高精度整数运算改进版

/* Name: 高精度整数运算改进版 Copyright:始发于goal00001111的专栏；允许自由转载，但必须注明作者和出处 Author: goal00001111 Date: 15-12-08 08:18 Description: 高精度整数运算：加减乘除，乘方，阶乘 。上次写了一个用字符串存储高精度整数的四则运算算法，虽然可以实现功能，但时间复杂度和空间复杂度都不够理想，这次出了个改进版，将原来的用字符串存储改成用整型数组存储，而且改进了乘法，除法和乘方的算法，更快更高效！ */#include<iostream>#include<string>using namespace std;const unsigned int MAX = 10000; //整型数组的最大长度 const long long WIDTHMAX = 1000000000; //整型数组val[MAX]的元素上限 const unsigned int WIDTH = 9; //输出整型数组val[MAX]的元素时的格式宽度，即整型数组val[MAX]的元素的最多位数 typedef struct node{ long long val[MAX]; //用来存储高精度整数 unsigned int size; //整型数组的实际长度 } BigInt;BigInt StrToBigInt(string s);void PrintBigInt(const BigInt & a);int ComPareBigInt(const BigInt & a, const BigInt & b);BigInt MulBigInt(const BigInt & a, const BigInt & b);BigInt AddBigInt(const BigInt & a, const BigInt & b);BigInt SubBigInt(BigInt a, BigInt b);BigInt DivBigInt(const BigInt & a, const BigInt & b);BigInt FacBigInt(unsigned int n);void PowBigInt(BigInt & c, const BigInt & a, unsigned int n);void PowBigInt_2(BigInt & c, const BigInt & a, unsigned int n);BigInt HalfBigInt(BigInt a);int main(){ string s; BigInt a, b, c; cin >> s; a = StrToBigInt(s); cin >> s; b = StrToBigInt(s); cout << " "; PrintBigInt(a); cout << "+ "; PrintBigInt(b); c = AddBigInt(a, b); cout << "= "; PrintBigInt(c); cout << endl; cout << " "; PrintBigInt(a); cout << "- "; PrintBigInt(b); c = SubBigInt(a, b); cout << "= "; PrintBigInt(c); cout << endl; cout << " "; PrintBigInt(a); cout << "* "; PrintBigInt(b); c = MulBigInt(a, b); cout << "= "; PrintBigInt(c); cout << endl; cout << " "; PrintBigInt(a); cout << "/ 2 " << endl; c = HalfBigInt(a); cout << "= "; PrintBigInt(c); cout << endl; cout << " "; PrintBigInt(a); cout << "/ "; PrintBigInt(b); c = DivBigInt(a, b); cout << "= "; PrintBigInt(c); cout << endl; unsigned int n; cin >> n; //cout << n << "! = ";// c = FacBigInt(n);// PrintBigInt(c);// cout << c.size << endl; cout << endl; cout << " "; PrintBigInt(a); cout << "^" << n << " = "; PowBigInt(c, a, n); PrintBigInt(c); cout << endl; cout << " "; PrintBigInt(a); cout << "^" << n << " = "; PowBigInt_2(c, a, n); PrintBigInt(c); cout << endl; system("pause"); return 0;}/*函数名称：PrintBigInt函数功能：输出用整型数组表示的高精度整数输入参数：const BigInt & a：用整型数组表示的高精度整数输出参数：无*/void PrintBigInt(const BigInt & a){ cout << a.val[a.size-1]; for (int i=a.size-2; i>=0; i--) { unsigned w = WIDTHMAX / 10; while (w > 0) { if (a.val[i] >= w) break; cout << 0; w /= 10; } cout << a.val[i]; } cout << endl; }/*函数名称：StrToBigInt函数功能：把元素为数字的字符串转换为用整型数组表示的高精度整数输入参数：string s：存储数字的字符串 输出参数：BigInt：返回用整型数组表示的高精度整数*/BigInt StrToBigInt(string s){ BigInt a; a.size = 0; int i = s.size(); unsigned long long sum = 0; while ( i>=WIDTH) { for (int j=i-WIDTH; j<i; j++) sum = sum * 10 + (s[j] - '0'); a.val[a.size++] = sum; sum = 0; i -= WIDTH; } if (i > 0) { for (int j=0; j<i; j++) sum = sum * 10 + (s[j] - '0'); a.val[a.size++] = sum; } return a;}/*函数名称：AddBigInt函数功能：高精度整数加法输入参数：const BigInt & a：用整型数组表示的高精度整数加数 const BigInt & b：用整型数组表示的高精度整数加数 输出参数：BigInt：返回用整型数组表示的高精度整数和 */BigInt AddBigInt(const BigInt & a, const BigInt & b){ //逆序计算a+b，则从低位开始计算 BigInt c; unsigned long long carry = 0; unsigned int i = 0; c.size = 0; while (i < a.size && i < b.size) { c.val[c.size++] = (a.val[i] + b.val[i] + carry) % WIDTHMAX; carry = (a.val[i] + b.val[i] + carry) / WIDTHMAX; i++; } while (i < a.size) { c.val[c.size++] = (a.val[i] + carry) % WIDTHMAX; carry = (a.val[i] + carry) / WIDTHMAX; i++; } while (i < b.size) { c.val[c.size++] = (b.val[i] + carry) % WIDTHMAX; carry = (b.val[i] + carry) / WIDTHMAX; i++; } if (carry > 0) c.val[c.size++] = carry; return c;}/*函数名称：SubBigInt函数功能：高精度整数减法输入参数：BigInt a：用整型数组表示的高精度整数被减数 BigInt b：用整型数组表示的高精度整数减数 输出参数：BigInt：返回用整型数组表示的高精度整数差 */BigInt SubBigInt(BigInt a, BigInt b){ BigInt c; c.size = 0; if (ComPareBigInt(a, b) == 0) { c.size = 1; c.val[0] = 0; return c; } bool flag = false; if (ComPareBigInt(a, b) < 0)//交换，并得到一个负号 { flag = true; BigInt temp = a; a = b; b = temp; } unsigned int i = 0; while (i < b.size) { if (a.val[i] >= b.val[i]) c.val[c.size++] = a.val[i] - b.val[i]; else { a.val[i+1] -= 1; c.val[c.size++] = a.val[i] + WIDTHMAX - b.val[i]; } i++; } while (i < a.size) { if (a.val[i] < 0) { a.val[i+1] -= 1; a.val[i] += WIDTHMAX; } c.val[c.size++] = a.val[i]; i++; } //消除多余的高位0 while (c.val[c.size-1] == 0) c.size--; if (flag)//如果是负数，加上负号 c.val[c.size-1] = -c.val[c.size-1]; return c;}/*函数名称：ComPareBigInt函数功能：比较两个高精度整数的大小 输入参数：BigInt a：用整型数组表示的高精度整数被减数 BigInt b：用整型数组表示的高精度整数减数 输出参数：int：a > b返回1，a = b返回0，a < b返回-1*/int ComPareBigInt(const BigInt & a, const BigInt & b){ if (a.size > b.size) return 1; if (a.size < b.size) return -1; for (int i=a.size-1; i>=0; i--) { if (a.val[i] > b.val[i]) return 1; if (a.val[i] < b.val[i]) return -1; } return 0;}/*函数名称：MulBigInt函数功能：高精度整数乘法输入参数：const BigInt & a：用整型数组表示的高精度整数被乘数 const BigInt & b：用整型数组表示的高精度整数乘数 输出参数：BigInt：返回用整型数组表示的高精度整数乘积 */BigInt MulBigInt(const BigInt & a, const BigInt & b){ if (a.size == 1 && a.val[0] == 0) return a; if (b.size == 1 && b.val[0] == 0) return b; BigInt c; for (int i=0; i<MAX; i++) //全部赋初值为0 c.val[i] = 0; for (int i=0, j=0; i<b.size; i++) { for (j=0; j<a.size; j++) { c.val[i+j] += a.val[j] * b.val[i]; c.val[i+j+1] += c.val[i+j] / WIDTHMAX; c.val[i+j] %= WIDTHMAX; } c.size = i + j; if (c.val[c.size] != 0)//最高位有进位 c.size++; } return c;}/*老版本： BigInt MulBigInt2(const BigInt & a, const BigInt & b){ if (a.size == 1 && a.val[0] == 0) return a; if (b.size == 1 && b.val[0] == 0) return b; BigInt c, tc; unsigned long long carry = 0; c.size = 0; for (int i=0, j=0; i<b.size; i++) { for (j=0; j<i; j++)//先在临时和tc的低位补足0 tc.val[j] = 0; carry = 0; for (j=0; j<a.size; j++) { tc.val[i+j] = (a.val[j] * b.val[i] + carry) % WIDTHMAX; carry = (a.val[j] * b.val[i] + carry) / WIDTHMAX; } tc.size = i + j; if (carry > 0) tc.val[tc.size++] = carry; //累加到c中 c = AddBigInt(tc, c); } return c;}*//*函数名称：FacBigInt函数功能：高精度整数阶乘输入参数：unsigned int n：正整数 输出参数：BigInt：返回用整型数组表示的高精度整数阶乘 */BigInt FacBigInt(unsigned int n){ BigInt s, c; c.size = s.size = 1; s.val[0] = 1; for (unsigned long long i=2; i<=n; i++) { c.val[0] = i; s = MulBigInt(s, c); } return s;}/*函数名称：PowBigInt函数功能：递归高效算法求高精度整数幂输入参数：BigInt & c：存储高精度整数幂的整型数组 const BigInt & a：用整型数组表示的高精度整数底数 long long n： 指数*/void PowBigInt(BigInt & c, const BigInt & a, unsigned int n){ if (n == 1) { c = a; return ; } if (n == 0 || (a.size == 1 && a.val[0] == 1)) { c.size = c.val[0] = 1; return ; } PowBigInt(c, a, n/2); //递归求高精度整数幂 c = MulBigInt(c, c); //a^n = a^(n/2)*a^(n/2)*f(a) if (n % 2 == 1) //其中f(a) = 1(n%2==0)或f(a) = a(n%2==1) c = MulBigInt(a, c);}/*函数名称：PowBigInt_2函数功能：非递归高效算法求高精度整数幂输入参数：BigInt & c：存储高精度整数幂的整型数组 const BigInt & a：用整型数组表示的高精度整数底数 long long n： 指数*/void PowBigInt_2(BigInt & c, const BigInt & a, unsigned int n){ int stack[MAX] = {0}; int top = 0; while (n > 0) //利用一个栈来存储n的状态：奇数还是偶数 { stack[top++] = n % 2; n /= 2; } c.size = c.val[0] = 1; for (int i=top-1; i>=0; i--) { c = MulBigInt(c, c); //a^n = a^(n/2)*a^(n/2)*f(a) if (stack[i] == 1) //其中f(a) = 1(n%2==0)或f(a) = a(n%2==1) c = MulBigInt(a, c); }}/*函数名称：DivBigInt函数功能：二分法实现高精度整数除法输入参数：const BigInt & a：用整型数组表示的高精度整数被除数 const BigInt & b：用整型数组表示的高精度整数除数 输出参数：BigInt：返回用整型数组表示的高精度整数商 */BigInt DivBigInt(const BigInt & a, const BigInt & b){ BigInt high, low, mid, one, c; if ((a.size == 1 && a.val[0] == 0) || (b.size == 1 && b.val[0] == 0)) { c.size = 1; c.val[0] = 0; return c; } one.size = 1; //值为1的高精度整数 one.val[0] = 1; high = a; //上界 low.size = 1; //下界 low.val[0] = 0; while (ComPareBigInt(low, high) < 0) { mid = HalfBigInt(AddBigInt(high, low)); //中间数 c = MulBigInt(mid, b); if (ComPareBigInt(c, a) == 0) return mid; else if (ComPareBigInt(c, a) < 0) low = AddBigInt(mid, one); else high = SubBigInt(mid, one); } c = MulBigInt(low, b); if (ComPareBigInt(c, a) <= 0) return low; else return SubBigInt(low, one);}/*函数名称：HalfBigInt函数功能：高精度整数求半 输入参数：BigInt a：用整型数组表示的高精度整数输出参数：BigInt：返回用整型数组表示的高精度整数的一半 */BigInt HalfBigInt(BigInt a){ BigInt c; c.size = a.size; for (int i=a.size-1; i>0; i--) { c.val[i] = a.val[i] / 2; if (a.val[i] % 2 == 1) a.val[i-1] += WIDTHMAX; } c.val[0] = a.val[0] / 2; if (c.size > 0 && c.val[c.size-1] == 0) c.size--; return c;}