Wheel Factorization
May 8, 2009
Here is a simple function to find factors by trial division:
(define (td-factors n)
(let loop ((n n) (x 2) (fs '()))
(cond ((< n (* x x)) (reverse (cons n fs)))
((zero? (modulo n x)) (loop (/ n x) x (cons x fs)))
(else (loop n (+ x 1) fs)))))
The list of totatives
of a number is calculated by counting down from the number to zero and including in the output all those numbers that are coprime to the input:
(define (totatives n)
(let loop ((i n) (ts '()))
(cond ((zero? i) ts)
((= (gcd i n) 1)
(loop (- i 1) (cons i ts)))
(else (loop (- i 1) ts)))))
It is easier to work with the gaps between spokes than with the absolute positions of the spokes. For instance, the 2,3,5-wheel with spokes at 1, 7, 11, 13, 17, 19, 23 and 29 can be viewed as a wheel with gaps 6, 4, 2, 4, 2, 4, 6, and 2 (which wraps back around to 1). Diffs
makes this calculation:
(define (diffs xs)
(let loop ((x (car xs)) (xs (cdr xs)) (ds '()))
(if (null? xs) (reverse ds)
(loop (car xs) (cdr xs) (cons (- (car xs) x) ds)))))
The wheel
function builds a wheel, including its starting tail:
(define (wheel n)
(let* ((ps (primes n))
(t (apply * (cdr (reverse ps))))
(ts (totatives t))
(ds (diffs ts)))
(append (diffs ps)
(cycle (append (cdr ds)
(list 2)
(list (car ds)))))))
Wheel
uses last-pair
and cycle
, and primes
from a previous exercise:
(define (last-pair xs)
(if (null? (cdr xs)) xs
(last-pair (cdr xs))))
(define (cycle xs)
(set-cdr! (last-pair xs) xs) xs)
Finally, wheel-factors
builds a 2,3,5,7-wheel and loops in the same manner as trial division. The call (wheel 11)
builds a 2,3,5,7-wheel; 11 is the next prime after 7, and is needed to compute the starting point of the wheel:
(define wheel-factors
(let ((w (wheel 11)))
(lambda (n)
(let loop ((n n) (i 2) (fs '()) (w w))
(cond ((< n (* i i)) (reverse (cons n fs)))
((zero? (modulo n i))
(loop (quotient n i) i (cons i fs) w))
(else (loop n (+ i (car w)) fs (cdr w))))))))
Here we see the two factorization functions in action:
> (td-factors 600851475143)
(71 839 1471 6857)
> (wheel-factors 600851475143)
(71 839 1471 6857)
This program can be run at http://codepad.org/3kwNA12j.
[…] Praxis – Wheel Factorization By Remco Niemeijer In today’s Programming Praxis problem we’re supposed to factor numbers both the naive way and using […]
My Haskell solution (see http://bonsaicode.wordpress.com/2009/05/08/programming-praxis-%E2%80%93-wheel-factorization/ for a version with comments):
Thanks to a suggestion from programmingpraxis I was able to speed up my program. Since tdFactors already more or less assumes a sorted list of integers as input we can stop looking as soon as x * x > r. The new code becomes:
There isn’t much point in showing the code to compute the wheel; once it is calculated, it can be converted to a static (i.e., compile-time) value:
My FORTH solution; modified brute force factorization we did in (I think a future) exercise to use the factor wheel. Cool thing is the defining word “cycle” to create a new data type — the cyclic constant array.
public class PrimeCalc {
public static class PrimeWheel {
// http://primes.utm.edu/glossary/xpage/WheelFactorization.html
static final long[] PRIMES = { 2, 3, 5, 7 };
static final int WHEEL_FACTOR = 210; // 2 * 3 * 5 * 7;
static final long[] SIEVES = { 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, 211 };
public long calcFactor(long n) {
for (long p : PRIMES) {
if (n % p == 0) {
return p;
}
}
long lim = (long) StrictMath.sqrt(n);
for (long s = 0; s <= lim; s+= WHEEL_FACTOR) {
for (long sieve : SIEVES) {
long m = s + sieve;
if (n % m == 0) {
return m;
}
}
}
return n;
}
public boolean isPrime(long n) {
if (n < 2) {
return false;
}
return calcFactor(n) == n;
}
}
public static void main(String[] args) {
PrimeWheel pw = new PrimeWheel();
long num = 600851475143L;
java.util.List factors = new java.util.ArrayList();
while (true) {
long f = pw.calcFactor(num);
factors.add(f);
if (f == num) {
break;
} else {
num /= f;
}
}
System.out.println(factors);
// [71, 839, 1471, 6857]
}
}