Exercise 1.17: The exponentiation algorithms in this section are based on performing exponentiation by means of repeated multiplication. In a similar way, one can perform integer multiplication by means of repeated addition. The following multiplication procedure (in which it is assumed that our language can only add, not multiply) is analogous to the expt procedure:
(define (mult a b) (if (= b 0) 0 (+ a (mult a (- b 1)))))
This algorithm takes a number of steps that is linear in
b. Now suppose we include, together
with addition, operations
double, which doubles an integer, and
halve, which divides an
(even) integer by 2. Using these, design a multiplication procedure analogous to
that uses a logarithmic number of steps.
(define (double x) (+ x x)) (define (halve x) (/ x 2)) (define (fast-mult a b) (cond ((= b 0) 0) ((even? b) (double (fast-mult a (halve b)))) (else (+ a (fast-mult a (- b 1))))))
The difference in between \( \Theta(n) \) and \( \Theta(log(n)) \) is apparent for large numbers.
On my system:
> (mult 2 (expt 20 8))
Takes forever, I had to interrupt it.
> (fast-mult 2 (expt 20 8)) 51200000000
The answer is almost instantaneous.