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分享高精度 | “朝闻道”知识分享大赛
这是我参加朝闻道知识分享大赛的第九篇文章。
引言：
在进行计算的过程中，常需要用到极大的整数，例如几百万位的大整数之间的加减乘除的运算。而C++等语言中，能够存储的数据范围是有限的。int: -231 到 231 - 1，long long：-263 到 263 - 1
当输入数据在此范围内，通过定义变量即可完成。 而当输入数据大于此范围，要怎么做呢？
高精度算法应运而生。涉及加，减，乘，除，乘方，取模，开方等运算。对于一般情况下的高精度算法，可以看做对数字计算过程的模拟：竖式计算。
高精度加法
- 以字符串形式读入数字，并逐位转换成数字倒序存储；
- 进行高精度加法计算；
- 倒序输出。
#include<iostream>
#include<vector>
using namespace std;
const int N = 1e6 + 10;
vector<int> add(vector<int>&A,vector<int>&B)
{
vector<int> C;
int t = 0;
for (int i = 0; i < A.size()|| i < B.size(); i++)
{
if(i < A.size()) t += A[i];
if(i < B.size()) t += B[i];
C.push_back(t%10);
t /= 10;
}
if(t) C.push_back(1);
return C;
}
int main()
{
string a,b;
vector<int>A,B;
cin >> a >> b;
for(int i = a.size()-1;i >= 0;i--) A.push_back(a[i]-'0');
for(int i = b.size()-1;i >= 0;i--) B.push_back(b[i]-'0');
auto C = add(A,B);
for (int i = C.size() - 1; i >= 0; i--) printf("%d",C[i]);
return 0;
}
高精度减法
- 判断正负
- 基本和加法一模一样，只不过从进位变成退位
- 退位后更新结果长度
#include<bits/stdc++.h>
using namespace std;
bool cmp(vector<int>&A,vector<int>&B)
{
if(A.size()!= B.size()) return A.size() > B.size();
for(int i = A.size() - 1;i >= 0;i--)
{
if(A[i] != B[i]) return A[i] > B[i];
}
return true;
}
vector<int> sub(vector<int> &A,vector<int> &B)
{
vector<int> C;
for(int i = 0,t = 0;i < A.size();i++)
{
t = A[i] - t;
if(i < B.size()) t -=B[i];
C.push_back((t+10)%10);
if(t < 0) t = 1;
else t = 0;
}
while(C.size() > 1 && C.back() == 0) C.pop_back();
return C;
}
int main()
{
string a,b;
vector<int> A,B;
cin >> a >> b;
for(int i = a.size()-1;i >= 0;i--) A.push_back(a[i]-'0');
for(int i = b.size()-1;i >= 0;i--) B.push_back(b[i]-'0');
if(cmp(A,B))
{
auto C = sub(A,B);
for(int i = C.size()-1;i >=0;i--) printf("%d",C[i]);
}
else
{
auto C = sub(B,A);
printf("-");
for(int i = C.size()-1;i >=0;i--) printf("%d",C[i]);
}
return 0;
}
高精度乘法
- 进位的运算
- 最高位进位的处理，可能不止进一位
#include<bits/stdc++.h>
using namespace std;
vector<int> mul(vector<int>&A,int b)
{
vector<int>C;
int t = 0;
for(int i = 0; i < A.size() || t;i++)
{
if(i < A.size()) t += A[i] * b;
C.push_back(t % 10);
t /= 10;
}
while(C.size()>1 && C.back()==0) C.pop_back();
return C;
}
int main()
{
string a;
int b;
cin >> a >> b;
vector<int>A;
for(int i = a.size() - 1;i >= 0;i--) A.push_back(a[i]-'0');
auto C = mul(A,b);
for(int i = C.size()-1;i >= 0;i--) printf("%d",C[i]);
return 0;
}
高精度除法
#include<bits/stdc++.h>
using namespace std;
vector<int> div(vector<int>&A,int b,int &r)
{
vector<int>C;
r = 0;
for(int i = A.size() - 1;i >= 0; i--)
{
r = r*10+A[i];
C.push_back(r/b);
r %= b;
}
reverse(C.begin(),C.end());
while(C.size() > 1 && C.back()==0) C.pop_back();
return C;
}
int main()
{
string a;
int b;
cin >> a >> b;
vector<int>A;
for(int i = a.size() -1;i>=0;i--) A.push_back(a[i]-'0');
int r;
auto C = div(A,b,r);
for(int i = C.size() - 1;i >= 0;i--) printf("%d",C[i]);
cout << endl << r << endl;
return 0;
}
高精度封装模板
最后，祭出比较全的高精度模板，见到高精度的题目，直接套这个封装好的模板，嘎嘎快，加减乘除取模读入等等等，都可以随意使用，输出也是无敌的！
#include <bits/stdc++.h>
using namespace std;
const int base = 1000000000;
const int base_digits = 9;
struct bigint {
std::vector<int> z;
int sign;
bigint() :
sign(1) {
}
bigint(long long v) {
*this = v;
}
bigint(const std::string &s) {
read(s);
}
void operator=(const bigint &v) {
sign = v.sign;
z = v.z;
}
void operator=(long long v) {
sign = 1;
if (v < 0)
sign = -1, v = -v;
z.clear();
for (; v > 0; v = v / base)
z.push_back(v % base);
}
bigint operator+(const bigint &v) const {
if (sign == v.sign) {
bigint res = v;
for (int i = 0, carry = 0; i < (int) std::max(z.size(), v.z.size()) || carry; ++i) {
if (i == (int) res.z.size())
res.z.push_back(0);
res.z[i] += carry + (i < (int) z.size() ? z[i] : 0);
carry = res.z[i] >= base;
if (carry)
res.z[i] -= base;
}
return res;
}
return *this - (-v);
}
bigint operator-(const bigint &v) const {
if (sign == v.sign) {
if (abs() >= v.abs()) {
bigint res = *this;
for (int i = 0, carry = 0; i < (int) v.z.size() || carry; ++i) {
res.z[i] -= carry + (i < (int) v.z.size() ? v.z[i] : 0);
carry = res.z[i] < 0;
if (carry)
res.z[i] += base;
}
res.trim();
return res;
}
return -(v - *this);
}
return *this + (-v);
}
void operator*=(int v) {
if (v < 0)
sign = -sign, v = -v;
for (int i = 0, carry = 0; i < (int) z.size() || carry; ++i) {
if (i == (int) z.size())
z.push_back(0);
long long cur = z[i] * (long long) v + carry;
carry = (int) (cur / base);
z[i] = (int) (cur % base);
//asm("divl %%ecx" : "=a"(carry), "=d"(a[i]) : "A"(cur), "c"(base));
}
trim();
}
bigint operator*(int v) const {
bigint res = *this;
res *= v;
return res;
}
friend std::pair<bigint, bigint> divmod(const bigint &a1, const bigint &b1) {
int norm = base / (b1.z.back() + 1);
bigint a = a1.abs() * norm;
bigint b = b1.abs() * norm;
bigint q, r;
q.z.resize(a.z.size());
for (int i = a.z.size() - 1; i >= 0; i--) {
r *= base;
r += a.z[i];
int s1 = b.z.size() < r.z.size() ? r.z[b.z.size()] : 0;
int s2 = b.z.size() - 1 < r.z.size() ? r.z[b.z.size() - 1] : 0;
int d = ((long long) s1 * base + s2) / b.z.back();
r -= b * d;
while (r < 0)
r += b, --d;
q.z[i] = d;
}
q.sign = a1.sign * b1.sign;
r.sign = a1.sign;
q.trim();
r.trim();
return std::make_pair(q, r / norm);
}
friend bigint sqrt(const bigint &a1) {
bigint a = a1;
while (a.z.empty() || a.z.size() % 2 == 1)
a.z.push_back(0);
int n = a.z.size();
int firstDigit = (int) sqrt((double) a.z[n - 1] * base + a.z[n - 2]);
int norm = base / (firstDigit + 1);
a *= norm;
a *= norm;
while (a.z.empty() || a.z.size() % 2 == 1)
a.z.push_back(0);
bigint r = (long long) a.z[n - 1] * base + a.z[n - 2];
firstDigit = (int) sqrt((double) a.z[n - 1] * base + a.z[n - 2]);
int q = firstDigit;
bigint res;
for(int j = n / 2 - 1; j >= 0; j--) {
for(; ; --q) {
bigint r1 = (r - (res * 2 * base + q) * q) * base * base + (j > 0 ? (long long) a.z[2 * j - 1] * base + a.z[2 * j - 2] : 0);
if (r1 >= 0) {
r = r1;
break;
}
}
res *= base;
res += q;
if (j > 0) {
int d1 = res.z.size() + 2 < r.z.size() ? r.z[res.z.size() + 2] : 0;
int d2 = res.z.size() + 1 < r.z.size() ? r.z[res.z.size() + 1] : 0;
int d3 = res.z.size() < r.z.size() ? r.z[res.z.size()] : 0;
q = ((long long) d1 * base * base + (long long) d2 * base + d3) / (firstDigit * 2);
}
}
res.trim();
return res / norm;
}
bigint operator/(const bigint &v) const {
return divmod(*this, v).first;
}
bigint operator%(const bigint &v) const {
return divmod(*this, v).second;
}
void operator/=(int v) {
if (v < 0)
sign = -sign, v = -v;
for (int i = (int) z.size() - 1, rem = 0; i >= 0; --i) {
long long cur = z[i] + rem * (long long) base;
z[i] = (int) (cur / v);
rem = (int) (cur % v);
}
trim();
}
bigint operator/(int v) const {
bigint res = *this;
res /= v;
return res;
}
int operator%(int v) const {
if (v < 0)
v = -v;
int m = 0;
for (int i = z.size() - 1; i >= 0; --i)
m = (z[i] + m * (long long) base) % v;
return m * sign;
}
void operator+=(const bigint &v) {
*this = *this + v;
}
void operator-=(const bigint &v) {
*this = *this - v;
}
void operator*=(const bigint &v) {
*this = *this * v;
}
void operator/=(const bigint &v) {
*this = *this / v;
}
bool operator<(const bigint &v) const {
if (sign != v.sign)
return sign < v.sign;
if (z.size() != v.z.size())
return z.size() * sign < v.z.size() * v.sign;
for (int i = z.size() - 1; i >= 0; i--)
if (z[i] != v.z[i])
return z[i] * sign < v.z[i] * sign;
return false;
}
bool operator>(const bigint &v) const {
return v < *this;
}
bool operator<=(const bigint &v) const {
return !(v < *this);
}
bool operator>=(const bigint &v) const {
return !(*this < v);
}
bool operator==(const bigint &v) const {
return !(*this < v) && !(v < *this);
}
bool operator!=(const bigint &v) const {
return *this < v || v < *this;
}
void trim() {
while (!z.empty() && z.back() == 0)
z.pop_back();
if (z.empty())
sign = 1;
}
bool isZero() const {
return z.empty() || (z.size() == 1 && !z[0]);
}
bigint operator-() const {
bigint res = *this;
res.sign = -sign;
return res;
}
bigint abs() const {
bigint res = *this;
res.sign *= res.sign;
return res;
}
long long longValue() const {
long long res = 0;
for (int i = z.size() - 1; i >= 0; i--)
res = res * base + z[i];
return res * sign;
}
friend bigint gcd(const bigint &a, const bigint &b) {
return b.isZero() ? a : gcd(b, a % b);
}
friend bigint lcm(const bigint &a, const bigint &b) {
return a / gcd(a, b) * b;
}
void read(const std::string &s) {
sign = 1;
z.clear();
int pos = 0;
while (pos < (int) s.size() && (s[pos] == '-' || s[pos] == '+')) {
if (s[pos] == '-')
sign = -sign;
++pos;
}
for (int i = s.size() - 1; i >= pos; i -= base_digits) {
int x = 0;
for (int j = std::max(pos, i - base_digits + 1); j <= i; j++)
x = x * 10 + s[j] - '0';
z.push_back(x);
}
trim();
}
friend std::istream& operator>>(std::istream &stream, bigint &v) {
std::string s;
stream >> s;
v.read(s);
return stream;
}
friend std::ostream& operator<<(std::ostream &stream, const bigint &v) {
if (v.sign == -1)
stream << '-';
stream << (v.z.empty() ? 0 : v.z.back());
for (int i = (int) v.z.size() - 2; i >= 0; --i)
stream << std::setw(base_digits) << std::setfill('0') << v.z[i];
return stream;
}
static std::vector<int> convert_base(const std::vector<int> &a, int old_digits, int new_digits) {
std::vector<long long> p(std::max(old_digits, new_digits) + 1);
p[0] = 1;
for (int i = 1; i < (int) p.size(); i++)
p[i] = p[i - 1] * 10;
std::vector<int> res;
long long cur = 0;
int cur_digits = 0;
for (int i = 0; i < (int) a.size(); i++) {
cur += a[i] * p[cur_digits];
cur_digits += old_digits;
while (cur_digits >= new_digits) {
res.push_back(int(cur % p[new_digits]));
cur /= p[new_digits];
cur_digits -= new_digits;
}
}
res.push_back((int) cur);
while (!res.empty() && res.back() == 0)
res.pop_back();
return res;
}
typedef std::vector<long long> vll;
static vll karatsubaMultiply(const vll &a, const vll &b) {
int n = a.size();
vll res(n + n);
if (n <= 32) {
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
res[i + j] += a[i] * b[j];
return res;
}
int k = n >> 1;
vll a1(a.begin(), a.begin() + k);
vll a2(a.begin() + k, a.end());
vll b1(b.begin(), b.begin() + k);
vll b2(b.begin() + k, b.end());
vll a1b1 = karatsubaMultiply(a1, b1);
vll a2b2 = karatsubaMultiply(a2, b2);
for (int i = 0; i < k; i++)
a2[i] += a1[i];
for (int i = 0; i < k; i++)
b2[i] += b1[i];
vll r = karatsubaMultiply(a2, b2);
for (int i = 0; i < (int) a1b1.size(); i++)
r[i] -= a1b1[i];
for (int i = 0; i < (int) a2b2.size(); i++)
r[i] -= a2b2[i];
for (int i = 0; i < (int) r.size(); i++)
res[i + k] += r[i];
for (int i = 0; i < (int) a1b1.size(); i++)
res[i] += a1b1[i];
for (int i = 0; i < (int) a2b2.size(); i++)
res[i + n] += a2b2[i];
return res;
}
bigint operator*(const bigint &v) const {
std::vector<int> a6 = convert_base(this->z, base_digits, 6);
std::vector<int> b6 = convert_base(v.z, base_digits, 6);
vll a(a6.begin(), a6.end());
vll b(b6.begin(), b6.end());
while (a.size() < b.size())
a.push_back(0);
while (b.size() < a.size())
b.push_back(0);
while (a.size() & (a.size() - 1))
a.push_back(0), b.push_back(0);
vll c = karatsubaMultiply(a, b);
bigint res;
res.sign = sign * v.sign;
for (int i = 0, carry = 0; i < (int) c.size(); i++) {
long long cur = c[i] + carry;
res.z.push_back((int) (cur % 1000000));
carry = (int) (cur / 1000000);
}
res.z = convert_base(res.z, 6, base_digits);
res.trim();
return res;
}
};
int main() {
bigint a,b;
cin >> a >> b;
cout << a + b << endl;
cout << a - b << endl;
cout << a * b << endl;
cout << a / b << endl;
cout << a % b << endl;
return 0;
}
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