Flow chart of an algorithm ( Euclid's algorithm ) for calculating the greatest common divisor ( g. c. d.

In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.

More precisely, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function.

" 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.

" 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.

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.

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 ), in order for the target machines to produce output from given input ( perhaps null ).

If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.

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.

Now it is easy to convince oneself that the set X could not possibly be measurable for any rotation-invariant countably additive finite measure on S. Hence one couldn't expect to find an algorithm to find a point in each orbit, without using the axiom of choice.

This algorithm is often more numerically reliable than the naïve algorithm for large sets of data, although it can be worse if much of the data is very close to but not precisely equal to the mean and some are quite far away from it.

For a particularly robust two-pass algorithm for computing the variance, first compute and subtract an estimate of the mean, and then use this algorithm on the residuals.

note that the above online algorithm III is a special case of an algorithm that works for any partition of the sample into sets,:

An example of the online algorithm for kurtosis implemented as described is:

The first approach is to compute the statistical moments by separating the data into bins and then computing the moments from the geometry of the resulting histogram, which effectively becomes a one-pass algorithm for higher moments.

The algorithm described by AES is a symmetric-key algorithm, meaning the same key is used for both encrypting and decrypting the data.

ALGOL ( short for ALGOrithmic Language ) is a family of imperative computer programming languages originally developed in the mid 1950s which greatly influenced many other languages and was the standard method for algorithm description used by the ACM, in textbooks, and academic works for the next 30 years and more.

For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise.

: For an example of the simple algorithm " Add m + n " described in all three levels see Algorithm examples.

The design and strength of all key lengths of the AES algorithm ( i. e., 128, 192 and 256 ) are sufficient to protect classified information up to the SECRET level.

In fact, any increase or decrease in radiation intensity that is observed to the same extent at all pixels chosen for correction is eliminated by the correction algorithm.

In all, the Blowfish encryption algorithm will run 521 times to generate all the subkeys-about 4KB of data is processed.

That briefly means that there is no known algorithm that efficiently solves all instances of SAT, and it is generally believed ( but not proven, see P versus NP problem ) that no such algorithm can exist.

Although this problem seems easier, it has been shown that if there is a practical ( randomized polynomial-time ) algorithm to solve this problem, then all problems in NP can be solved just as easily.

After one pair of Bézout coefficients ( x, y ) has been computed ( using extended Euclid or some other algorithm ), all pairs may be found using the formula

In a symmetric key algorithm ( e. g., DES and AES ), the sender and receiver must have a shared key set up in advance and kept secret from all other parties ; the sender uses this key for encryption, and the receiver uses the same key for decryption.

Derivatives of Shor's algorithm are widely conjectured to be effective against all mainstream public-key algorithms including RSA, Diffie-Hellman and elliptic curve cryptography.

However, consider the short but never halting algorithm which systematically lists and runs all possible programs ; whenever one of them halts its probability gets added to the output ( initialized by zero ).

The simplest checksum algorithm is the so-called longitudinal parity check, which breaks the data into " words " with a fixed number n of bits, and then computes the exclusive or of all those words.

A variant of the previous algorithm is to add all the " words " as unsigned binary numbers, discarding any overflow bits, and append the two's complement of the total as the checksum.

A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem.

The algorithm looks for all nodes in the array which are connected to the start node by a path of the target color, and changes them to the replacement color.

There are many ways in which the flood-fill algorithm can be structured, but they all make use of a queue or stack data structure, explicitly or implicitly.

However, if the shape is complex with many features, the algorithm spends a large amount of time tracing the edges of the region trying to ensure that all can be painted.

The algorithm is as follows: eliminate x from all equations below, and then eliminate y from all equations below.

If a large, b-bit number is the product of two primes that are roughly the same size, then no algorithm has been published that can factor in polynomial time, i. e., that can factor it in time O ( b < sup > k </ sup >) for some constant k. There are published algorithms that are faster than O (( 1 + ε )< sup > b </ sup >) for all positive ε, i. e., sub-exponential.

) is NP-complete, thus it is expected that no algorithm can be both correct and fast ( polynomial-time ) on all cases.

In information theory and computer science, a code is usually considered as an algorithm which uniquely represents symbols from some source alphabet, by encoded strings, which may be in some other target alphabet.

A collation algorithm such as the Unicode collation algorithm defines an order through the process of comparing two given character strings and deciding which should come before the other.

A standard algorithm for collating any collection of strings composed of any standard Unicode symbols is the Unicode Collation Algorithm.

In computer science, the Earley parser is an algorithm for parsing strings that belong to a given context-free language.

; Markov algorithm: a string rewriting system that uses grammar-like rules to operate on strings of symbols.

In a genetic algorithm, a population of strings ( called chromosomes or the genotype of the genome ), which encode candidate solutions ( called individuals, creatures, or phenotypes ) to an optimization problem, is evolved toward better solutions.

Jaffe continued to explore the musical and technical possibilities of the algorithm in Silicon Valley Breakdown, for computer-generated plucked strings ( 1982 ), as well as in later works such as Telegram to the President, 1984 for string quartet and tape, and Grass for female chorus and tape ( 1987 ).

searches for any of a list of fixed strings using the Aho – Corasick string matching algorithm.

* LZW ( algorithm ) ( 1978 paradigm ) Data Compression-given strings s and t, will compressing s with an LZ78 method add t to the dictionary?

A 5-bit code gives 2 < sup > 5 </ sup > = 32 possible combinations of bits, so when the 33rd dictionary word is created, the algorithm will have to switch at that point from 5-bit strings to 6-bit strings ( for all code values, including those which were previously output with only five bits ).

It is a kind of dictionary-matching algorithm that locates elements of a finite set of strings ( the " dictionary ") within an input text.

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input.

In theoretical computer science, a Markov algorithm is a string rewriting system that uses grammar-like rules to operate on strings of symbols.

A single seed number is run through a fixed algorithm the appropriate number of times and creates a sequence of numbers determining each planet's complete composition ( position in the galaxy, prices of commodities, and even name and local details — text strings are chosen numerically from a lookup table and assembled to produce unique descriptions for each planet ).

; Markov algorithm: a string rewriting system that uses grammar-like rules to operate on strings of symbols.

BGI ’ s computational biologists developed the first successful algorithm, based on graph theory, for aligning billions of 25 to 75-base pair strings produced by next-generation sequencers, specifically Illumina ’ s Genome Analyzer, during de novo sequencing.

gave an exact algorithm to sort binary and ternary strings.

It comprises a symbol set made up of 32 different characters, as well as an algorithm for encoding arbitrary strings using 8-bit characters into the Base32 alphabet.