, e.g. ZPP.
Reduction of complexity
This technique involves solving a difficult problem by transforming it
into a better known problem for which we have (hopefully)
asymptotically optimal algorithms. The goal is to find a reducing
algorithm whose complexity is not dominated by the resulting reduced
algorithm's. For example, one selection algorithm for finding the
median in an unsorted list involves first sorting the list (the
expensive portion) and then pulling out the middle element in the
sorted list (the cheap portion). This technique is also known as
transform and conquer.
For optimization problems there is a more specific classification of
algorithms; an algorithm for such problems may fall into one or more
of the general categories described above as well as into one of the
When searching for optimal solutions to a linear function bound to
linear equality and inequality constraints, the constraints of the
problem can be used directly in producing the optimal solutions. There
are algorithms that can solve any problem in this category, such as
the popular simplex algorithm. Problems that can be solved with
linear programming include the maximum flow problem for directed
graphs. If a problem additionally requires that one or more of the
unknowns must be an integer then it is classified in integer
programming. A linear programming algorithm can solve such a problem
if it can be proved that all restrictions for integer values are
superficial, i.e., the solutions satisfy these restrictions anyway. In
the general case, a specialized algorithm or an algorithm that finds
approximate solutions is used, depending on the difficulty of the
When a problem shows optimal substructures – meaning the optimal
solution to a problem can be constructed from optimal solutions to
subproblems – and overlapping subproblems, meaning the same
subproblems are used to solve many different problem instances, a
quicker approach called dynamic programming avoids recomputing
solutions that have already been computed. For example,
Floyd–Warshall algorithm, the shortest path to a goal from a vertex
in a weighted graph can be found by using the shortest path to the
goal from all adjacent vertices.
Dynamic programming and memoization
go together. The main difference between dynamic programming and
divide and conquer is that subproblems are more or less independent in
divide and conquer, whereas subproblems overlap in dynamic
programming. The difference between dynamic programming and
straightforward recursion is in caching or memoization of recursive
calls. When subproblems are independent and there is no repetition,
memoization does not help; hence dynamic programming is not a solution
for all complex problems. By using memoization or maintaining a table
of subproblems already solved, dynamic programming reduces the
exponential nature of many problems to polynomial complexity.
The greedy method
A greedy algorithm is similar to a dynamic programming algorithm in
that it works by examining substructures, in this case not of the
problem but of a given solution. Such algorithms start with some
solution, which may be given or have been constructed in some way, and
improve it by making small modifications. For some problems they can
find the optimal solution while for others they stop at local optima,
that is, at solutions that cannot be improved by the algorithm but are
not optimum. The most popular use of greedy algorithms is for finding
the minimal spanning tree where finding the optimal solution is
possible with this method. Huffman Tree, Kruskal, Prim, Sollin are
greedy algorithms that can solve this optimization problem.
The heuristic method
In optimization problems, heuristic algorithms can be used to find a
solution close to the optimal solution in cases where finding the
optimal solution is impractical. These algorithms work by getting
closer and closer to the optimal solution as they progress. In
principle, if run for an infinite amount of time, they will find the
optimal solution. Their merit is that they can find a solution very
close to the optimal solution in a relatively short time. Such
algorithms include local search, tabu search, simulated annealing, and
genetic algorithms. Some of them, like simulated annealing, are
non-deterministic algorithms while others, like tabu search, are
deterministic. When a bound on the error of the non-optimal solution
is known, the algorithm is further categorized as an approximation
By field of study
See also: List of algorithms
Every field of science has its own problems and needs efficient
algorithms. Related problems in one field are often studied together.
Some example classes are search algorithms, sorting algorithms, merge
algorithms, numerical algorithms, graph algorithms, string algorithms,
computational geometric algorithms, combinatorial algorithms, medical
algorithms, machine learning, cryptography, data compression
algorithms and parsing techniques.
Fields tend to overlap with each other, and algorithm advances in one
field may improve those of other, sometimes completely unrelated,
fields. For example, dynamic programming was invented for optimization
of resource consumption in industry, but is now used in solving a
broad range of problems in many fields.
Complexity class and Parameterized complexity
Algorithms can be classified by the amount of time they need to
complete compared to their input size:
Constant time: if the time needed by the algorithm is the same,
regardless of the input size. E.g. an access to an array element.
Linear time: if the time is proportional to the input size. E.g. the
traverse of a list.
Logarithmic time: if the time is a logarithmic function of the input
size. E.g. binary search algorithm.
Polynomial time: if the time is a power of the input size. E.g. the
bubble sort algorithm has quadratic time complexity.
Exponential time: if the time is an exponential function of the input
size. E.g. Brute-force search.
Some problems may have multiple algorithms of differing complexity,
while other problems might have no algorithms or no known efficient
algorithms. There are also mappings from some problems to other
problems. Owing to this, it was found to be more suitable to classify
the problems themselves instead of the algorithms into equivalence
classes based on the complexity of the best possible algorithms for
The adjective "continuous" when applied to the word "algorithm" can
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; or
An algorithm in the form of a differential equation that operates
continuously on the data, running on an analog computer.
See also: Software patent
Algorithms, by themselves, are not usually patentable. In the United
States, a claim consisting solely of simple manipulations of abstract
concepts, numbers, or signals does not constitute "processes" (USPTO
2006), and hence algorithms are not patentable (as in Gottschalk v.
Benson). However, practical applications of algorithms are sometimes
patentable. For example, in Diamond v. Diehr, the application of a
simple feedback algorithm to aid in the curing of synthetic rubber was
deemed patentable. The patenting of software is highly controversial,
and there are highly criticized patents involving algorithms,
especially data compression algorithms, such as Unisys' LZW patent.
Additionally, some cryptographic algorithms have export restrictions
(see export of cryptography).
Researcher, Andrew Tutt, argues that algorithms should be overseen by
a specialist regulatory agency, similar to FDA. His academic work
emphasizes that the rise of increasingly complex algorithms calls for
the need to think about the effects of algorithms today. Due to the
nature and complexity of algorithms, it will prove to be difficult to
hold algorithms accountable under criminal law. Tutt recognizes that
while some algorithms will be beneficial to help meet technological
demand, others should not be used or sold if they fail to meet safety
requirements. Thus, for Tutt, algorithms will require "closer forms of
federal uniformity, expert judgment, political independence, and
pre-market review to prevent the introduction of unacceptably
dangerous algorithms into the market". The issue of algorithmic
accountability (the responsibility of algorithm designers to
provide evidence of potential or realised harms) is of particular
relevance in the field of dynamic and non-linearly programmed systems,
e.g. artificial neural networks, deep learning, and genetic algorithms
(see Explainable AI).
History: Development of the notion of "algorithm"
Ancient Near East
Algorithms were used in ancient Greece. Two examples are the Sieve of
Eratosthenes, which was described in Introduction to
Nicomachus,:Ch 9.2 and the Euclidean algorithm, which was first
Euclid's Elements (c. 300 BC).:Ch 9.1 Babylonian clay
tablets describe and employ algorithmic procedures to compute the time
and place of significant astronomical events.
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 and the abacus evolved (Dilson, p. 16–41). Tally marks
appear prominently in unary numeral system arithmetic used in Turing
machine and Post–
Turing machine computations.
Manipulation of symbols as "place holders" for numbers: algebra
The work of the ancient Greek geometers (Euclidean algorithm), the
Indian mathematician Brahmagupta, and the Persian mathematician
Al-Khwarizmi (from whose name the terms "algorism" and "algorithm" are
derived), and Western European mathematicians culminated in Leibniz's
notion of the calculus ratiocinator (ca 1680):
A good century and a half ahead of his time, Leibniz proposed an
algebra of logic, an algebra that would specify the rules for
manipulating logical concepts in the manner that ordinary algebra
specifies the rules for manipulating numbers.
Mechanical contrivances with discrete states
The clock: Bolter credits the invention of the weight-driven clock as
"The key invention [of Europe in the Middle Ages]", in particular the
verge escapement that provides us with the tick and tock of a
mechanical clock. "The accurate automatic machine" led immediately
to "mechanical automata" beginning in the 13th century and finally to
"computational machines"—the difference engine and analytical
Charles Babbage and Countess Ada Lovelace, mid-19th
century. Lovelace is credited with the first creation of an
algorithm intended for processing on a computer – Babbage's
analytical engine, the first device considered a real Turing-complete
computer instead of just a calculator – and is sometimes called
"history's first programmer" as a result, though a full implementation
of Babbage's second device would not be realized until decades after
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 maps.
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 or any other simple logical argument".
This machine he displayed in 1870 before the Fellows of the Royal
Society. Another logician John Venn, 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. 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 (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 (ca 1870s) was in use, as was the use of Hollerith
cards in the 1890 U.S. census. Then came the teleprinter (ca. 1910)
with its punched-paper use of
Baudot code on tape.
Telephone-switching networks of electromechanical relays (invented
1835) was behind the work of
George Stibitz (1937), the inventor of
the digital adding device. As he worked in Bell Laboratories, he
observed the "burdensome' use of mechanical calculators with gears.
"He went home one evening in 1937 intending to test his idea... When
the tinkering was over, Stibitz had constructed a binary adding
Davis (2000) observes the particular importance of the
electromechanical relay (with its two "binary states" open and
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
Gottlob Frege (1879), and
Giuseppe Peano (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
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 and
Bertrand Russell in their
Principia Mathematica (1910–1913).
The paradoxes: At the same time a number of disturbing paradoxes
appeared in the literature, in particular the Burali-Forti paradox
Russell paradox (1902–03), and the Richard Paradox.
The resultant considerations led to Kurt Gödel's paper (1931)—he
specifically cites the paradox of the liar—that completely reduces
rules of recursion to numbers.
Effective calculability: In an effort to solve the
Entscheidungsproblem defined precisely by Hilbert in 1928,
mathematicians first set about to define what was meant by an
"effective method" or "effective calculation" or "effective
calculability" (i.e., a calculation that would succeed). In rapid
succession the following appeared: 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 (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'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
Emil Post (1936) and
Alan Turing (1936–37, 1939)
Here is a remarkable coincidence[according to whom?] of two men not
knowing each other but describing a process of men-as-computers
working on computations—and they yield virtually identical
Emil Post (1936) described the actions of a "computer" (human being)
"...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 terminates only when it comes to the direction of type (C )
[i.e., STOP]". See more at Post–Turing machine
Alan Turing's statue at Bletchley Park
Alan Turing'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[who?] might conjecture that all were
Turing—his model of computation is now called a Turing
machine—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
"The behaviour of the computer at any moment is determined by the
symbols which he is observing, and his "state of mind" at that moment.
We may suppose that there is a bound B to the number of symbols or
squares which the computer can observe at one moment. If he wishes to
observe more, he must use successive observations. We will also
suppose that the number of states of mind which need be taken into
account is finite...
"Let us imagine that the operations performed by the computer to be
split up into 'simple operations' which are so elementary that it is
not easy to imagine them further divided."
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. Though it is fairly easy
to get an intuitive grasp of this idea, it is nevertheless desirable
to have some more definite, mathematical expressible definition . . .
[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
S. C. Kleene (1943)
J. Barkley Rosser defined an 'effective [mathematical] method' in the
following manner (italicization added):
"'Effective method' is used here in the rather special sense of a
method each step of which is precisely determined and which is certain
to produce the answer in a finite number of steps. With this special
meaning, three different precise definitions have been given to date.
[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
Rosser's footnote No. 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
Principia Mathematica and Related Systems I (1931); and (3) Post
(1936) and Turing (1936–37) in their mechanism-models of
Stephen C. Kleene
Stephen C. Kleene defined as his now-famous "Thesis I" known as the
Church–Turing thesis. But he did this in the following context
(boldface in original):
"12. Algorithmic theories... In setting up a complete algorithmic
theory, what we do is to describe a procedure, performable for each
set of values of the independent variables, which procedure
necessarily terminates and in such manner that from the outcome we can
read a definite answer, "yes" or "no," to the question, "is the
predicate value true?"" (
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
(especially the Church–Turing thesis) and philosophy of mind
(especially arguments about artificial intelligence). For more, see
Garbage in, garbage out
Introduction to Algorithms
Introduction to Algorithms (textbook)
List of algorithms
List of algorithm general topics
List of important publications in theoretical computer science –
Theory of computation
Computational complexity theory
^ "Any classical mathematical algorithm, for example, can be described
in a finite number of English words" (Rogers 1987:2).
^ Well defined with respect to the agent that executes the algorithm:
"There is a computing agent, usually human, which can react to the
instructions and carry out the computations" (Rogers 1987:2).
^ "an algorithm is a procedure for computing a function (with respect
to some chosen notation for integers) ... this limitation (to
numerical functions) results in no loss of generality", (Rogers
^ "An algorithm has zero or more inputs, i.e., quantities which are
given to it initially before the algorithm begins" (Knuth 1973:5).
^ "A procedure which has all the characteristics of an algorithm
except that it possibly lacks finiteness may be called a
'computational method'" (Knuth 1973:5).
^ "An algorithm has one or more outputs, i.e. quantities which have a
specified relation to the inputs" (Knuth 1973:5).
^ Whether or not a process with random interior processes (not
including the input) is an algorithm is debatable. Rogers opines that:
"a computation is carried out in a discrete stepwise fashion, without
use of continuous methods or analogue devices . . . carried forward
deterministically, without resort to random methods or devices, e.g.,
dice" Rogers 1987:2.
^ a b c Cooke, Roger L. (2005). The History of Mathematics: A Brief
Course. John Wiley & Sons. ISBN 9781118460290.
Kleene 1943 in Davis 1965:274
^ Rosser 1939 in Davis 1965:225
^ "Al-Khwarizmi biography". www-history.mcs.st-andrews.ac.uk.
^ "Etymology of algorithm". Chambers Dictionary. Retrieved December
^ Hogendijk, Jan P. (1998). "al-Khwarzimi". Pythagoras. 38 (2): 4–5.
Archived from the original on April 12, 2009.
^ Oaks, Jeffrey A. "Was al-Khwarizmi an applied algebraist?".
University of Indianapolis. Retrieved May 30, 2008.
^ Brezina, Corona (2006). Al-Khwarizmi: The Inventor Of Algebra. The
Rosen Publishing Group. ISBN 978-1-4042-0513-0.
^ Foremost mathematical texts in history, according to Carl B. Boyer.
^ Oxford English Dictionary, Third Edition, 2012 s.v.
^ Stone 1973:4
^ Stone simply requires that "it must terminate in a finite number of
steps" (Stone 1973:7–8).
^ Boolos and Jeffrey 1974,1999:19
^ cf Stone 1972:5
^ Knuth 1973:7 states: "In practice we not only want algorithms, we
want good algorithms ... one criterion of goodness is the length of
time taken to perform the algorithm ... other criteria are the
adaptability of the algorithm to computers, its simplicity and
^ cf Stone 1973:6
^ Stone 1973:7–8 states that there must be, "...a procedure that a
robot [i.e., computer] can follow in order to determine precisely how
to obey the instruction." Stone adds finiteness of the process, and
definiteness (having no ambiguity in the instructions) to this
^ Knuth, loc. cit
^ Minsky 1967, p. 105
^ Gurevich 2000:1, 3
^ Sipser 2006:157
^ Knuth 1973:7
^ Chaitin 2005:32
^ Rogers 1987:1–2
^ In his essay "Calculations by Man and Machine: Conceptual Analysis"
Seig 2002:390 credits this distinction to Robin Gandy, cf Wilfred
Seig, et al., 2002 Reflections on the foundations of mathematics:
Essays in honor of Solomon Feferman, Association for Symbolic Logic,
A. K Peters Ltd, Natick, MA.
^ cf Gandy 1980:126, Robin Gandy Church's Thesis and Principles for
Mechanisms appearing on pp. 123–148 in J. Barwise et al. 1980 The
Kleene Symposium, North-Holland Publishing Company.
^ A "robot": "A computer is a robot that performs any task that can be
described as a sequence of instructions." cf Stone 1972:3
^ Lambek's "abacus" is a "countably infinite number of locations
(holes, wires etc.) together with an unlimited supply of counters
(pebbles, beads, etc). The locations are distinguishable, the counters
are not". The holes have unlimited capacity, and standing by is an
agent who understands and is able to carry out the list of
instructions" (Lambek 1961:295). Lambek references Melzak who defines
his Q-machine as "an indefinitely large number of locations . . . an
indefinitely large supply of counters distributed among these
locations, a program, and an operator whose sole purpose is to carry
out the program" (Melzak 1961:283). B-B-J (loc. cit.) add the
stipulation that the holes are "capable of holding any number of
stones" (p. 46). Both Melzak and Lambek appear in The Canadian
Mathematical Bulletin, vol. 4, no. 3, September 1961.
^ If no confusion results, the word "counters" can be dropped, and a
location can be said to contain a single "number".
^ "We say that an instruction is effective if there is a procedure
that the robot can follow in order to determine precisely how to obey
the instruction." (Stone 1972:6)
^ cf Minsky 1967: Chapter 11 "Computer models" and Chapter 14 "Very
Simple Bases for Computability" pp. 255–281 in particular
^ cf Knuth 1973:3.
^ But always preceded by IF–THEN to avoid improper subtraction.
^ However, a few different assignment instructions (e.g. DECREMENT,
INCREMENT and ZERO/CLEAR/EMPTY for a Minsky machine) are also required
for Turing-completeness; their exact specification is somewhat up to
the designer. The unconditional GOTO is a convenience; it can be
constructed by initializing a dedicated location to zero e.g. the
instruction " Z ← 0 "; thereafter the instruction IF Z=0 THEN GOTO
xxx is unconditional.
^ Knuth 1973:4
^ Stone 1972:5. Methods for extracting roots are not trivial: see
Methods of computing square roots.
^ Leeuwen, Jan (1990). Handbook of Theoretical Computer Science:
Algorithms and complexity.
Volume A. Elsevier. p. 85.
John G. Kemeny
John G. Kemeny and
Thomas E. Kurtz
Thomas E. Kurtz 1985 Back to Basic: The History,
Corruption, and Future of the Language, Addison-Wesley Publishing
Company, Inc. Reading, MA, ISBN 0-201-13433-0.
^ Tausworthe 1977:101
^ Tausworthe 1977:142
^ Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff
and Chapter 9.1
^ cf Tausworthe 1977
^ Heath 1908:300; Hawking's Dover 2005 edition derives from Heath.
^ " 'Let CD, measuring BF, leave FA less than itself.' This is a neat
abbreviation for saying, measure along BA successive lengths equal to
CD until a point F is reached such that the length FA remaining is
less than CD; in other words, let BF be the largest exact multiple of
CD contained in BA" (Heath 1908:297)
^ For modern treatments using division in the algorithm, see Hardy and
Wright 1979:180, Knuth 1973:2 (
Volume 1), plus more discussion of
Euclid's algorithm in Knuth 1969:293–297 (
Euclid covers this question in his Proposition 1.
^ "Euclid's Elements, Book VII, Proposition 2". Aleph0.clarku.edu.
Retrieved May 20, 2012.
^ Knuth 1973:13–18. He credits "the formulation of algorithm-proving
in terms of assertions and induction" to R. W. Floyd, Peter Naur, C.
A. R. Hoare, H. H. Goldstine and J. von Neumann. Tausworth 1977
Euclid example and extends Knuth's method in section
9.1 Formal Proofs (pages 288–298).
^ Tausworthe 1997:294
^ cf Knuth 1973:7 (Vol. I), and his more-detailed analyses on pp.
1969:294–313 (Vol II).
^ Breakdown occurs when an algorithm tries to compact itself. Success
would solve the Halting problem.
^ Kriegel, Hans-Peter; Schubert, Erich; Zimek, Arthur (2016). "The
(black) art of runtime evaluation: Are we comparing algorithms or
implementations?". Knowledge and Information Systems. 52 (2):
341–378. doi:10.1007/s10115-016-1004-2. ISSN 0219-1377.
^ Gillian Conahan (January 2013). "Better Math Makes Faster Data
^ Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price,
"ACM-SIAM Symposium On Discrete Algorithms (SODA) Archived July 4,
2013, at the Wayback Machine., Kyoto, January 2012. See also the sFFT
^ Kowalski 1979
^ Knapsack Problems Hans Kellerer Springer.
^ Carroll, Sue; Daughtrey, Taz (July 4, 2007). Fundamental Concepts
for the Software Quality Engineer. American Society for Quality.
pp. 282 et seq. ISBN 978-0-87389-720-4.
^ For instance, the volume of a convex polytope (described using a
membership oracle) can be approximated to high accuracy by a
randomized polynomial time algorithm, but not by a deterministic one:
see Dyer, Martin; Frieze, Alan; Kannan, Ravi (January 1991), "A Random
Algorithm for Approximating the
Volume of Convex
Bodies", J. ACM, New York, NY, USA: ACM, 38 (1): 1–17,
CiteSeerX 10.1.1.145.4600 , doi:10.1145/102782.102783 .
George B. Dantzig
George B. Dantzig and Mukund N. Thapa. 2003. Linear Programming 2:
Theory and Extensions. Springer-Verlag.
^ Tsypkin (1971). Adaptation and learning in automatic systems.
Academic Press. p. 54. ISBN 978-0-08-095582-7.
^ Tutt, Andrew (March 15, 2016). "An FDA for Algorithms".
Administrative Law Review. 69. SSRN 2747994 .
^ Algorithmic accountability. Applying the concept to different
Web Foundation. 2017.
^ Aaboe, Asger (2001), Episodes from the Early History of Astronomy,
New York: Springer, pp. 40–62, ISBN 0-387-95136-9
^ Davis 2000:18
^ Bolter 1984:24
^ Bolter 1984:26
^ Bolter 1984:33–34, 204–206.
^ All quotes from W.
Stanley Jevons 1880 Elementary Lessons in Logic:
Deductive and Inductive, Macmillan and Co., London and New York.
Republished as a googlebook; cf Jevons 1880:199–201. Louis Couturat
1914 the Algebra of Logic, The Open Court Publishing Company, Chicago
and London. Republished as a googlebook; cf Couturat 1914:75–76
gives a few more details; interestingly he compares this to a
typewriter as well as a piano. Jevons states that the account is to be
found at Jan . 20, 1870 The Proceedings of the Royal Society.
^ Jevons 1880:199–200
^ All quotes from
John Venn 1881 Symbolic Logic, Macmillan and Co.,
London. Republished as a googlebook. cf Venn 1881:120–125. The
interested reader can find a deeper explanation in those pages.
^ Bell and Newell diagram 1971:39, cf. Davis 2000
^ * Melina Hill, Valley News Correspondent, A Tinkerer Gets a Place in
History, Valley News West Lebanon NH, Thursday March 31, 1983, page
^ Davis 2000:14
^ van Heijenoort 1967:81ff
^ van Heijenoort's commentary on Frege's Begriffsschrift, a formula
language, modeled upon that of arithmetic, for pure thought in van
^ Dixon 1906, cf.
^ cf. footnote in
Alonzo Church 1936a in Davis 1965:90 and 1936b in
Kleene 1935–6 in Davis 1965:237ff,
Kleene 1943 in Davis 1965:255ff
^ Church 1936 in Davis 1965:88ff
^ cf. "Formulation I", Post 1936 in Davis 1965:289–290
^ Turing 1936–7 in Davis 1965:116ff
^ Rosser 1939 in Davis 1965:226
Kleene 1943 in Davis 1965:273–274
Kleene 1952:300, 317
^ Turing 1936–7 in Davis 1965:289–290
^ Turing 1936 in Davis 1965, Turing 1939 in Davis 1965:160
^ Hodges, p. 96
^ Turing 1936–7:116
^ a b Turing 1936–7 in Davis 1965:136
^ Turing 1939 in Davis 1965:160
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Readings and Examples, McGraw–Hill Book Company, New York.
Blass, Andreas; Gurevich, Yuri (2003). "Algorithms: A Quest for
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Theoretical Computer Science. 81. Includes an excellent
bibliography of 56 references.
Boolos, George; Jeffrey, Richard (1999) .
Logic (4th ed.). Cambridge University Press, London.
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Turing machines where
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Undecidable, p. 89ff. The first expression of "Church's Thesis".
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notion of "effective calculability" in terms of "an algorithm", and he
uses the word "terminates", etc.
Church, Alonzo (1936b). "A Note on the Entscheidungsproblem". The
Journal of Symbolic Logic. 1 (1): 40–41. doi:10.2307/2269326.
JSTOR 2269326. Church, Alonzo (1936). "Correction to a Note
on the Entscheidungsproblem". The Journal of Symbolic Logic. 1 (3):
101–102. doi:10.2307/2269030. JSTOR 2269030. Reprinted in
The Undecidable, p. 110ff. Church shows that the
Entscheidungsproblem is unsolvable in about 3 pages of text and 3
pages of footnotes.
Daffa', Ali Abdullah al- (1977). The Muslim contribution to
mathematics. London: Croom Helm. ISBN 0-85664-464-1.
Davis, Martin (1965). The Undecidable: Basic Papers On Undecidable
Propositions, Unsolvable Problems and Computable Functions. New York:
Raven Press. ISBN 0-486-43228-9. Davis gives commentary
before each article. Papers of Gödel, Alonzo Church, Turing, Rosser,
Emil Post are included; those cited in the article are
listed here by author's name.
Davis, Martin (2000). Engines of Logic: Mathematicians and the Origin
of the Computer. New York: W. W. Nortion.
ISBN 0-393-32229-7. Davis offers concise biographies of
Leibniz, Boole, Frege, Cantor, Hilbert,
Gödel and Turing with von
Neumann as the show-stealing villain. Very brief bios of Joseph-Marie
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This article incorporates public domain material from
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Kleene's definition of "general recursion" (known now as mu-recursion)
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Kleene refined his definition of "general
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posit "Thesis I" (p. 274); he would later repeat this thesis (in
Kleene 1952:300) and name it "Church's Thesis"(
Kleene 1952:317) (i.e.,
the Church thesis).
Kleene, Stephen C. (1991) . Introduction to Metamathematics
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ISBN 0-7204-2103-9. Excellent—accessible,
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Knuth, Donald (1969).
Volume 2/Seminumerical Algorithms, The Art of
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A. A. Markov
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Translations, 1961; available from the Office of Technical Services,
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Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College
library. U.S. Dept. of Commerce, Office of Technical Services, number
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(First ed.). Prentice-Hall, Englewood Cliffs, NJ.
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algorithm—an effective procedure..." in chapter 5.1 Computability,
Effective Procedures and Algorithms. Infinite machines.
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Journal of Symbolic Logic. 1 (3): 103–105. doi:10.2307/2269031.
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or erasing marks and going from box to box and eventually halting, as
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of "effective method": "...a method each step of which is precisely
predetermined and which is certain to produce the answer in a finite
number of steps... a machine which will then solve any problem of the
set with no human intervention beyond inserting the question and
(later) reading the answer" (p. 225–226, The Undecidable)
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**>Mathematical Algorithms: 2100 Patentability, Manual of Patent
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Look up algorithm in Wiktionary, the free dictionary.
Wikibooks has a book on the topic of: Algorithms
At Wikiversity, you can learn more and teach others about
the Department of Algorithm
Hazewinkel, Michiel, ed. (2001) , "Algorithm", Encyclopedia of
Mathematics, Springer Science+Business Media B.V. / Kluwer Academic
Publishers, ISBN 978-1-55608-010-4
Algorithms at Curlie (based on DMOZ)
Weisstein, Eric W. "Algorithm". MathWorld.
Dictionary of Algorithms and Data Structures—National Institute of
Standards and Technology
Algorithms and Data Structures by Dr Nikolai Bezroukov
OpenGenus Cosmos - Largest crowd-sourced Algorithm
The Stony Brook
Algorithm Repository—State University of New York at
University of Tennessee
University of Tennessee and Oak Ridge National
Collected Algorithms of the ACM—Association for Computing Machinery
The Stanford GraphBase—Stanford University
University of Iowa
University of Iowa and State University of New York at
Library of Efficient Datastructures and Algorithms (LEDA)—previously
from Max-Planck-Institut für Informatik
Algorithms Course Materials. Jeff Erickson. University of Illinois.
BNF: cb119358199 (data)