Karatsuba algorithm: Difference between revisions

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The '''Karatsuba''' [[multiplication algorithm]], a technique for quickly multiplying large numbers, was discovered by A. Karatsuba and published together with Yu. Ofman in 1962. Its [[time complexity]] is [[Big O notation|Θ]]<math>(n^{\log_23})</math>. This makes it faster than the classical [[Biglong O notationmultiplication|Θclassical]] Θ(''n''²) algorithm (<math>\log_23 \approx 1.585</math>), although for small numbers it is not as fast.

==ExampleAlgorithm==

The speed of this algorithm relies partly on the ability to do [[arithmetic shift|shift]]s faster than a standard multiply, with shifts typically taking [[linear time]] on a practical computer.

To multiply two ''n''-digit numbers ''x'' and ''y'' represented in base ''W'', where ''n'' is even and equal to 2''m'' (if ''n'' is odd instead, or the numbers are not of the same length, this can be corrected by adding zeros at the left end of ''x'' and/or ''y''), we can write

We assume two ''n''-digit numbers ''x'' and ''y'', represented in [[radix|base]] ''B'', where ''B'' is a convenient value to shift with, like [[binary numeral system|two]] in most computers [[as of 2007|today]]. Choosing a number ''m'' less than ''n'', we can write

: ''x'' = ''x''₁ ''W''^''m'' + ''x''₂

: ''y'' = ''y''₁ ''W''^''m'' + ''y''₂

with :''mx''-digit numbers= ''x''₁, ''xB''<subsup>2</sub>, ''ym''<sub>1</subsup> and+ ''yx''₂. The product is given by

: ''xy'' = ''xy''₁ ''WB''^''m'' + ''xy''₂

: ''xy'' =where ''x''₁₂''y''₁ ''W''^2''m'' +and (''x''₁''y''₂ +are ''x''₂''y''₁)less than ''WB''^''m''; +it ''x''₂''y''₂is easily seen that there is a unique representation. We now have

so we need to quickly determine the numbers :''xxy''₁ = (''yx''₁, ''xB''<sub>1</subsup>''ym''<sub>2</subsup> + ''x''₂)(''y''₁ and ''xB''<subsup>2''m''</subsup> + ''y''₂. The heart of Karatsuba's method lies in the observation that this can be done with only three rather than four multiplications:)

::= ''x''₁''y''₁''B''^2''m'' + (''x''₁''y''₂ + ''x''₂''y''₁)''B''^''m'' + ''x''₂''y''₂

The standard method is to multiply the four subproducts seperately, then finally shift and add; this is O(n²). However, Karatsuba observed that we can compute ''xy'' in only three multiplications:

# compute ''x''₁''y''₁, call the result ''A''

# compute ''x''₂''y''₂, call the result ''B''

# compute (''x''₁ + ''x''₂)(''y''₁ + ''y''₂), call the result ''C''

# compute ''C'' - ''A'' - ''B''; this number is equal to ''x''₁''y''₂ + ''x''₂''y''₁.

#:Let compute''X'' = ''x''₁''y''₁, call the result ''A''

To compute these three products of ''m''-digit numbers, we can employ the same trick again, effectively using [[recursion]]. Once the numbers are computed, we need to add them together, which takes about ''n'' operations.

#:Let compute''Y'' = ''x''₂''y''₂, call the result ''B''

:Let ''yZ'' = (''yx''₁ + ''Wx''<supsub>2</sub>)(''my''<sub>1</supsub> + ''y''₂) - X - Y

We note that

:''Z'' = (''x''₁''y''₁ + ''x''₁''y''₂ + ''x''₂''y''₁ + ''x''₂''y''₂) - ''x''₁''y''₁ + ''x''₂''y''₂

#::= compute (''x''₁ + ''xy''₂)( + ''yx''₁₂ + ''y''₂₁), call the result ''C''

Thus ''xy'' = ''X'' ''b''^2m + ''Y'' + ''Z'' ''b''^m; we have cut down the number of multiplications from four to three.

To compute these three products of ''m''-digit numbers, we can employ theKaratsuba same trickmultiplication again, effectively using [[recursion]]. OnceThe theshifts, numbers are computed,additions weand needsubtractions totake addlinear themtime togetherrespectively, whichand takescan aboutbe ''n''considered operationsnegligible.

Karatsuba multiplication works best when the two operands are of similar size; thus while one can theoretically choose ''m'' arbitrarily, the best time bound would be acheived by choosing ''m'' such that the subproducts are roughly equal, that is setting ''m'' to be about ''n''/2.

==Example==

Say we want to compute the product of 1234 and 5678. We choose ''B'' = [[decimal|10]] and ''m'' = 2. We have

:12 34 = 12 &sdot; 10² + 34

:56 78 = 56 &sdot; 10⁵ + 78

:''X'' = 12 &sdot; 56 = 672

:''Y'' = 34 &sdot; 78 = 2652

:''Z'' = (12 + 34)(56 + 78) - ''X'' - ''Y'' = 46 &sdot; 134 - 672 - 2652 = 2840

:''X'' 10^2&sdot;2 + ''Y'' + ''Z'' 10² = 672 0000 + 2652 + 2840 00 = '''706652''' = 1234 &sdot; 5678

==Complexity==

Line 26 ⟶ 47:

:''T''(''n'') = 3 ''T''(''n''/2) + ''cn'' + ''d''

for some constants ''c'' and ''d'', and this [[recurrence relation]] can be solved using the [[master theorem]], giving a time complexity of Θ(''n''^{lnlog(3)/lnlog(2)}). The number lnlog(3)/lnlog(2) is approximately 1.585, so this method is significantly faster than [[long multiplication]] as ''n'' grows large. Because of the overhead of recursion, however, Karatsuba's multiplication is typically slower than long multiplication for small values of ''n''; typical implementations therefore switch to long multiplication if ''n'' is below some threshold when computing the subproducts.

Implementations differ greatly on what the crossover point is when Karatsuba becomes faster than the classical algorithm, but generally numbers over 2³²⁰ are faster with Karatsuba. [http://www.swox.com/gmp/manual/Karatsuba-Multiplication.html][http://ozark.hendrix.edu/~burch/proj/karat/comment1.html] [[Toom-Cook multiplication]] is a faster generalization of Karatsuba's, and is further surpassed by the asymptoticallyfastest fasterknown, the [[Schönhage-Strassen algorithm]] around 2^33,000.

<!--

Implementations differ greatly on what the crossover point is when Karatsuba becomes faster than the classical algorithm, but generally numbers over 2³²⁰ are faster with Karatsuba. [http://www.swox.com/gmp/manual/Karatsuba-Multiplication.html][http://ozark.hendrix.edu/~burch/proj/karat/comment1.html] Karatsuba is surpassed by the asymptotically faster [[Schönhage-Strassen algorithm]] around 2^33,000.

[[[ What's this doing here? ]]]

==An Objective Caml implementation==

Line 85 ⟶ 109:

in

karatsuba_aux (Array.length decompa -1) 0 (Array.length decompa-1) 0;;

-->

==References==