Stein’s Algorithm for finding GCD

3.5

Stein’s algorithm or binary GCD algorithm is an algorithm that computes the greatest common divisor of two non-negative integers. Stein’s algorithm replaces division with arithmetic shifts, comparisons, and subtraction.

Examples:

Input: a = 17, b = 34 
Output : 17

Input: a = 50, b = 49
Output: 1

Algorithm to find GCD using Stein’s algorithm gcd(a,b)

  1. If both a and b are 0, gcd is zero gcd(0, 0) = 0.
  2. gcd(a, 0) = a and gcd(0, b) = b because everything divides 0.
  3. If a and b are both even, gcd(a, b) = 2*gcd(a/2, b/2) because 2 is a common divisor. Multiplication with 2 can be done with bitwise shift operator.
  4. If a is even and b is odd, gcd(a, b) = gcd(a/2, b). Similarly, if a is odd and b is even, then
    gcd(a, b) = gcd(a, b/2). It is because 2 is not a common divisor.
  5. If both a and b are odd, then gcd(a, b) = gcd(|a-b|/2, b). Note that difference of two odd numbers is even
  6. Repeat steps 3–5 until a = b, or until a = 0. In either case, the GCD is power(2, k) * b, where power(2, k) is 2 raise to the power of k and k is the number of common factors of 2 found in step 2.

Iterative Implementation

// Iterative C++ program to implement Stein's Algorithm
#include<bits/stdc++.h>
using namespace std;

//Function to implement Stein's Algorithm
int gcd(int a, int b)
{
    /* GCD(0, b) == b; GCD(a,0) == a, GCD(0,0) == 0 */
    if (a == 0)
        return b;
    if (b == 0)
        return a;

    /*Finding K, where K is the greatest power of 2
      that divides both a and b. */
    int k;
    for (k = 0; ((a | b) & 1) == 0; ++k)
    {
        a >>= 1;
        b >>= 1;
    }

    /* Dividing a by 2 until a becomes odd */
    while ((a & 1) == 0)
        a >>= 1;

    /* From here on, 'a' is always odd. */
    do
    {
        /* If b is even, remove all factor of 2 in b */
        while ((b & 1) == 0)
            b >>= 1;

        /* Now a and b are both odd. Swap if necessary
           so a <= b, then set b = b - a (which is even).*/
        if (a > b)
            swap(a, b);   // Swap u and v.

        b = (b - a);
    }   while (b != 0);

    /* restore common factors of 2 */
    return a << k;
}

// Driver code
int main()
{
    int  a = 34, b = 17;
    printf("Gcd of given numbers is %d\n", gcd(a,b));
    return 0;
}

Output:

Gcd of given numbers is 17

Recursive Implementation

//Recursive C++ program to implement Stein's Algorithm
#include<bits/stdc++.h>
using namespace std;

//Function to implement Stein's Algorithm
int gcd(int a, int b)
{
    if (a == b)
        return a;

    /* GCD(0,b) == b; GCD(a,0) == a, GCD(0,0) == 0 */
    if (a == 0)
        return b;
    if (b == 0)
        return a;

    // look for factors of 2
    if (~a & 1 )        // a is even
    {
        if (b & 1)      // b is odd
            return gcd(a >> 1, b);
        else            // both a and b are even
            return gcd(a >> 1, b >> 1) << 1;
    }

    if (~b & 1)         // a is odd, b is even
        return gcd(a, b >> 1);

    // reduce larger number
    if (a > b)
        return gcd((a - b) >> 1, b);

    return gcd((b - a) >> 1, a);
}

// Driver code
int main()
{
    int  a=34, b=17;
    printf("Gcd of given numbers is %d\n", gcd(a,b));
    return 0;
}

Output:

Gcd of given numbers is 17

Time Complexity: O(N*N) where N is the number of bits in the larger number.

You may also like – Basic and Extended Euclidian Algorithm

References:

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