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{{short description|Algorithm for integer multiplication}}
{{Infobox algorithm
[[File:Karatsuba_multiplication.svg|thumb|300px|Karatsuba multiplication of az+b and cz+d (boxed), and 1234 and 567. Magenta arrows denote multiplication, amber denotes addition, silver denotes subtraction and light cyan denotes left shift. (A), (B) and (C) show recursion used to obtain intermediate values.]]▼
|class = [[Multiplication algorithm]]
The '''Karatsuba algorithm''' is a fast [[multiplication algorithm]]. It was discovered by [[Anatoly Karatsuba]] in 1960 and published in 1962.<ref name="kara1962">▼
|image = <!-- filename only, no "File:" or "Image:" prefix, and no enclosing [[brackets]] -->
|caption =
|data =
|time = <!-- Worst time big-O notation -->
|best-time =
|average-time =
|space = <!-- Worst-case space complexity; auxiliary space
(excluding input) if not specified -->
}}
▲[[File:Karatsuba_multiplication.svg|thumb|300px|Karatsuba multiplication of az+b and cz+d (boxed), and 1234 and 567 with z=100. Magenta arrows denote multiplication, amber denotes addition, silver denotes subtraction and
▲The '''Karatsuba algorithm''' is a fast [[multiplication algorithm]] for [[Integer|integers]]. It was discovered by [[Anatoly Karatsuba]] in 1960 and published in 1962.<ref name="kara1962">
{{cite journal
| author = A. Karatsuba and Yu. Ofman
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:<math>z_0 = x_0 y_0.</math>
These formulae require four multiplications and were known to [[Charles Babbage]].<ref>Charles Babbage, Chapter VIII – Of the Analytical Engine, Larger Numbers Treated, [https://archive.org/details/bub_gb_Fa1JAAAAMAAJ/page/n142 <!-- pg=125 --> Passages from the Life of a Philosopher], Longman Green, London, 1864; page 125.</ref> Karatsuba observed that <math>xy</math> can be computed in only three multiplications, at the cost of a few extra additions. With <math>z_0</math> and <math>z_2</math> as before and <math>z_3=(x_1 + x_0) (y_1 + y_0),</math> one can observe that
:<math>
\begin{align}
z_1 &= x_1 y_0 + x_0 y_1 \\
&= (x_1 + x_0) (y_0 + y_1) - x_1 y_1 - x_0 y_0 \\
&=
\end{align}
</math>
Thus only three multiplications are required for computing <math>z_0, z_1</math> and <math>z_2.</math>
===Example===
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: ''z''<sub>1</sub> = ('''12''' + '''345''') '''×''' ('''6''' + '''789''') − ''z''<sub>2</sub> − ''z''<sub>0</sub> = 357 '''×''' 795 − 72 − 272205 = 283815 − 72 − 272205 = 11538
We get the result by just adding these three partial results, shifted accordingly (and then taking carries into account by decomposing these three inputs in base ''1000''
: result = ''z''<sub>2</sub> · (''B''<sup>''m''</sup>)<sup>''2''</sup> + ''z''<sub>1</sub> · (''B''<sup>''m''</sup>)<sup>''1''</sup> + ''z''<sub>0</sub> · (''B''<sup>''m''</sup>)<sup>''0''</sup>, i.e.
: result = 72 · ''1000''<sup>2</sup> + 11538 · ''1000'' + 272205 = '''83810205'''.
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Since one can extend any inputs with zero digits until their length is a power of two, it follows that the number of elementary multiplications, for any ''n'', is at most <math>3^{ \lceil\log_2 n \rceil} \leq 3 n^{\log_2 3}\,\!</math>.
Since the additions, subtractions, and digit shifts (multiplications by powers of ''B'') in Karatsuba's basic step take time proportional to ''n'', their cost becomes negligible as ''n'' increases. More precisely, if ''
:<math>T(n) = 3 T(\lceil n/2\rceil) + cn + d</math>
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for some constants ''c'' and ''d''. For this [[recurrence relation]], the [[master theorem (analysis of algorithms)|master theorem for divide-and-conquer recurrences]] gives the [[big O notation|asymptotic]] bound <math>T(n) = \Theta(n^{\log_2 3})\,\!</math>.
It follows that, for sufficiently large ''n'', Karatsuba's algorithm will perform fewer shifts and single-digit additions than longhand multiplication, even though its basic step uses more additions and shifts than the straightforward formula. For small values of ''n'', however, the extra shift and add operations may make it run slower than the longhand method.
==Implementation==
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This computation of <math>(x_0 - x_1)</math> and <math>(y_1 - y_0)</math> will produce a result in the range of <math>-B^m < \text{result} < B^m</math>. This method may produce negative numbers, which require one extra bit to encode signedness, and would still require one extra bit for the multiplier. However, one way to avoid this is to record the sign and then use the absolute value of <math>(x_0 - x_1)</math> and <math>(y_1 - y_0)</math> to perform an unsigned multiplication, after which the result may be negated when both signs originally differed. Another advantage is that even though <math>(x_0 - x_1)(y_1 - y_0)</math> may be negative, the final computation of <math>z_1</math> only involves additions.
==References==
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