For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations, see Algorithm examples.

Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist.

For the frequent case of propositional logic, the problem is decidable but Co-NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks.

For example, algorithms are known for factoring an n-bit integer using just over 2n qubits ( Shor's algorithm ).

For this reason cryptologists tend to look at indicators that an algorithm or key length shows signs of potential vulnerability, to move to longer key sizes or more difficult algorithms.

For example, machines, unlike humans, can support a wide selection of learning algorithms, and controversy has arisen over the relative ethical merits of these options.

For the case of power-of-two, Papadimitriou ( 1979 ) argued that the number of complex-number additions achieved by Cooley – Tukey algorithms is optimal under certain assumptions on the graph of the algorithm ( his assumptions imply, among other things, that no additive identities in the roots of unity are exploited ).

For other points of view on online inputs to algorithms, see streaming algorithm ( focusing on the amount of memory needed to accurately represent past inputs ), dynamic algorithm ( focusing on the time complexity of maintaining solutions to problems with online inputs ) and online machine learning.

For this method of analysis, the offline algorithm knows in advance which edges will fail and the goal is to minimize the ratio between the online and offline algorithms ' performance.

For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions.

For this reason, many online gambling sites provide descriptions of their shuffling algorithms and the sources of randomness used to drive these algorithms, with some gambling sites also providing auditors ' reports of the performance of their systems.

For more details on the sequences of operations used for various quantum algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch-Jozsa algorithm, amplitude amplification, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction.

The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L < sup > 2 </ sup >, where L is the number of bits in the number to be factored ; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 10 < sup > 4 </ sup > qubits without error correction.

For example, loss of a packet in audio application may result in loss of a fraction of a second of audio data, which can be made unnoticeable with suitable error concealment algorithms.

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.

* For specific optimization problems and problem instances, other optimization algorithms may find better solutions than genetic algorithms ( given the same amount of computation time ).

For points in two and three dimensions, output-sensitive algorithms are known that compute the convex hull in time O ( n log h ).

For commodity processors, this means installing an ever-increasing amount of high-speed cache memory and using increasingly sophisticated algorithms to avoid " cache misses ".

For instance, once an image is digitized it can become data that sophisticated algorithms can act upon.

For computer screens, where each individual pixel can mean the difference between legible and illegible characters, some digital fonts use hinting algorithms to make readable bitmaps at small sizes.

For example, given sequence data structures, e. g. singly linked list, vector etc., and algorithms to operate on them, e. g., etc., a direct approach would implement each algorithm specifically for each data structure, giving combinations to implement.

For example, feature extraction algorithms attempt to reduce a large-dimensionality feature vector into a smaller-dimensionality vector that is easier to work with and encodes less redundancy, using mathematical techniques such as principal components analysis ( PCA ).

For the GCHQ this means that it shares information with, and gets information from, the National Security Agency ( NSA ) in the US.

For a historically oriented list of NSA encryption products ( most of them Type 1 ), see NSA encryption systems.

For instance, programs such as Carnivore and NarusInsight have been used by the FBI and NSA to eavesdrop on the systems of internet service providers.

For lawyers, reflecting perhaps their parochial preferences, there has been a special fascination since then in the role played by the Supreme Court in that transformation -- the manner in which its decisions altered in `` the switch in time that saved nine '', President Roosevelt's ill-starred but in effect victorious `` Court-packing plan '', the imprimatur of judicial approval that was finally placed upon social legislation.

For last-ditch emergencies SAC has alternate command posts on KC-135 jet tankers.

Even so astute a commentator as Harold Clurman of The Nation has said that `` Waiting For Godot '' is `` the concentrate of the contemporary European mood of despair ''.

For he has the pride that comes of self-acceptance and the humility, perhaps of the same genesis, not to impose himself upon another.

For several generations much fiction has appeared dealing with the steprelationship.

For the truth formerly experienced by the community no longer has existential status in the community, nor does any answer elaborated by philosophers or theoriticians.

For the figure of Vincent Berger Malraux has obviously drawn on his studies of T. E. Lawrence ( though Berger fights on the side of the Turks instead of against them ), and like both Lawrence and Malraux himself he is a fervent admirer of Nietzsche.

For the sad truth is that while one might write well without having read Bartleby The Scrivener, one is more likely, to write well if one has `` read it, and much else.

For it is the family that, in China, has always provided social security for the indigent, the sick, the down-and-out members of the clan.

For ten years a small group of European and U.S. critics has been calling attention to the half-forgotten Austrian expressionist Egon Schiele, who died 42 years ago at the age of 28.

For something, clearly, has gone very, very seriously wrong in Soviet-Chinese relations, which were never easy, and have now deteriorated.

For it is such a distinguished place, with such fine works of art and such a big library, that there can be little doubt but that the owner has become depraved by all this culture.

For the past 40 years Congress has advocated a carefully planned, balanced and competitive railway system.

For me it has more of both elements than the majority of its competitors.

For some time this writer has been suggesting a Junior Judging Class for Intermediates over 16 and under 20 years of age who are ineligible to compete in the Junior Class.

For an example let's dream up an engine that has a final combustion chamber volume of 5 cubic inches and a cylinder volume of 45 cubic inches.

For this reason, U.S. Camera has prepared this special U.S.A. vacation feature.

For one thing, the driver usually sees less and has less fun than his passengers since it becomes pretty necessary for him to keep at least one eye on the road.

For the last two years, this frontier of the arts has produced a number of so-called `` non-dramas '' which have left indelible, bittersweet impressions on the psyche of this veteran theatregoer.

For any pencil in a plane containing a Af-fold secant of **zg has an image regulus which meets the plane of the pencil in Af lines, namely the images of the lines of the pencil which pass through the intersection of **zg and the multiple secant, plus an additional component to account for the intersections of the images of the general lines of the pencil.

For the industry of this model, the effect of such public pressures in the past has been to hold the price well below the short-run profit-maximizing price ( given the wage rate and the level of GNP ), and even below the entry-limited price ( but not below average cost ).

For critics, Hardy has had no poetic periods -- one does not speak of early Hardy or late Hardy, or of the London or Max Gate period, but simply of Hardy, as of a poetic monolith.

For example Test 3, E 1 refers to a specimen which has been washed by procedure `` 3 ( '' ( at 160-degrees-F ) for a total of 60 minutes in the machine, has been dried in a tumble dryer by procedure `` E '' and has been subjected to restorative forces on the Tension Presser by procedure `` 1 ''.