So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming.

This algorithm is due to Knuth, who cites Welford, and it has been thoroughly analyzed.

On any given run of the algorithm, it has a probability of at most 1 / 3 that it will give the wrong answer.

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.

The PPSZ algorithm has a runtime of for 3-SAT with a single satisfying assignment.

In the setting with many satisfying assignments the randomized algorithm by Schöning has a better bound.

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

If one key cannot be deduced from the other, the asymmetric key algorithm has the public / private key property and one of the keys may be made public without loss of confidentiality.

The Finite Field Diffie-Hellman algorithm has roughly the same key strength as RSA for the same key sizes.

A message encrypted with an elliptic key algorithm using a 109-bit long key has been broken by brute force.

The basic concepts are moderately straightforward, but intractable to calculate ; and a single elegant algorithm or approach has been elusive for more general purpose renderers.

Church subsequently modified his methods to include use of Herbrand – Gödel recursion and then proved ( 1936 ) that the Entscheidungsproblem is unsolvable: There is no generalized " effective calculation " ( method, algorithm ) that can determine whether or not a formula in either the recursive-or λ-calculus is " valid " ( more precisely: no method to show that a well formed formula has a " normal form ").

When an order has been defined in this way, a sorting algorithm can be used to put a list of any number of items into that order.

Problems are nonetheless still common when the algorithm has to encompass more than one language.

A generation of cryptanalysts has cut its teeth analyzing ( that is trying to " crack ") the DES algorithm.

" An astonishing share of the open literature in cryptography in the 1970s and 1980s dealt with the DES, and the DES is the standard against which every symmetric key algorithm since has been compared.

The security algorithm has been broken.

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

( Earley's original algorithm included a look-ahead in the state ; later research showed this to have little practical effect on the parsing efficiency, and it has subsequently been dropped from most implementations.

An arbitrary PID has much the same " structural properties " of a Euclidean domain ( or, indeed, even of the ring of integers ), but knowing an explicit algorithm for Euclidean division, and thus also for greatest common divisor computation, gives a concreteness which is useful for algorithmic applications.

The Euclidean algorithm has been generalized further to other mathematical structures, such as knots and multivariate polynomials.

The algorithm has many theoretical and practical applications.

The FFT has been described as " the most important numerical algorithm of our lifetime ".

Bruun's algorithm ( above ) is another method that was initially proposed to take advantage of real inputs, but it has not proved popular.

) Thus far, no published FFT algorithm has achieved fewer than complex-number additions ( or their equivalent ) for power-of-two.

* Differential Dynamic Programming, a second-order algorithm for trajectory optimization.

For some versions of the algorithm, it is possible to prove that it is convergent ( i. e. it is able to find the global optimum in finite time ).

For instance, in the original version, the RSA algorithm was used to encrypt session keys ; RSA's security depends upon the one-way function nature of mathematical integer factoring.

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.

However, the question has arisen whether an algorithm should be used to determine in what order to connect the dots to maintain the " automatic " nature of the method.

This sorting algorithm is probabilistic in nature.

Algorithms for pattern recognition depend on the type of label output, on whether learning is supervised or unsupervised, and on whether the algorithm is statistical or non-statistical in nature.

It is interdisciplinary in nature, borrowing, adapting and enhancing method and theory from numerous other disciplines such as computer science ( e. g. algorithm and software design, database design and theory ), geoinformation science ( spatial statistics and modeling, geographic information systems ), artificial intelligence research ( supervised classification, fuzzy logic ), ecology ( point pattern analysis ), applied mathematics ( graph theory, probability theory ) and statistics.

Since an algorithm's performance time may vary with different inputs of the same size, one commonly uses the worst-case time complexity of an algorithm, denoted as T ( n ), which is defined as the maximum amount of time taken on any input of size n. Time complexities are classified by the nature of the function T ( n ).

Depending on the nature of the encryption algorithm used, users may be able to verify that a cookie is authentic.

The algorithm is iterative in nature.

The temperature parameter is modified as the algorithm progresses to alter the nature of the search.

The property of self-stabilization enables a distributed algorithm to recover from a transient fault regardless of its nature.

Also, the nature of the ABN algorithm means that any 9-digit number can be made into a valid ABN.

The set of remembered associations may be a fixed-size set controlled by a replacement algorithm or a fixed set, depending on the nature of the function and its use.

A non-memoized version of the above, given the nature of the recursive algorithm involved, would require n + 1 invocations of to arrive at a result, and each of these invocations, in turn, has an associated cost in the time it takes the function to return the value computed.

The document classification scheme ( or indexing algorithm ) in use determines the nature of the document retrieval system.

The nature of this detection algorithm depends on whether or not the target fluctuations observed by each bistatic pair within the multistatic system are mutually correlated.

They are still theoretically intriguing, however, due to the locality and Hebbian nature of their training algorithm, as well as their parallelism and the resemblance of their dynamics to simple physical processes.

Unfortunately, this mathematical modification introduces new issues of its own, namely that the exact nature of the original problem has been destroyed, and visible artifacts of this kludge will continue to haunt the algorithm.

One of the two checksums was additive, and the other was Boolean in nature ( executing EOR instructions ), making for an easy to understand algorithm for other programmers to understand and emulate.

The court moderated that assertion by agreeing that not all patent applications involving formulas are patent-ineligible by saying, " Yet it is equally clear that a process is not unpatentable simply because it contains a law of nature or a mathematical algorithm.

One of the biggest problems with the. BSP file format is that large open areas do not work well due to the nature of the partitioning algorithm used.