Binary GCD algorithm

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The binary GCD algorithm is an algorithm which computes the greatest common divisor of two nonnegative integers. It gains a measure of efficiency over the ancient Euclidean algorithm by replacing divisions and multiplications with shifts, which are cheaper when operating on the binary representation used by modern computers. This is particularly critical on embedded platforms that have no direct processor support for division. While the algorithm was first published by the German Josef Stein in 1961, it may have been known in first-century China.[1]

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[edit] Algorithm

This algorithm is also known as Stein's Algorithm.

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 defined.
  2. If u and v are both even, then gcd(u, v) = 2·gcd(u/2, v/2), because 2 is a common divisor.
  3. If u is even and v is odd, then gcd(u, v) = gcd(u/2, v), because 2 is not a common divisor. Similarly, if u is odd and v is even, then gcd(u, v) = gcd(u, v/2).
  4. If u and v are both odd, and u ≥ v, then gcd(u, v) = gcd((u−v)/2, v). If both are odd and u < v, then gcd(u, v) = 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.[2]
  5. Repeat steps 3–4 until u = v, or (one more step) until u = 0. In either case, the result is 2kv, where k is the number of common factors of 2 found in step 2.

Since this definition is tail-recursive, a loop can be used to replace the recursion.

The algorithm requires O((log2 uv)2) worst-case time, or in other words time proportional to the square of the number of bits in u and v together. Although each step reduces at least one of the operands by at least a factor of 2, the subtract and shift operations do not take constant time for very large integers (although they're still quite fast in practice, requiring about one operation per word of the representation).

An extended version of binary GCD, analogous to the extended Euclidean algorithm, is given in The Art of Computer Programming,[3] along with pointers to other versions.

[edit] Implementation

[edit] Implementation in C

Following is an implementation of the algorithm in C, taking two non-negative 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,x) := x */
     if (u == 0 || v == 0)
       return u | v;
 
     /* 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 {
         while ((v & 1) == 0)  /* Loop X */
           v >>= 1;
 
         /* Now u and v are both odd, so diff(u, v) is even.
            Let u = min(u, v), v = diff(u, v)/2. */
         if (u < v) {
             v -= u;
         } else {
             unsigned int diff = u - v;
             u = v;
             v = diff;
         }
         v >>= 1;
     } while (v != 0);
 
     return u << shift;
 }

[edit] Implementation in assembly

This version of binary GCD in ARM assembly (using GNU Assembler syntax) highlights the benefit of branch predication. The loop to implement step 2 consists of three instructions, all predicated. Step 3 consists of two loops, each 2 instructions long (one of the instructions being predicated); however, after the first iteration r0 is kept odd and need not be tested, and only one of the loops is executed. (On cores that implement the clz instruction, steps 2 and 3 can be completed without looping.) Finally, step 4 takes four instructions of which three are predicated.

Since u and v are guaranteed even or odd at certain points, their least significant bits (LSBs) need not be stored in the registers but considered part of the program state. The evenness tests then become a side effect of the bit shifts, since the LSB can be placed in the carry flag. Thus the code works with u/2 and v/2, which are known at the start of each loop to be even or odd.

                             @ Arguments arrive in registers r0 and r1
gcd:
   subs    r3, r0, r0        @ Power-of-two counter = 0, carry flag = 1
   orrs    r2, r0, r1        @ Logical-OR r0 and r1, set flags on result

                             @ Carry flag remains set. If r0 and r1 are
                             @ both zero, this loop does nothing and the
                             @ code exits with r0 = 0.
remove_twos_loop:
   movnes  r2, r2, lsr #1    @ Shift r2 right if >0, carry flag = LSB
   addcc   r3, r3, #1        @ If the LSB was 0 then add 1 to the counter
   bcc     remove_twos_loop  @ and loop to try the next bit (terminates)
   movs    r2, r0, lsr r3    @ else divide r0 by 2^r3 to r2, and test r2
   movnes  r0, r1, lsr r3    @ If r2 > 0 then divide r1 to r0, and test r0
   bxeq    lr                @ If either is zero, return result (r0) = 0
                             @ (for if r2 is zero, r0 must be zero)

                             @ From this point on:-
                             @ u > 0; v > 0; r0 = u / 2; r2 = v / 2.
                             @ The LSBs of u and v are tested in the carry
                             @ flag, then memorized by the program state.
                             @ Now the LSB of either r0 or r2 is 1,
                             @ and u and v are considered to be even.
check_two_r0:
   movs    r0, r0, lsr #1    @ divide u by 2 by shifting r0 right
   bcc     check_two_r0      @ repeat until u is odd (loop terminates)
check_two_r2:                @ Loop X:
   movs    r2, r2, lsr #1    @ divide v by 2 by shifting r2 right
   bcc     check_two_r2      @ repeat until v is odd (loop terminates)

   subs    r2, r2, r0        @ v := v0 - u0 (even, possibly zero)
   rsblt   r2, r2, #0        @ v := |v0 - u0| (if v0 < u0 then v := 0 - v)
   sublt   r0, r0, r2        @ u := min(u0, v0) (if v0 < u0, u := u - v)
   bne     check_two_r2      @ u remains odd. if v > 0 then loop.
                             @ This loop can be shown to terminate.

                             @ if v = 0, the carry flag is set (from subs)
   adc     r0, r0, r0        @ Restore u; u = 2(r0) + 1
   mov     r0, r0, lsl r3    @ multiply u by 2^r3 by shifting left
   bx      lr                @ return to caller with result in r0.

[edit] Efficiency

Brigitte 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.[1][2][4]

Real computers, however, operate on more than one bit at a time. A real GCD subroutine's speed depends substantially on details of implementation, the CPU architecture, and compiler optimizations. With implementations of binary GCD similar to the above, the gain in speed over the Euclidean algorithm is always less in practice than in theory. For example, Knuth (1998) reports a 15% gain.[1] According to one comparison, the greatest gain was about 60%, while on some popular architectures even good implementations of binary GCD were marginally slower than the Euclidean algorithm.[5]

One way to optimize the implementation is to unroll the loop marked "Loop X", taking several bits at a time with the aid of a lookup table.[5] With a 256-byte lookup table (8 bits), the implementation turned to be between 82.5% and 163% faster than the Euclidean algorithm, depending on CPU and compiler. Even with a small 16-byte lookup table (4 bits), the gains are in the range of 54% to 116%.

The lookup-table approach finds its logical conclusion in the use of special CPU instructions such as "CTZ" that count leading or trailing binary zeros in a number, allowing all trailing zero bits to be removed in a single step instead of one at a time.[6] Of course, this optimization is possible only on platforms where such instructions are available.

This code features a plethora of data-dependent branches; using CTZ instructions removes some of them, and the others may be rewritten, either using conditional instructions along the model of the ARM code, or using the various formulations of conditional operations using mixtures of Boolean algebra and arithmetic.

Although this algorithm outperforms the traditional Euclidean algorithm, its asymptotic performance is the same, and it is considerably more complex.

[edit] See also

[edit] References

  1. ^ a b Donald Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms (3rd Edition). Addison-Wesley.
  2. ^ 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 u−v must be divisible by four. Specifically, assuming u ≥ v, if ((u xor v) and 2) = 2, then gcd(u, v) = gcd((u+v)/4, v), and otherwise gcd(u, v) = gcd((u−v)/4, v).
  3. ^ Knuth (1998), answer to exercise 39 of section 4.5.2, p. 646
  4. ^ Notes on Programming by Alexander Stepanov
  5. ^ a b Faster implementations of binary GCD algorithm (download GCD.zip)
  6. ^ PowerPC Compiler Writer's Guide, section 5.10: Count Trailing Zeros. Gives an instruction sequence for counting the trailing zeros in a word, on a platform providing only the complementary "Count Leading Zeros" instruction.

[edit] External links

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