The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields " yes " ( or true ) ( more accurately the number b in location B is greater than or equal to the number a in location A ) THEN the algorithm specifies B ← B − A ( meaning the number b − a replaces the old b ).

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.

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.

The concept of algorithm is also used to define the notion of decidability.

In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension.

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 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.

Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm.

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 ).

is the length of time taken to perform the algorithm.

Simulation of an algorithm: computer ( computor ) language: Knuth advises the reader that " the best way to learn an algorithm is to try it.

This algorithm is much less prone to loss of precision due to massive cancellation, but might not be as efficient because of the division operation inside the loop.

This is a serious problem with Algorithm I and is due to catastrophic cancellation in the subtraction of two similar numbers at the final stage of the algorithm.

In 2002, a theoretical attack, termed the " XSL attack ", was announced by Nicolas Courtois and Josef Pieprzyk, purporting to show a weakness in the AES algorithm due to its simple description .< ref >

There is a simple randomized algorithm due to that runs in time where n is the number of clauses and succeeds with high probability to correctly decide 3-Sat.

A divide and conquer algorithm for triangulations in two dimensions is due to Lee and Schachter which was improved by Guibas and Stolfi and later by Dwyer.

Another prime-size FFT is due to L. I. Bluestein, and is sometimes called the chirp-z algorithm ; it also re-expresses a DFT as a convolution, but this time of the same size ( which can be zero-padded to a power of two and evaluated by radix-2 Cooley – Tukey FFTs, for example ), via the identity.

Another algorithm for approximate computation of a subset of the DFT outputs is due to Shentov et al.

If the algorithm has terminated due to a maximum number of generations, a satisfactory solution may or may not have been reached.

The nearest neighbour algorithm is easy to implement and executes quickly, but it can sometimes miss shorter routes which are easily noticed with human insight, due to its " greedy " nature.

For density estimation, the area / volume that is closer to a particular centroid than to any other is inversely proportional to the density ( due to the density matching property of the algorithm ).

However, they are not expected to take as long as the previous projects due to the discovery of an improved algorithm.

Given a desired backbone conformation, side chain packing can be designed using variations of the dead-end elimination algorithm ; however, attempts to design proteins of novel folds have difficulty using this method due to an absence of plausible backbone models.

This slowly propagates through the network until it reaches infinity ( in which case the algorithm corrects iself, due to the relaxation property of Bellman-Ford ).

Jess ( at least versions 5. 0 and later ) also adds a backward chaining algorithm on top of the Rete network, but it cannot be said to fully implement Rete II, in part due to the fact that no full specification is publicly available.

In 2010, Netflix canceled a running contest to improve the company's recommendation algorithm due to privacy concerns: under the terms of the competition, contestants were given access to customer rental data, which the company had purportedly anonymized.

Its original version, due to Gary L. Miller, is deterministic, but the determinism relies on the unproven generalized Riemann hypothesis ; Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm.

A faster randomized minimum spanning tree algorithm based in part on Borůvka's algorithm due to Karger, Klein, and Tarjan runs in expected time.

compress has fallen out of favor in particular user-groups because it makes use of the LZW algorithm, which was covered by a Unisys patent — because of this, gzip and bzip2 increased in popularity on Linux-based operating systems due to their alternative algorithms, along with better file compression.

The VEGAS algorithm, due to G. P. Lepage, is a method for reducing error in Monte Carlo simulations by using a known or approximate probability distribution function to concentrate the search in those areas of the graph that make the greatest contribution to the final integral.

The two penalties are computed together in a relatively complex algorithm, and computing the actual penalties due is somewhat challenging.

The classic spanning tree algorithm, depth-first search ( DFS ), is due to Robert Tarjan.

The Shiloach-Vishkin algorithm, due to Yossi Shiloach and Uzi Vishkin, is the basis for many parallel implementations.

This is mostly due to the fact that the assembly algorithm needs to compare every read with every other read ( an operation that has a complexity of O (< var > n </ var >< sup > 2 </ sup >) but can be reduced to O (< var > n </ var > log (< var > n </ var >)).

** Knuth – Morris – Pratt algorithm

** Fisher – Yates shuffle, also known as the Knuth Shuffle, an algorithm for randomly permuting sets

The Fisher – Yates shuffle, popularized by Donald Knuth, is a simple ( a few lines of code ) and efficient ( O ( n ) on an n-card deck, assuming constant time for fundamental steps ) algorithm for doing this.

It is important to realize that the word problem is in fact solvable for many groups G. For example, polycyclic groups have solvable word problems since the normal form of an arbitrary word in a polycyclic presentation is readily computable ; other algorithms for groups may, in suitable circumstances, also solve the word problem, see the Todd – Coxeter algorithm and the Knuth – Bendix completion algorithm.

Donald Knuth and Ronald W. Moore refined the algorithm in 1975 and Judea Pearl proved its optimality in 1982.

As another example of a divide and conquer algorithm that did not originally involve computers, Knuth gives the method a post office typically uses to route mail: letters are sorted into separate bags for different geographical areas, each of these bags is itself sorted into batches for smaller sub-regions, and so on until they are delivered.

In 1977, Donald Knuth demonstrated that the codebreaker can solve the pattern in five moves or fewer, using an algorithm that progressively reduced the number of possible patterns.

* 1970-Knuth – Bendix completion algorithm developed by Donald Knuth and Peter B. Bendix

* 1976-Knuth – Morris – Pratt algorithm developed by Donald Knuth and Vaughan Pratt and independently by J. H. Morris

The Knuth – Bendix completion algorithm ( named after Donald Knuth and Peter Bendix ) is an algorithm for transforming a set of equations ( over terms ) into a confluent term rewriting system.

The Knuth – Bendix algorithm also can perform coset enumeration, and unlike the Todd – Coxeter algorithm, it can sometimes solve the word problem for infinite groups.

* the Todd – Coxeter algorithm and Knuth – Bendix algorithm for coset enumeration

# REDIRECT Knuth – Morris – Pratt algorithm

# REDIRECT Knuth – Morris – Pratt algorithm

# REDIRECT Knuth – Morris – Pratt algorithm

# REDIRECT Knuth – Morris – Pratt algorithm