April 2024
Intermediate to advanced
832 pages
29h 31m
English
Content preview from Design and Analysis of Algorithms
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208 Design and Analysis of Algorithms
9.2 A MULTIPLICATION ALGORITHM
Let us consider a multiplication algorithm which requires fewer single-digit multiplications.
Each N-digit integer can be written as the sum of four N/2-digit multiplications.
xy = (a10
N
/2 + b) * (c10
N
/2 + d) = ac10
N
+ (ad + bc) 10
N
/2 + bd
Note, if the numbers are represented in a numbering system other than decimal, then that base would be
substituted for the 10. Also note that four multiplications are required. The following identity,
(ad + bc) = (a + b)(c + d) − ac − bd
helps to reduce the number of multiplications to three. This shows that the central product can be
replaced by a single product plus two additions of size N/2 numbers and two subtractions of size N
numbers. ...
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