Binary GCD algorithm

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The binary GCD algorithm, also known as Stein's algorithm, is an algorithm that computes the greatest common divisor of two nonnegative integers. Stein's algorithm uses simpler arithmetic operations than the conventional Euclidean algorithm; it replaces division with arithmetic shifts, comparisons, and subtraction. Although the algorithm was first published by the Israeli physicist and programmer Josef Stein in 1967,[1] it may have been known in 1st-century China.[2]

Algorithm[edit]

The algorithm reduces the problem of finding the GCD by repeatedly applying these identities:

  1. gcd(0, v) = v, because everything divides zero, and v is the largest number that divides v. Similarly, gcd(u, 0) = u. gcd(0, 0) is not typically defined, but it is convenient to set gcd(0, 0) = 0.
  2. If u and v are both even, then gcd(uv) = 2·gcd(u/2, v/2), because 2 is a common divisor.
  3. If u is even and v is odd, then gcd(uv) = gcd(u/2, v), because 2 is not a common divisor. Similarly, if u is odd and v is even, then gcd(uv) = gcd(uv/2).
  4. If u and v are both odd, and u ≥ v, then gcd(uv) = gcd((u − v)/2, v). If both are odd and u < v, then gcd(uv) = gcd((v − u)/2, u). These are combinations of one step of the simple Euclidean algorithm, which uses subtraction at each step, and an application of step 3 above. The division by 2 results in an integer because the difference of two odd numbers is even.[3]
  5. Repeat steps 2–4 until u = v, or (one more step) until u = 0. In either case, the GCD is 2kv, where k is the number of common factors of 2 found in step 2.

The algorithm requires O(n2)[4] worst-case time, where n is the number of bits in the larger of the two numbers. Although each step reduces at least one of the operands by at least a factor of 2, the subtract and shift operations take linear time for very large integers (although they're still quite fast in practice, requiring about one operation per word of the representation).

An extended binary GCD, analogous to the extended Euclidean algorithm, is given by Knuth along with pointers to other versions.[5]

Implementation[edit]

Recursive version in C[edit]

Following is a recursive implementation of the algorithm in C. The implementation is similar to the description of the algorithm given above. It use two arguments u and v. All but one of the recursive calls are tail recursive.

unsigned int gcd(unsigned int u, unsigned int v)
{
    // simple cases (termination)
    if (u == v)
        return u;
 
    if (u == 0)
        return v;
 
    if (v == 0)
        return u;
 
    // look for factors of 2
    if (~u & 1) // u is even
    {
        if (v & 1) // v is odd
            return gcd(u >> 1, v);
        else // both u and v are even
            return gcd(u >> 1, v >> 1) << 1;
    }
 
    if (~v & 1) // u is odd, v is even
        return gcd(u, v >> 1);
 
    // reduce larger argument
    if (u > v)
        return gcd((u - v) >> 1, v);
 
    return gcd((v - u) >> 1, u);
}

Iterative version in C[edit]

Following is an implementation of the algorithm in C, taking two (non-negative) integer arguments u and v. It first removes all common factors of 2 using identity 2, then computes the GCD of the remaining numbers using identities 3 and 4, and combines these to form the final answer.

unsigned int gcd(unsigned int u, unsigned int v)
{
  int shift;
 
  /* GCD(0,v) == v; GCD(u,0) == u, GCD(0,0) == 0 */
  if (u == 0) return v;
  if (v == 0) return u;
 
  /* Let shift := lg K, where K is the greatest power of 2
        dividing both u and v. */
  for (shift = 0; ((u | v) & 1) == 0; ++shift) {
         u >>= 1;
         v >>= 1;
  }
 
  while ((u & 1) == 0)
    u >>= 1;
 
  /* From here on, u is always odd. */
  do {
       /* remove all factors of 2 in v -- they are not common */
       /*   note: v is not zero, so while will terminate */
       while ((v & 1) == 0)  /* Loop X */
           v >>= 1;
 
       /* Now u and v are both odd. Swap if necessary so u <= v,
          then set v = v - u (which is even). For bignums, the
          swapping is just pointer movement, and the subtraction
          can be done in-place. */
       if (u > v) {
         unsigned int t = v; v = u; u = t;}  // Swap u and v.
       v = v - u;                       // Here v >= u.
     } while (v != 0);
 
  /* restore common factors of 2 */
  return u << shift;
}

Efficiency[edit]

Akhavi and Vallée proved that binary GCD can be about 60% more efficient (in terms of the number of bit operations) on average than the Euclidean algorithm.[6][7][8] However, although this algorithm modestly outperforms the traditional Euclidean algorithm in real implementations (see next paragraph), its asymptotic performance is the same, and binary GCD is considerably more complex to code given the widespread availability of a division instruction in all modern microprocessors. (Note however that the division instruction may take a significant number of cycles to execute, relative to the other machine instructions.[9][10])

Real computers operate on more than one bit at a time, and even assembly language implementations of binary GCD have to compete against carefully designed hardware circuits for integer division. Overall, Knuth (1998) reports a 15% gain over Euclidean GCD,[2][clarification needed] and according to one comparison, the greatest gain was about 60%,[clarification needed] while on some popular architectures[which?] even good implementations of binary GCD were marginally slower than the Euclidean algorithm.[11][original research?][full citation needed]

In general, with implementations of binary GCD similar to the above C code, the gain in speed over the Euclidean algorithm is always less in practice than in theory. The reason is that the code uses many data-dependent branches.[citation needed][clarification needed] Many branches may be removed by computing min(a,b) and |a-b| using mixtures of Boolean algebra and arithmetic.

The only data-dependent branch that these techniques do not remove is the loop condition marked Loop X, which can be replaced by a single count trailing zeros (CTZ) operation and shift. Depending on platform, CTZ may be performed either by a single hardware instruction, by an equivalent instruction sequence, or with the aid of a lookup table.[11][original research?][full citation needed]

Historical description[edit]

An algorithm for computing the GCD of two numbers was described in the ancient Chinese mathematics book The Nine Chapters on the Mathematical Art. The original algorithm was used to reduce a fraction. The description reads:

"If possible halve it; otherwise, take the denominator and the numerator, subtract the lesser from the greater, and do that alternately to make them the same. Reduce by the same number."

This just looks like a normal Euclidian algorithm, but the ambiguity lies in the phrase "if possible halve it".[citation needed] The traditional interpretation is that this only works when 'both' numbers to begin with are even, under which the algorithm is just a slightly inferior Euclidean algorithm (for using subtraction instead of division). But the phrase may as well mean dividing by 2 should 'either' of the numbers become even, in which case it is the binary GCD algorithm.

See also[edit]

References[edit]

  1. ^ Stein, J. (1967), "Computational problems associated with Racah algebra", Journal of Computational Physics 1 (3): 397–405, doi:10.1016/0021-9991(67)90047-2, ISSN 0021-9991 
  2. ^ a b Knuth, Donald (1998), Seminumerical Algorithms, The Art of Computer Programming 2 (3rd ed.), Addison-Wesley, ISBN 0-201-89684-2 
  3. ^ In fact, the algorithm might be improved by the observation that if both u and v are odd, then exactly one of u + v or uv must be divisible by four. Specifically, assuming u ≥ v, if ((u xor vand 2) = 2, then gcd(uv) = gcd((u + v)/4, v), and otherwise gcd(uv) = gcd((u − v)/4, v).
  4. ^ http://gmplib.org/manual/Binary-GCD.html
  5. ^ Knuth 1998, p. 646, answer to exercise 39 of section 4.5.2
  6. ^ Akhavi, Ali; Vallée, Brigitte (2000), "AverageBit-Complexity of Euclidean Algorithms", Proceedings ICALP'00, Lecture Notes Computer Science 1853: 373–387, CiteSeerX: 10.1.1.42.7616 
  7. ^ Brent, Richard P. (2000), "Twenty years' analysis of the Binary Euclidean Algorithm", Millenial Perspectives in Computer Science: Proceedings of the 1999 Oxford-Microsoft Symposium in honour of Professor Sir Antony Hoare (Palgrave, NY): 41–53  proceedings edited by J. Davies, A. W. Roscoe and J. Woodcock.
  8. ^ Notes on Programming by Alexander Stepanov
  9. ^ Jon Stokes (2007). Inside the Machine: An Illustrated Introduction to Microprocessors and Computer Architecture. No Starch Press. p. 117. ISBN 978-1-59327-104-6. 
  10. ^ Robert Reese, J. W. Bruce, Bryan A. Jones (2009). Microcontrollers: From Assembly Language to C Using the PIC24 Family. Cengage Learning. p. 217. ISBN 1-58450-633-4. 
  11. ^ a b Faster implementations of binary GCD algorithm (download GCD.zip)

Further reading[edit]

External links[edit]