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<0)?-u:u;
        v=(v<0)?-v:v;
        
        if(u==0)
            return v;
        if(v==0)
            return u;
        
        int k=0;
        while((u & 0x01)==0 && (v & 0x01) == 0){
            u>>=1; //divide by 2
            v>>=1;
            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)==1){
            //v is odd
            t = u;
        }
        //process t as a possible even number
        while(t != 0){
            while((t & 0x01)==0){
                //do until t is not even 
                t>>=1;
            }
            if(t>0) //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*(1<<k);
    }
    
    public static void print(int m,int n,int gcd){
        m = (m<0)?-m:m;
        n = (n<0)?-n:n;
        System.out.format("gcd of %d and %d is: %d%n",m,n,gcd);
    }
    
    public static void main(String[] args) {
        int m = -18;
        int n= 12;
        print(m,n,gcdBinary(m,n));
        
        //co-prime
        m = 15;
        n= 28;
        print(m,n,gcdBinary(m,n));
                
        m = 6;
        n= 3;
        print(m,n,gcdBinary(m,n));
        
        m = 6;
        n= 3;
        print(m,n,gcdBinary(m,n));
        
        m = 6;
        n= 0;
        print(m,n,gcdBinary(m,n));
        
        m = 0;
        n= 6;
        print(m,n,gcdBinary(m,n));
        
        m = 0;
        n= 0;
        print(m,n,gcdBinary(m,n));
        
        m = 1;
        n= 1;
        print(m,n,gcdBinary(m,n));
        
        m = 3;
        n= 3;
        print(m,n,gcdBinary(m,n));
        
        m = 2;
        n= 2;
        print(m,n,gcdBinary(m,n));
        
        m = 1;
        n= 4;
        print(m,n,gcdBinary(m,n));
        
        m = 4;
        n= 1;
        print(m,n,gcdBinary(m,n));
        
        m = 10;
        n= 14;
        print(m,n,gcdBinary(m,n));
        
        m = 14;
        n= 10;
        print(m,n,gcdBinary(m,n));
        
        m = 10;
        n= 4;
        print(m,n,gcdBinary(m,n));
        
    
        m = 273;
        n= 24;
        print(m,n,gcdBinary(m,n));
        
        m = 120;
        n= 23;
        print(m,n,gcdBinary(m,n));        
        
    }
}