用二进制方法求两个整数的最大公约数(GCD)

二进制GCD算法基本原理是:
 先用移位的方式对两个数除2，直到两个数不同时为偶数。然后将剩下的偶数（如果有的话）做同样的操作，这样做的原因是如果u和v中u为偶数,v为奇数，则有gcd(u,v)=gcd(u/2,v)。到这时，两个数都是奇数，将两个数相减(因为gcd(u,v) = gcd(u-v,v))，得到的是偶数t，对t也移位直到t为奇数。每次将最大的数用t替换。

二进制GCD算法优点是只需用减法和二进制移位运算，不像Euclid's算法需要用除法，这在某些嵌入式系统中可能排上用场。

本例实现参考了<<计算机编程的艺术>>第二卷中介绍的算法。

public class GCD_Binary {
    /**
     * solve gcd using binary method
     * @param u
     * @param v
     * @return gcd(u,v)
     */
    public static int gcdBinary(int u,int v){
        u=(u<)?-u:u;
        v=(v<)?-v:v;

        if(u==)
            return v;
        if(v==)
            return u;

        int k=;
        while((u & 0x01)== && (v & 0x01) == ){
            u>>=; //divide by 2
            v>>=;
            k++;
        }
        //at this time, there is at least one number is odd between m and n
        int t=-v; //set it negative for later comparison of (t>0)
        if((v & 0x01)==){
            //v is odd
            t = u;
        }
        //process t as a possible even number
        while(t != ){
            while((t & 0x01)==){
                //do until t is not even
                t>>=;
            }
            if(t>) //u > v (the max is replaced by |t|)
                u=t;
            else //u<v (the max is replaced by |t|)
                v=-t;
            //now u and v are all odd, then u-v is even
            t = u-v;
        }
        return u*(<<k);
    }

    public static void print(int m,int n,int gcd){
        m = (m<)?-m:m;
        n = (n<)?-n:n;
        System.out.format("gcd of %d and %d is: %d%n",m,n,gcd);
    }

    public static void main(String[] args) {
        int m = -;
        int n= ;
        print(m,n,gcdBinary(m,n));

        //co-prime
        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));

        m = ;
        n= ;
        print(m,n,gcdBinary(m,n));        

    }
}