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)); } }