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.

Although a digital filter can be an IIR filter if the algorithm implementing it includes feedback, it is also possible to easily implement a filter whose impulse truly goes to zero after N time steps ; this is called a finite impulse response ( FIR ) filter.

Although the basic idea is recursive, most traditional implementations rearrange the algorithm to avoid explicit recursion.

Although the problem is not known to be in NC, parallel algorithms with time superior to the Euclidean algorithm exist ; the best known deterministic algorithm is by Chor and Goldreich, which ( in the CRCW-PRAM model ) can solve the problem in O ( n / log n ) time with n < sup > 1 + ε </ sup > processors.

Although this has similarities with a structured architecture, the connections between peers are not determined by any algorithm.

Although physical modelling was not a new concept in acoustics and synthesis, having been implemented using finite difference approximations of the wave equation by Hiller and Ruiz in 1971, it was not until the development of the Karplus-Strong algorithm, the subsequent refinement and generalization of the algorithm into the extremely efficient digital waveguide synthesis by Julius O. Smith III and others, and the increase in DSP power in the late 1980s that commercial implementations became feasible.

Although the purpose of Grover's algorithm is usually described as " searching a database ", it may be more accurate to describe it as " inverting a function ".

Although there is no compulsory algorithm for creating the 8. 3 name from an LFN, Windows uses the following convention:

Although these pseudo-random sequences pass statistical pattern tests for randomness, by knowing the algorithm and the conditions used to initialize it, called the " seed ", the output can be predicted.

Although it would be possible in all-digital systems to eliminate the need for the flywheel algorithm by adding a frame delay to allow the timecode to be decoded prior to the processing of the frame, this is not done in most practical systems because

Although not defined by the Rete algorithm, some engines provide extended functionality to support greater control of truth maintenance.

( Although the PFA is distinct from the Cooley – Tukey algorithm, Good's 1958 work on the PFA was cited as inspiration by Cooley and Tukey in their famous 1965 paper, and there was initially some confusion about whether the two algorithms were different.

Although he reached inconsistent results, an improved formula studied by Hartle, J. Garcia, and E. Elizalde includes the technique of the zeta regularization algorithm

Although more practical than Karnaugh mapping when dealing with more than four variables, the Quine – McCluskey algorithm also has a limited range of use since the problem it solves is NP-hard: the runtime of the Quine – McCluskey algorithm grows exponentially with the number of variables.

Although computer scientists can also focus their work and research on specific areas ( such as algorithm and data structure development and design, software engineering, information theory, database theory, computational complexity theory, human-computer interaction, numerical analysis, programming language theory, computer graphics, and computer vision ), their foundation is the theoretical study of computing from which these other fields derive.

Although the terms " Toom-3 " and " Toom – Cook " are sometimes incorrectly used interchangeably, Toom-3 is only a single instance of the Toom – Cook algorithm, where k = 3.

Although not a hidden surface removal algorithm as such, it implicitly solves the hidden surface removal problem by finding the nearest surface along each view-ray.

Although of little practical use, it is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm.

Although seen more often as an educational device, pancake sorting also appears in applications in parallel processor networks, in which it can provide an effective routing algorithm between processors.

Although incorrect security markings immediately indicated a hoax, there were several features of the code which suggested it might be leaked source code for the Skipjack encryption algorithm, which was still classified at the time.

Although a longbow achieves comparable accuracy and faster shooting rate than an average crossbow, crossbows release more kinetic energy and can be used effectively after a week of training, while a comparable single-shot skill with a longbow takes years of strength training to overcome the draw strength of the longbow, as well as years of practice needed to use it with skill.

Although it is the optimal symbol-by-symbol coding for such probability distributions, Golomb coding achieves better compression capability for the geometric distribution because it does not consider input symbols independently, but rather implicitly groups the inputs.

Although limited to one soprano voice, Bach achieves a variety of movements.

Sean Day-Lewis wrote, “ Although Piers Haggard's direction achieves much verisimilitude and the story is certainly enough to command some addiction ; I did not feel exactly grabbed ; the genre has moved some way since the 1950s and the Professor moves a little slowly for the 1970s ”.

Although both are approximately the same length, Banagher Bridge achieves the crossing with fewer spans ( six as opposed to 16 ).

Although the Allies were again repulsed, these persistent attacks on Blenheim eventually bore fruit, panicking Clérambault into making the worst French error of the day.

Trailing zeros after the decimal point are not necessary, although in science, engineering and statistics they can be retained to indicate a required precision or to show a level of confidence in the accuracy of the number: Although 0. 080 and 0. 08 are numerically equal, in engineering 0. 080 suggests a measurement with an error of up to one part in two thousand (± 0. 0005 ), while 0. 08 suggests a measurement with an error of up to one in two hundred ( see significant figures ).

Although the vowel was originally written in the English orthography as " oo " ( in spellings such as coola or koolah ), it was changed to " oa " possibly due to an error.

Although conversion from wavelengths to metres introduces an additional error in the overall length due to measurement error in determining the refractive index and the frequency, the measurement of frequency is one of the most accurate measurements available.

:" Although theoretical calculations can be made today, the eventually attainable range of chemical reaction cycles, error rates, speed of operation, and thermodynamic efficiencies of such bottom-up manufacturing systems cannot be reliably predicted at this time.

Although Halley's naming error was pointed out by Guillaume Le Gentil in 1756, the name continues to be used.

Although both the consumer desktop and the workstation benefit from CPUs designed around the multicore concept ( essentially, multiple processors on a die, the application of which IBM's POWER4 was a pioneer ), modern workstations typically use multiple multicore CPUs, error correcting memory and much larger on-die caches than those found on " consumer-level " CPUs.

( Although Cruden appears to have made an error in writing this, since his source, Holinshed, does not date Borthwick that early, and Caldwell agrees ) By 1511 Edinburgh was the principal foundry in Scotland, supplanting Stirling Castle, with Scottish and European smiths working under Borthwick, who by 1512 was appointed " master melter of the king's guns ".

Although core memory had been superseded by semiconductor memory by the end of the 1970s, memory is still occasionally called " core "; in particular, a file recording the contents of memory after a system error is usually called a core dump.

Although scientists had warned of the dangers of such an accident for some time, the government blamed the workers who had put out the fire for ' an error of judgement ', rather than the political pressure for fast-tracking the megaton bomb.

Although some scholars have tried to argue that " Titius " is derived from Titus Tatius, Otis Chapman has proposed that " Ticius " is a scribal error for what the poet intended to read as Turnus.

" Although this small " velocity " was measured, it was considered far too small to be used as evidence of speed relative to the aether, and it was understood to be within the range of an experimental error that would allow the speed to actually be zero.

Although The Netherlands were neutral in the First World War, seven bombs hit Goes and Kloetinge, due to an error by a British airplane.

Although Eddy used terms such as " Science ", " Health ", " error ", " shadow ", " belief ", " Christ " and others used by Quimby, these terms are also to be found in the Bible and in the common language of the day.

Although Malvern attempted to fix the error, the village changed the name to Malverne as so many people thought the name was spelled with the " e ".

Although the name " Uwchland " is the result of a spelling error made by the Postal Service in the 19th century when establishing the 19480 ZIP code, it has yet to be corrected.

Although the crash was originally blamed on pilot error, it was subsequently found to have been caused by the build-up of slush towards the end of the runway, causing deceleration of the aircraft and preventing safe flying speed from being attained.

Although it is often dismissed as inaccurate, it is hard to pinpoint a passage where he is categorically in error ... his account of events has stood up remarkably well ... His interpretation of the Glorious Revolution also remains the essential starting point for any discussion of that episode ... What has not survived, or has become subdued, is Macaulay's confident belief in progress.

Although it has also been translated as " The Book of the Abacus ", writes that this is an error: the intent of the book is to describe methods of doing calculations without aid of an abacus, and as confirms, for centuries after its publication the algorismists ( followers of the style of calculation demonstrated in Liber Abaci ) remained in conflict with the abacists ( traditionalists who continued to use the abacus in conjunction with Roman numerals ).

Although the proof uses probability, the final conclusion is determined for certain, without any possible error.

Although analytical solutions to this problem are scarce, there are upper and lower bounds to these functions including the famous Shannon lower bound ( SLB ), which in the case of squared error and memoryless sources, states that for arbitrary sources with finite differential entropy,

Although the error was corrected in later printings, it was picked up by some later editions.

Although Yeong-ha later finds out, he refuses to correct the error and instead emphasizes it when he realizes that it enrages Hikaru, who takes it as a direct affront to Sai.