高精度 | “朝闻道”知识分享大赛

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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;
}