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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''.
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| 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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