经典算法（2）- 用欧几里得算法求两个整数的最大公约数(GCD)

求两个整数的GCD有两个方法：采用欧几里得算法(Euclid's Algorithm)和二进制GCD算法, 这里实现的是欧几里得算法。

欧几里得算法基本原理很简单，即:
 m = q1.n + r1
 m2= q2.n2 + r2

    ....

 mi = qi.ni + ri
其中m2=n, n2=r1....

gcd(m,n) = gcd(m2,n2) ＝ gcd(mi,ni)....直到ri=0（因为0<=ri<ni，所以ri可以收敛到0）。

实现:

/**
 * 
 * @author ljs 2011-5-17
 * 
 * solve gcd(m,n) using Euclid's Algorithm
 *
 */
public class GCD_Euclid {
	//Euclid's Algorithm to solve gcd(greatest common divisor)
	public static int gcd(int m,int n){
		m = (m<0)?-m:m;
		n = (n<0)?-n:n;
		
		if(n==0)
			return m;
		/*
		//this swap is not needed, since: m % n=m when m<n; so the next recursion will change to gcd(n,m)
		if(m<n){
			int tmp = n;
			n=m;
			m=tmp;
		}	
		*/
		return gcd(n,m%n);
	}
	//an implementation without recursion
	public static int gcdNoTailRecursion(int m,int n){	
		m = (m<0)?-m:m;
		n = (n<0)?-n:n;
		
		while(n!=0){
			int remainder = m%n;
			m = n;
			n = remainder;
		}
		return m;
	}
	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,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		//co-prime
		m = 15;
		n= 28;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
				
		m = 6;
		n= 3;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 6;
		n= 3;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 6;
		n= 0;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 0;
		n= 6;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 0;
		n= 0;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 1;
		n= 1;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 3;
		n= 3;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 2;
		n= 2;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 1;
		n= 4;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 4;
		n= 1;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 10;
		n= 14;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 14;
		n= 10;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 10;
		n= 4;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		
		m = 273;
		n= 24;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		m = 120;
		n= 23;
		print(m,n,gcd(m,n));
		print(m,n,gcdNoTailRecursion(m,n));
		
		
	}
}