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In mathematics, an addition chain for computing a positive integer ''n'' can be given by a sequence of natural numbers ''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'': : ''v'' =(''v''0,...,''v''''s''), with ''v''0 = 1 and ''v''''s'' = ''n'' :for each 0< ''i'' ≤ ''s'' holds: ''v''''i'' = ''v''''j'' + ''v''''k'', with ''w''''i''=(''j,k'') and 0 ≤ ''j,k'' ≤ ''i'' − 1 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 Knuth.〔D. E. Knuth, ''The Art of Computer Programming'', Vol 2, "Seminumerical Algorithms", Section 4.6.3, 3rd edition, 1997〕 ==Examples== As an example: ''v'' = (1,2,3,6,12,24,30,31) is an addition chain for 31 of length 7, since :2 = 1 + 1 :3 = 2 + 1 :6 = 3 + 3 :12 = 6 + 6 :24 = 12 + 12 :30 = 24 + 6 :31 = 30 + 1 Addition chains can be used for addition-chain exponentiation: so for example we only need 7 multiplications to calculate 531: :52 = 51 × 51 :53 = 52 × 51 :56 = 53 × 53 :512 = 56 × 56 :524 = 512 × 512 :530 = 524 × 56 :531 = 530 × 51 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「addition chain」の詳細全文を読む スポンサード リンク
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