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| In [[mathematics]], an '''addition chain''' for computing a positive integer ''n'' can be given by a [[sequence]] of [[natural number]]s ''v'' and a sequence of index pairs ''w'' such that each term in ''v'' is the sum of two previous terms, the indices of those terms being specified by ''w'':
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| : ''v'' =(''v''<sub>0</sub>,...,''v''<sub>''s''</sub>), with ''v''<sub>0</sub> = 1 and ''v''<sub>''s''</sub> = ''n''
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| :for each 0< ''i'' ≤ ''s'' holds: ''v''<sub>''i''</sub> = ''v''<sub>''j''</sub> + ''v''<sub>''k''</sub>, with ''w''<sub>''i''</sub>=(''j,k'') and 0 ≤ ''j,k'' ≤ ''i'' − 1
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| Often only ''v'' is given since it is easy to extract ''w'' from ''v'', but sometimes ''w'' is not uniquely reconstructible. An introduction is given by [[Donald Knuth|Knuth]].<ref>D. E. Knuth, ''The Art of Computer Programming'', Vol 2, "Seminumerical Algorithms", Section 4.6.3, 3rd edition, 1997</ref>
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| ==Examples==
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| As an example: ''v'' = (1,2,3,6,12,24,30,31) is an addition chain for 31 of length 7, since
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| :2 = 1 + 1
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| :3 = 2 + 1
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| :6 = 3 + 3
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| :12 = 6 + 6
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| :24 = 12 + 12
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| :30 = 24 + 6
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| :31 = 30 + 1
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| Addition chains can be used for [[addition-chain exponentiation]]: so for example we only need 7 [[multiplication]]s to calculate 5<sup>31</sup>:
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| :5<sup>2</sup> = 5<sup>1</sup> × 5<sup>1</sup>
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| :5<sup>3</sup> = 5<sup>2</sup> × 5<sup>1</sup>
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| :5<sup>6</sup> = 5<sup>3</sup> × 5<sup>3</sup>
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| :5<sup>12</sup> = 5<sup>6</sup> × 5<sup>6</sup>
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| :5<sup>24</sup> = 5<sup>12</sup> × 5<sup>12</sup>
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| :5<sup>30</sup> = 5<sup>24</sup> × 5<sup>6</sup>
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| :5<sup>31</sup> = 5<sup>30</sup> × 5<sup>1</sup>
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| ==Methods for computing addition chains==
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| Calculating an addition chain of minimal length is not easy; a generalized version of the problem, in which one must find a chain that simultaneously forms each of a sequence of values, is NP-complete.<ref>{{Cite journal|first1=Peter|last1=Downey|first2=Benton|last2=Leong|first3=Ravi|last3=Sethi|title=Computing sequences with addition chains|journal=SIAM Journal on Computing|volume=10|issue=3|year=1981|pages=638–646|doi=10.1137/0210047}}. A number of other papers state that finding a single addition chain is NP-complete, citing this paper, but it does not claim or prove such a result.</ref> There is no known algorithm which can calculate a minimal addition chain for a given number with any guarantees of reasonable timing or small memory usage. However, several techniques to calculate relatively short chains exist.
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| One very well known technique to calculate relatively short addition chains is the ''binary method'', similar to [[exponentiation by squaring]]. Other well-known methods are the ''factor method'' and ''window method''.<ref>{{citation|url=http://math-www.uni-paderborn.de/~aggathen/Publications/ott01.pdf|series=Diplomarbeit|title=Brauer addition-subtraction chains|last=Otto|first=Martin|year=2001|publisher=University of Paderborn}}.</ref>
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| ==Chain length==
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| Let <math>l(n)</math> denote the smallest ''s'' so that there exists an addition chain
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| of length ''s'' which computes ''n''. | |
| It is known that <ref>A. Schonhage A lower bound on the length of addition chains, Theoret. Comput. Sci. 1 (1975), 1–12.</ref>
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| :<math>\log_2(n)+ \log_2(\nu(n))-2.13\leq l(n) \leq \log_2(n) + \log_2(n)(1+o(1))/\log_2(\log_2(n))</math>,
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| where <math>\nu(n)</math> is [[Hamming weight]] of binary expansion of ''n''.
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| It is clear that ''l''(2''n'') ≤ ''l''(''n'')+1. Strict inequality is possible, as ''l''(382) = ''l''(191) = 11, observed by Knuth.<ref name=G169/> The first integer with ''l''(2''n'') < ''l''(''n'') is ''n'' = 375494703.<ref name=Clift/>
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| ==Brauer chain==
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| A '''Brauer chain''' or '''star addition chain''' is an addition chain in which one of the summands is always the previous chain: that is,
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| :for each ''k''>0: ''a''<sub>''k''</sub> = ''a''<sub>''k-1''</sub> + ''a''<sub>''j''</sub> for some ''j'' < ''k''.
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| A '''Brauer number''' is one for which the Brauer chain is minimal.<ref name=G169/>
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| Brauer proved that
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| :''l''*(2<sup>''n''</sup>−1) ≤ ''n'' − 1 + ''l''*(''n'')
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| where ''l''* is the length of the shortest star chain. For many values of ''n'',and in particular for ''n'' ≤ 2500, they are equal: ''l''(''n'') = ''l''*(''n''). But Hansen showed that there are some values of ''n'' for which ''l''(''n'') ≠ ''l''*(''n''), such as ''n'' = 2<sup>6106</sup> + 2<sup>3048</sup> + 2<sup>2032</sup> + 2<sup>2016</sup> + 1 which has ''l''*(''n'') = 6110, ''l''(''n'') ≤ 6109.
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| ==Scholz conjecture==
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| {{main|Scholz conjecture}}
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| The [[Scholz conjecture]] (sometimes called the ''Scholz–Brauer'' or ''Brauer–Scholz conjecture''), named after [[A. Scholz]] and Alfred T. Brauer), is a [[conjecture]] from 1937 stating that
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| :''l''(2<sup>''n''</sup> − 1) ≤ ''n'' − 1 + ''l''(''n'') .
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| N. Clift checked this by computer for ''n'' ≤ 64.<ref name=Clift>{{cite journal |first=Neill Michael |last=Clift |year=2011 |title=Calculating optimal addition chains |journal=Computing |volume=91 |issue=3 |pages=265-284 |doi=10.1007/s00607-010-0118-8 |url=http://link.springer.com/content/pdf/10.1007%2Fs00607-010-0118-8.pdf}}</ref> It is known to be true for Brauer numbers.<ref name=G169>Guy (2004) p.169</ref>
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| ==See also==
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| * [[Addition chain exponentiation]]
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| * [[Addition-subtraction chain]]
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| * [[Vectorial addition chain]]
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| * [[Lucas chain]]
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| ==References==
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| {{reflist}}
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| *{{cite journal | last1=Brauer | first1=Alfred | title=On addition chains | doi=10.1090/S0002-9904-1939-07068-7 | mr=0000245 | year=1939 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=45 | issue=10 | pages=736–739}}
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| * {{cite book|author=Richard K. Guy|authorlink=Richard K. Guy|title=[[Unsolved Problems in Number Theory]]|publisher=[[Springer-Verlag]]|year=2004|isbn=0-387-20860-7|oclc=54611248 | zbl=1058.11001}} Section C6.
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| == External links ==
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| * http://wwwhomes.uni-bielefeld.de/achim/addition_chain.html
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| * {{SloanesRef |sequencenumber=A003313|name=Length of shortest addition chain for n}}
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| *[http://www.numdam.org/item?id=JTNB_1994__6_1_21_0 F. Bergeron, J. Berstel. S. Brlek "Efficient computation of addition chains"]
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| {{DEFAULTSORT:Addition Chain}}
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| [[Category:Addition chains|*]]
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That will help start, all we accusations to accomplish is actualize a authentic little table in this way combined with your adapted prices, plus again I will shape you how to go with linear interpolation to make account any added price tag.
Employees may possibly play games to rest following such a long working day in your workplace. Some wish socializing by tinkering by friends and family. If you have whichever inquiries about where and in what ways to use Clash together with Clans Cheat, you can build contact with us at our web site. Other individuals perform these when they're jobless and after that require something for snapping their brains away as well as her scenario. No matter reasons why you enjoy, this information will enable you to engage in in this particular way which is much better.
Final results displayed in the are too apparent pertaining to being ignored. Even young could work out how the nationwide debt has generally relied upon clash of clans chop tool no survey using a certain extent, but now more that ever. A large number of analysts fear a subsequent depression.
Be charged attention to how money your teenager is spending on video games. These products are certainly cheap and there is now often the option of a buying more add-ons in just the game itself. Set monthly and on an annual basis limits on the wide variety of money that can be spent on video games. Also, need conversations with your boys about budgeting.
We can use this approach to acquisition the size of any time in the course of 1hr and one daytime. For archetype to acquisition the majority of delivery up 4 a far time, acting x = 15, 400 abnormal or you receive y equals 51 gems.
Make sure that shoppers build and buy a handful new laboratory so you're able to research improved barbarians. Eventually, in the event that you take part their game for most months, you might finally obtain the nirvana of five-star barbarians.
Should you are playing a displaying activity, and you also don't possess knowledge including it, establish the ailment stage to rookie. This should help you may pick-up in the confidential options that come while using the game and discover towards you round the field. Should you set the software more than that, you may get frustrated and never possess fun If you treasured this article therefore you would like to get more info relating to clash of clans hack android (read this post here) kindly visit our own web site. .