大数类的简单实现

本文仅做参考，主要是体现大数实现原理。

 

大数操作是使用计算机计算经常遇到的问题。

使用VS2005，在32位的机器上

 

各个类型的大小：

sizeof(int): 4

sizeof(unsigned): 4

sizeof(long): 4

sizeof(long long): 8

sizeof(char): 1

 

因此，unsigned的二进制位数为：4*8=32位，能够表示的最大数为：4294967295 即

11111111 11111111 11111111 11111111

而 long long 最大能够表示的无符号数为2^64 -1=18446744073709551615

因此，实现一个大数类是很有必要的。

网上有很多以实现不错的大数类，我这里只是练习练习。

 

使用数组记录大数据：

 

 * 这里列举vector<char> dats表示大数

 *100000的二进制为:
 * 1111 01000010 01000000
 *将之存放到数组中为:
 * -----------------     -----------------        -----------------
 * |   第2个元素   |   |   第1个元素   |       |   第0个元素   |
 * -----------------     -----------------        -----------------
 * | | | | |1|1|1|1| | |1|0|0|0|0|1|0| | |1|0|0|0|0|0|0|
 * ----------------- ----------------- -----------------
 * 即:dats[0] = 64,dats[1] =66,dats[2]=15
 * 还原方法:((dats[2]*2^8)+dats[1])*2^8+dats[0] = 100000

 

 

定义一个大数类的结构：

class bigint 

{ 

    char sign;      //符号0:+, 1:-                   

    vector<unsigned> dats;

}

 

1.将string转到dats中,可以通过用2除来转为二进制。

例如：string numStr = ‘76543210’;

首先由于末位为0，为偶数，所以记录0，结果为0

7&1==1 有补位给6, 7>>1 = 3

16&1==0 无补位， 16>>1 = 8

5&1 ==1 有补位给4， 5>>1 = 2

14&1 ==0 无补位…

 

最后得到38271605

由于末位为5，为奇数，所以再记录1，结果为10

在次除2得到19135802

由于末位为2，为偶数，所以再记录0，结果为010

….

最后当末位为1，为奇数，再记录1，结果为100100011111111010011101010

 

当然也可以采用分块计算，即把numStr分成多个整数，如：

numStr = “755432109876543210”;

可以每次取8位放入到vector<unsigned> arr中，得到：

Arr = {75,54321098,76543210};

同样，也可以按照上面的除法，只是数组元素之间的补位值相当于10^8。

如，第一个数75&1==1，所以补位为1，也就说，

下一个数是154321098>>1.

 

 

2.加操作：

对应dats各块，进行加，数组的每个位置和用long long carry记录，则该为的值 = carry & (1<<32 -1)

而进位值为carry>>32.具体需要注意位对齐细节，以及最后的进位处理，可能会进位多次。还要注意数的符号。

 

3.减操作：

减法是通过反码来实现的，其实就是利用越界来实现。

如：正数A – 正数B，首先求正数B的反码

若当相减之后为正数，那就是（正数A + 正数B的反码）再加1.

若相减之后为负数，那么就是（正数A + 正数B的反码）再取反码。当然别忘了修改符号sign。

 

对于乘除，则需要对数组各个部分进行分开来看，考虑进位即可。

跟加法很相似，只是原来各位置相加变为一个位置对应多个位置的相加。

 

4.除法操作：

按照位运算来操纵。

 

贴上代码，代码只是半成品，未经过测试，不能直接使用。

/* bigInt.hmade by g_idea 20110301**/#include <string>#include <vector>#include <algorithm>#define JUDGECARRY 4294967296class bigInt{public:bigInt(){}bigInt(const std::string src) {if( !StrToBigInt(src) )throw "Input num Error!";}bigInt( int num ) {if( num < 0 ){sign = 1;dats.push_back(-num);}else{sign = 0;dats.push_back(num);}}bigInt( const bigInt& otherNum ):sign(otherNum.sign),dats(otherNum.dats){}~bigInt(){}public://重置void reset(){this->sign = 0;this->dats.clear();this->dats.push_back(0);}//左移bigInt operator<<(unsigned length) {bigInt result;result.sign = this->sign;result.dats = std::vector<unsigned>(length/32,0);length %= 32;long long carry = 0;for( size_t i = 0; i < dats.size(); ++i ){long long temp = dats[i];temp <<= length;result.dats.push_back( unsigned((temp^carry)&(JUDGECARRY-1)) );carry = temp>>32;}if( carry > 0 )result.dats.push_back(unsigned(carry));result.bigIntReserve();return result;}//右移bigInt operator>>(unsigned length) {bigInt result;result.sign = this->sign;int pos(length/32);length %= 32;unsigned selectBits = ((1<<length) - 1);long long carry = 0;result.dats = std::vector<unsigned>(dats.size() - pos);for( int i = int(dats.size()-1); i >= pos; --i ){result.dats[i-pos] = unsigned((((carry<<32)+dats[i])>>length)&(JUDGECARRY-1));carry = (dats[i]&selectBits);}VecReserve<unsigned>(result.dats);return result;}//逻辑与bigInt operator&( bigInt& right ){bigInt result;size_t length = this->dats.size()<right.dats.size()?dats.size():right.dats.size();for(size_t i=0; i< length; ++i){result.dats.push_back(dats[i] & right.dats[i] );}VecReserve<unsigned>(result.dats);return result;}//逻辑或bigInt operator|( bigInt& right ){bigInt result;result.sign = 0;std::vector<unsigned> *minPos = &dats;//指向位数较小std::vector<unsigned> *maxPos = &right.dats;//指向位数较大if( dats.size() > right.dats.size() ){maxPos = &dats;minPos = &right.dats;}for(int i=0; i< int(minPos->size()); ++i) {result.dats.push_back(minPos->at(i)|maxPos->at(i) );}//将剩余高位直接拷贝for(int i = int(minPos->size()); i < int(maxPos->size()); ++i){result.dats.push_back(maxPos->at(i));}return result;}//逻辑异或bigInt operator^( bigInt& right ){bigInt result;result.sign = 0;std::vector<unsigned> *minPos = &dats;//指向位数较小std::vector<unsigned> *maxPos = &right.dats;//指向位数较大if( dats.size() > right.dats.size() ){maxPos = &dats;minPos = &right.dats;}for(int i=0; i< int(minPos->size()); ++i) {result.dats.push_back(minPos->at(i)^maxPos->at(i) );}//将剩余高位直接拷贝for(int i = int(minPos->size()); i < int(maxPos->size()); ++i){result.dats.push_back(maxPos->at(i));}VecReserve<unsigned>(result.dats);return result;}bigInt bigInt::operator|=(bigInt right){*this=*this|right;return *this;}bigInt bigInt::operator&=(bigInt right){*this=*this&right;return *this;}bigInt bigInt::operator^=(bigInt right){*this=*this^right;return *this;}bigInt bigInt::operator<<=(unsigned right){*this=*this<<right;return *this;}bigInt bigInt::operator>>=(unsigned right){*this = *this>>right;return *this;}std::string GetDecimalStr(){if( dats.size() == 1 && dats.front() == 0 )return "0";int length = int(dats.size());std::string result = "0";//2^32 = 4294967296std::string two_32 = "6927694924";for( int i = length-1; i >= 0; --i ){unsigned long num = dats[i];std::string temp;while(num>0){temp.push_back( char(num%10 + '0'));num /= 10;}if( i == length-1 ){result = temp;continue;}result = strNumAdd(strNumMul(result,two_32),temp);}if( sign == 1 )result.push_back('-');std::reverse(result.begin(),result.end());return result;}// 计算n的p次方,如：3^7 = 3*3*3*3*3*3*3 = ((3*3)*(3*3)) * (3*3) * 3 bigInt Power(bigInt n, bigInt p) {bigInt ONE(1);bigInt odd(1); //oddk用来计算“剩下的”乘积while (p > ONE)// 一直计算到指数小于或等于1{if ( (p&ONE) == ONE )// 判断p是否奇数{odd = odd*n; // 若odd为奇数，则把“剩下的”乘起来}n = n*n; // 主体乘方p >>=1; // 指数除以2 }return n * odd; // 最后把主体和“剩下的”乘起来作为结果}////快速计算(n^p)%m的值，用到定理(X*Y)%Z=((X%Z)*(Y%Z))%Z////算法实现原理同Power一样//unsigned Montgomery(unsigned n,unsigned p,unsigned m)//{//unsigned k=1;//n%=m;//while(p!=1)//{//if( (p&1) == 1 )//k=(k*n)%m;//n=(n*n)%m;//p>>=1;// 指数除以2//}//return(n*k)%m;//}friend bigInt operator+(bigInt& left, bigInt& right);friend bigInt operator-(bigInt& left, bigInt& right);friend bigInt operator*(bigInt& left, bigInt& right);friend bigInt operator/(bigInt& left, bigInt& right);friend bool operator>(bigInt& left, bigInt& right);friend bool operator<(bigInt& left, bigInt& right);friend bool operator==(bigInt& left, bigInt& right);friend bool operator>=(bigInt& left, bigInt& right);friend bool operator<=(bigInt& left, bigInt& right);friend bool operator!=(bigInt& left, bigInt& right);friend bigInt operator+(bigInt& left, int& right);friend bigInt operator-(bigInt& left, int& right);friend bigInt operator*(bigInt& left, int& right);friend bigInt operator/(bigInt& left, int& right);friend bool operator>(bigInt& left, int& right);friend bool operator<(bigInt& left, int& right);friend bool operator==(bigInt& left, int& right);friend bool operator>=(bigInt& left, int& right);friend bool operator<=(bigInt& left, int& right);friend bool operator!=(bigInt& left, int& right);//friend bigInt operator%(bigInt& left, bigInt& right);//friend bigInt operator<<(bigInt& left, bigInt& right);//friend bigInt operator>>(bigInt& left, bigInt& right);//friend bigInt operator+=(bigInt& left, bigInt& right);//friend bigInt operator-=(bigInt& left, bigInt& right);//friend bigInt operator*=(bigInt& left, bigInt& right);//friend bigInt operator/=(bigInt& left, bigInt& right);//friend bigInt operator-(bigInt& right);private: char sign; //+:0, -:1std::vector<unsigned> dats;//低位到高位存放private://清除多余位void bigIntReserve(){VecReserve(this->dats);if( this->dats.empty() ){this->dats.push_back(0);this->sign = 0;}}//清除高位上的0template<typename _T>void VecReserve(std::vector<_T>& arr){while(!arr.empty()){if( arr.back() == 0 )arr.pop_back();elsebreak;}}//获取二进制总位数unsigned getBitsNum(){unsigned length = unsigned(this->dats.size()-1)*32;long long carry = 1;while( carry <= this->dats.back() ){carry<<=1;++length;}return length;}//正数比较大小int compPlus( bigInt&left, bigInt& right ){size_t lengthL = left.dats.size();size_t lengthR = right.dats.size();if( lengthL > lengthR )return 1;elseif( lengthL < lengthR )return -1;else{for( int i = int(lengthL-1); i >=0; --i ){if( left.dats[i] > right.dats[i] )return 1;if( left.dats[i] < right.dats[i] )return -1;}}return 0;}//比较大小:1:>, 0:==, -1:<int compVal( bigInt& right ){if( sign == 1 && right.sign == 0 )return -1;if( sign == 0 && right.sign == 1 )return 1;int comp = compPlus(*this,right);if( sign == 1 )comp = -comp;return comp;}//读取字符型输入数据bool StrToBigInt(const std::string src){this->sign = 0;//默认为正this->dats = std::vector<unsigned>(src.size()/9 +1,0);//分配空间,只多不少//将字符串转为数字std::vector<int> NumTemp;for( int i = int(src.size() - 1); i > 0; --i ){if( '0'<=src[0] && '9'>=src[0] )NumTemp.push_back(src[i]-'0');elsereturn false;}if( '0'<src[0] && '9'>=src[0] ){NumTemp.push_back(src[0]-'0');}elseif( '-' == src[0])this->sign = 1;elseif( '+'!= src[0] )return false;long long tempCarry = 1;//标记位置size_t Pos = 0;//对应数组元素位置while( !NumTemp.empty() ){if( tempCarry >= JUDGECARRY ){++Pos;tempCarry>>=32;}if( 1 == (1&NumTemp[0]) )//为奇数{this->dats[Pos] ^= tempCarry;}//将NumTemp表示的数除2for( int i = int(NumTemp.size()-1); i >=0; --i ){if( 1==(1&NumTemp[i]) && 0<i )NumTemp[i-1] += 10;NumTemp[i]>>=1;}tempCarry<<=1;VecReserve<int>(NumTemp);//清理为0元素}this->bigIntReserve();return true;}//取反void Negate( size_t length = 0 ){size_t i=0;for( ; i < dats.size(); ++i)dats[i] = ~dats[i];for( ; i < length; ++i )dats.push_back(~unsigned(0));}//加,取this的符号为结果符号bigInt Add(bigInt &right){std::vector<unsigned> *minPos = &dats;//指向位数较小std::vector<unsigned> *maxPos = &right.dats;//指向位数较大if( dats.size() > right.dats.size() ){maxPos = &dats;minPos = &right.dats;}bigInt result;long long carry = 0;for(int i=0; i< int(minPos->size()); ++i) { carry = (long long)minPos->at(i) + (long long)maxPos->at(i)+ carry;result.dats.push_back( unsigned( carry & (JUDGECARRY-1) ) );carry>>=32;}//将进位carry再与本数相加,可能会多次进位.for(int i = int(minPos->size()); i < int(maxPos->size()); ++i){carry += (long long)maxPos->at(i);result.dats.push_back(unsigned( carry&(JUDGECARRY-1) ));carry>>=32;}if( carry != 0 )result.dats.push_back(unsigned(carry));result.sign = this->sign;return result;}//减操作,其中:this和right均为正数bigInt Sub(bigInt &right){size_t maxLength = dats.size()>right.dats.size()?dats.size():right.dats.size();bigInt result(*this);bigInt subtrahend(right); //减数subtrahend.Negate(maxLength);result = result.Add(subtrahend);if(result.dats.size() > maxLength)//数组长度增加则为正,得到结果需再加1.{result.dats.pop_back();result = result.Add(bigInt(1));//再加1.}else//为负则再取反码{result.Negate();result.sign = 1;}result.bigIntReserve();return result;}//乘操作bigInt Mul(bigInt &right){bigInt result;result.dats = std::vector<unsigned>(dats.size()+right.dats.size()+1,0);size_t posl, posr;unsigned long long carry;for(posl=0; posl<dats.size(); ++posl){bigInt temp;temp.dats = std::vector<unsigned>(dats.size()+right.dats.size()+1,0);for(carry=posr=0; posr<right.dats.size(); ++posr){carry = (long long)dats[posl] * (long long)right.dats[posr] + carry;temp.dats[posr+posl] = (unsigned)carry&(JUDGECARRY-1);//posr+posl确定位置carry>>=32;}temp.dats[posr+posl]=(unsigned)carry;result = result.Add(temp);}VecReserve<unsigned>(result.dats);if( this->sign == right.sign )result.sign = 0;elseresult.sign = 1;return result;}//除操作,this为被除数,right为除数,qoutient为商,remainder为余数bool Div(bigInt &right,bigInt"ient, bigInt& remainder){if( right == bigInt(0) )return false;quotient.reset();remainder.reset();int test = compPlus(*this,right);if( test == -1 ){quotient = 0;remainder = *this;return true;}if( test == 0 ){quotient = 1;remainder = 0;return true;}bigInt dividend = *this;int dis = int(this->getBitsNum() - right.getBitsNum());while(dis>=0){bigInt temp = right<<unsigned(dis);int comp = compPlus(dividend,temp);if( comp >= 0 ){dividend = dividend.Sub(temp);quotient = quotient.Add(bigInt(1)<<unsigned(dis));}--dis;}remainder = dividend;VecReserve<unsigned>(remainder.dats);if( this->sign == right.sign )quotient.sign = 0;elsequotient.sign = 1;return true;}std::string strNumAdd( std::string left, std::string right ){size_t maxLength = (left.length() > right.length() ? left.length():right.length())+1;std::string result = left;result.resize(maxLength,'0');int carry = 0;size_t pos = 0;for( ; pos < right.length(); ++pos ){carry = result[pos]-'0' + right[pos] - '0' + carry;result[pos] = carry%10 + '0';carry /= 10;}for( ; carry > 0; ++pos ){carry = result[pos] - '0' + carry;result[pos] = carry%10 + '0';carry /= 10;}if( result[result.length()-1] == '0' )result.resize(result.length()-1);return result;}std::string strNumMul( std::string left, std::string right ){std::string result, tmp;size_t i,j;size_t carry;for( i = 0; i < right.length(); ++i ){tmp.resize( (left.length() + 1 + i),'0' );for( carry = j = 0; j< left.length(); j++ ){carry = (left[j] - '0') * (right[i] - '0' ) + carry;tmp[j+i] = char(carry%10 + '0');//j+i确定位置carry /= 10;}tmp[j+i] = char(carry + '0');result = strNumAdd(result,tmp);tmp.clear();}while( result[result.length()-1] == '0' )result.resize(result.length()-1);return result;}};//友元函数//加法bigInt operator+(bigInt& left, bigInt& right){if( left.sign == right.sign )return left.Add(right);if( left.sign == 0 && right.sign == 1 )return left.Sub(right);elsereturn right.Sub(left);}//减法bigInt operator-(bigInt& left, bigInt& right){if( left.sign == right.sign ){if( left.sign == 0 )return left.Sub(right);elsereturn right.Sub(left);}elsereturn left.Add(right);}bigInt operator*(bigInt& left, bigInt& right){bigInt zero(0);if( left == zero || right == zero)return zero;return left.Mul(right);}bigInt operator/(bigInt& left, bigInt& right){bigInt quotient;if( !left.Div(right,quotient,bigInt()) )throw "error: Divisor is Zero!";return quotient;}bool operator<(bigInt& left, bigInt& right){if( left.compVal(right) == -1 )return true;return false;}bool operator<=(bigInt& left, bigInt& right) {if(left.compVal(right) <= 0 )return true;return false;}bool operator>(bigInt& left, bigInt& right){if( left.compVal(right) == 1 )return true;return false;}bool operator>=(bigInt& left, bigInt& right){if( left.compVal(right) >= 0 )return true;return false;}bool operator==(bigInt& left, bigInt& right){if( left.compVal(right) == 0 )return true;return false;}bool operator!=(bigInt& right, bigInt& left) {if( left.compVal(left) == 0 )return false;return true;}bigInt operator+(bigInt& left, int& right){return left + bigInt(right);}bigInt operator-(bigInt& left, int& right){return left - bigInt(right);}bigInt operator*(bigInt& left, int& right){return left * bigInt(right);}bigInt operator/(bigInt& left, int& right){return left / bigInt(right);}bool operator>(bigInt& left, int& right){return left > bigInt(right);}bool operator<(bigInt& left, int& right){return left < bigInt(right);}bool operator==(bigInt& left, int& right){return left == bigInt(right);}bool operator>=(bigInt& left, int& right){return left >= bigInt(right);}bool operator<=(bigInt& left, int& right){return left <= bigInt(right);}bool operator!=(bigInt& left, int& right){return left != bigInt(right);}