最大公约数问题

最大公约数问题

 

以上内容摘自《编程之美》P150-154。

为了方便使用，下面是可拷贝的代码：

Math.h

#pragma once

class Math
{
public:
    Math(void);
    ~Math(void);

public :
    //编程之美P150-154

    //求最大公约数，欧几里德——辗转相除法
    static int Gcd1(int x, int y);

    //求最大公约数，欧几里德——辗转相除法（变相将除法变成了减法）
    static int Gcd2(int x, int y);

    static int Gcd3(int x, int y);

    inline static bool IsEven(int x);

    inline static int Absolute(int x);
};

Math.cpp

#include "Math.h"

Math::Math(void)
{
}

Math::~Math(void)
{
}

int Math::Gcd1(int x, int y)
{
    //y, x%y顺序不能错；
    return y ? Gcd1(y, x % y) : x;
}

int Math::Gcd2(int x, int y)
{
    //与Gcd1相同的方式，但由于x%y计算速度较x-y要慢，但效果相同，所以换用x - y
    // 但用减法和除法不同的是，比如和，%20=10，-20=70，也就是-4×=10
    // 也就是说迭代次数较Gcd1而言通常是增加了。
    return y ? Gcd1(y, x - y) : x;
}

int Math::Gcd3(int x, int y)
{
    if(x < y)
        return Gcd3(y, x);
    if(y == 0)
        return x;
    else
    {
        if(IsEven(x))
        {
            if(IsEven(y))
                return (Gcd3(x >> 1, y >> 1) << 1);
            else
                return Gcd3(x >> 1, y);
        }
        else
        {
            if(IsEven(y))
                return Gcd3(x, y >> 1);
            else
                return Gcd3(y, x - y);
        }
    }
}

bool Math::IsEven(int x)
{
    return !(bool)x & 0x0001;
}

int Math::Absolute(int x)
{
    return x < 0 ? -x : x;
}

Main.cpp

#include <stdafx.h>
#include <iostream>
#include "Math.h"

using namespace std;
int _tmain(const int & arg)
{
    cout<<"Math::Gcd1(42,30) = "<<Math::Gcd1(42,30)<<endl;
    cout<<"Math::Gcd1(30,42) = "<<Math::Gcd1(30,42)<<endl;
    cout<<"Math::Gcd1(50,50) = "<<Math::Gcd1(50,50)<<endl;
    cout<<"Math::Gcd1(0,0) = "<<Math::Gcd1(0,0)<<endl;
    cout<<"Math::Gcd1(-42,-30) = "<<Math::Gcd1(-42,-30)<<endl;
    cout<<"Math::Gcd1(-42,30) = "<<Math::Gcd1(-42,30)<<endl;

    cout<<"------------------------------"<<endl;

    cout<<"Math::Gcd2(42,30) = "<<Math::Gcd2(42,30)<<endl;
    cout<<"Math::Gcd2(30,42) = "<<Math::Gcd2(30,42)<<endl;
    cout<<"Math::Gcd2(50,50) = "<<Math::Gcd2(50,50)<<endl;
    cout<<"Math::Gcd2(0,0) = "<<Math::Gcd2(0,0)<<endl;
    cout<<"Math::Gcd2(-42,-30) = "<<Math::Gcd2(-42,-30)<<endl;
    cout<<"Math::Gcd2(-42,30) = "<<Math::Gcd2(-42,30)<<endl;

    cout<<"------------------------------"<<endl;

    cout<<"Math::Gcd3(42,30) = "<<Math::Gcd3(42,30)<<endl;
    cout<<"Math::Gcd3(30,42) = "<<Math::Gcd3(30,42)<<endl;
    cout<<"Math::Gcd3(50,50) = "<<Math::Gcd3(50,50)<<endl;
    cout<<"Math::Gcd3(0,0) = "<<Math::Gcd3(0,0)<<endl;
    cout<<"Math::Gcd3(-42,-30) = "<<Math::Gcd3(-42,-30)<<endl;
    cout<<"Math::Gcd3(-42,30) = "<<Math::Gcd3(-42,30)<<endl;

    return 0;
}

不过有一点值得一提，就是所谓性能最好效率最高的Gcd3不支持负数，也就是最后两行测试代码无法通过。但是限于对负数的最大公约数并没有定义，也就是说即便上面的Gcd1和Gcd2好像算出了负数，但它们的结果没有意义。