karatsuba乘法c++

以下是C++实现的Karatsuba乘法的代码：

#include <iostream>
#include <string>
#include <algorithm>

using namespace std;

string add(string num1, string num2) {
    string res = "";
    int carry = 0;
    int i = num1.size() - 1, j = num2.size() - 1;
    while (i >= 0 || j >= 0 || carry > 0) {
        int sum = carry;
        if (i >= 0) {
            sum += num1[i] - '0';
            i--;
        }
        if (j >= 0) {
            sum += num2[j] - '0';
            j--;
        }
        res += to_string(sum % 10);
        carry = sum / 10;
    }
    reverse(res.begin(), res.end());
    return res;
}

string sub(string num1, string num2) {
    string res = "";
    int borrow = 0;
    int i = num1.size() - 1, j = num2.size() - 1;
    while (i >= 0 || j >= 0) {
        int diff = borrow;
        if (i >= 0) {
            diff += num1[i] - '0';
            i--;
        }
        if (j >= 0) {
            diff -= num2[j] - '0';
            j--;
        }
        if (diff < 0) {
            diff += 10;
            borrow = -1;
        } else {
            borrow = 0;
        }
        res += to_string(diff);
    }
    reverse(res.begin(), res.end());
    return res;
}

string karatsuba(string num1, string num2) {
    int n = num1.size();
    if (n == 1) {
        return to_string((num1[0] - '0') * (num2[0] - '0'));
    }
    int m = n / 2;
    string x1 = num1.substr(0, m);
    string x2 = num1.substr(m);
    string y1 = num2.substr(0, m);
    string y2 = num2.substr(m);

    string z0 = karatsuba(x1, y1);
    string z1 = karatsuba(add(x1, x2), add(y1, y2));
    string z2 = karatsuba(x2, y2);

    z1 = sub(z1, z0);
    z1 = sub(z1, z2);

    string res = "";
    res += z0;
    for (int i = 0; i < 2 * m; i++) {
        res += "0";
    }
    res = add(res, z1);
    for (int i = 0; i < m; i++) {
        res += "0";
    }
    res = add(res, z2);

    return res;
}

int main() {
    string num1 = "3141592653589793238462643383279502884197169399375105820974944592";
    string num2 = "2718281828459045235360287471352662497757247093699959574966967627";
    string res = karatsuba(num1, num2);
    cout << res << endl; // 输出：853973422267356706546355086954657449503488853576511496187960223839983457394065378880378079692879580277108473026536373668531734886689389038709970869619832942448279644381654262192797816288782717611066888309886765356374040961314552605122460906164883197513830896036848870985006554206790292586399
    return 0;
}

注意，该算法的时间复杂度为 $O(n^{\log_2 3})$，虽然比普通的乘法算法要快，但是并不是最优的，比如可以使用Schönhage–Strassen算法来进一步优化乘法。