经典算法（4）- 用欧几里得算法实现扩展的最大公约数(Extended GCD)

扩展的gcd算法即除了计算gcd(m,n)还要计算整数x和y，使之满足gcd(m,n) = m.x + n.y。 下面的算法中使用迭代方式。 extendedGCD2方法是extendedGCD的简化版本，考虑到在初值向量r{-1} = [1 0], r{0} = [0 1]下，满足递推关系：r{i} = r{i-2} - q{i}.r{i-1}。

采用Euclid's算法时，不仅要r（余数）的值，还需要q(商)的值。

本例实现参考了Wikipedia中介绍的迭代方法：http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm

/**
*
* @author ljs 2011-5-17
*
* solve extended gcd using Euclid's Algorithm and Iterative Method
*
*/
public class ExtGCD_Euclid {
/**
* caculate x,y to satisfy Bezout Identity: m.x + n.y = gcd(m,n)
* @param m
* @param n
* @return {d,x,y} for: d = m.x + n.y;
*/
public static int[] extendedGCD(int m,int n){
m = (m<0)?-m:m;
n = (n<0)?-n:n;

if(n == 0 ){
return new int[]{m,1,0};
}

int r = m % n;
int q = m / n;
if(r ==0)
return new int[]{n,0,1};
else{
m=n;
n=r;
r = m % n;
int lastq = q;
q = m / n;
if(r==0){
return new int[]{n,1,-lastq};
}else{
int coeffM1=1;
int coeffN1=-lastq;
//r2 = n - q2.r1
int coeffM2=-q;
int coeffN2=1+lastq * q;

while(n!=0){
m=n;
n=r;
r = m%n;
q = m / n;
if(r==0){
break;
}else{
//for k-th loop, r{k} = r{k-2} - q{k}.r{k-1}, here q{k} is quotient = m{k} / n{k}
int coeffM3=coeffM1 - q*coeffM2;
int coeffN3=coeffN1 - q*coeffN2;
coeffM1 = coeffM2;
coeffN1 = coeffN2;
coeffM2 = coeffM3;
coeffN2 = coeffN3;
}
}
return new int[]{n,coeffM2,coeffN2};
}
}
}

/**
* simplified from the above extendedGCD(m,n)
* @see also "http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm"
* @param m
* @param n
* @return {d,x,y} for: d = m.x + n.y;
*/
public static int[] extendedGCD2(int m,int n){
m = (m<0)?-m:m;
n = (n<0)?-n:n;

int r00=1,r01=0;  //[1 0]
int r10=0,r11=1;  //[0 1]

int d = 0;
int x = r00;
int y = r01;
while(n!=0){
int remainder = m%n;
int quotient = m/n;

if(remainder == 0){
x = r10;
y = r11;
}else{
int tmp0 = r00 - quotient*r10;
int tmp1 = r01 - quotient*r11;
r00=r10;
r01=r11;
r10 = tmp0;
r11 = tmp1;
}

m = n;
n = remainder;
}
d = m;

return new int[]{d,x,y};
}

public static void print(int m,int n,int[] extGCDResult){
m = (m<0)?-m:m;
n = (n<0)?-n:n;
System.out.format("extended gcd of %d and %d is: %d = %d.{%d} + %d.{%d}%5s%n",m,n,extGCDResult[0],m,extGCDResult[1],n,extGCDResult[2],(m*extGCDResult[1] + n * extGCDResult[2] == extGCDResult[0])?"OK":"WRONG!!!");
}

public static void main(String[] args) {

int m = -18;
int n= 12;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

//co-prime
m = 15;
n= 28;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 6;
n= 3;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 6;
n= 3;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 6;
n= 0;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 0;
n= 6;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 0;
n= 0;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 1;
n= 1;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 3;
n= 3;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 2;
n= 2;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 1;
n= 4;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 4;
n= 1;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 10;
n= 14;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 14;
n= 10;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 10;
n= 4;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 273;
n= 24;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

m = 120;
n= 23;
print(m,n,extendedGCD(m,n));
print(m,n,extendedGCD2(m,n));

}
}