算法来源

#include <iostream>
#include <vector>
#include <string>
#include <algorithm>
using namespace std;

inline long long ksm(long long a, long long b)
{
    long long r = 1;
    while (b)
    {
        if (b & 1)
        {
            r *= a;
        }
        b >>= 1;
        a *= a;
    }
    return r;
}

const int CUTOFF = 8;//2 < CUTOFF < 9
const long long BASE = ksm(10, CUTOFF);

template<typename T>

inline void vec_reserve(vector<T> &v, size_t n)
{
    if (v.capacity() < n)
    {
        v.reserve(n);
    }
}

inline vector<long long> str_to_num(const string &s)
{
    vector<long long> res;
    int len = s.size(), first = len % CUTOFF;
    if (first != 0)
    {
        long long num = 0;
        for (register int i = 0; i < first; i++)
        {
            num = (num << 3) + (num << 1) + (s[i] - '0');
        }
        res.push_back(num);
    }
    for (register int i = first; i < len; i += CUTOFF)
    {
        long long num = 0;
        for (register int j = 0; j < CUTOFF; j++)
        {
            num = (num << 3) + (num << 1) + (s[i + j] - '0');
        }
        res.push_back(num);
    }
    reverse(res.begin(), res.end());
    return res.empty() ? vector<long long>{0} : res;
}

inline string num_to_str(const vector<long long> &num)
{
    if (num.empty())
    {
        return "0";
    }
    string res;
    res.reserve(num.size() * CUTOFF);
    bool leading = true;
    for (auto it = num.rbegin(); it != num.rend(); it++)
    {
        if (leading)
        {
            if (*it != 0)
            {
                res += to_string(*it);
                leading = false;
            }
        }
        else
        {
            char buf[10];
            snprintf(buf, sizeof(buf), ("%0" + to_string(CUTOFF) + "lld").c_str(), *it);
            res.append(buf, CUTOFF);
        }
    }
    return res.empty() ? "0" : res;
}

inline void add_inplace(vector<long long> &a, const vector<long long> &b)
{
    long long carry = 0;
    int max_size = max(a.size(), b.size());
    vec_reserve(a, max_size + 1);
    for (register int i = 0; i < max_size || carry; i++)
    {
        if (i == a.size()) a.push_back(0);
        long long bb = i < b.size() ? b[i] : 0;
        a[i] += bb + carry;
        carry = a[i] / BASE;
        a[i] %= BASE;
    }
}

inline vector<long long> add(const vector<long long> &a, const vector<long long> &b)
{
    vector<long long> res = a;
    add_inplace(res, b);
    return res;
}

inline void sub_inplace(vector<long long> &a, const vector<long long> &b)
{
    long long borrow = 0;
    for (register int i = 0; i < a.size(); i++)
    {
        long long bb = i < b.size() ? b[i] : 0;
        a[i] -= borrow + bb;
        borrow = a[i] < 0 ? 1 : 0;
        if (borrow)
        {
            a[i] += BASE;
        }
    }
    while (a.size() > 1 && a.back() == 0)
    {
        a.pop_back();
    }
}

inline vector<long long> multiply_simple(const vector<long long> &a, const vector<long long> &b)
{
    vector<long long> res(a.size() + b.size(), 0);
    for (register int i = 0; i < a.size(); i++)
    {
        long long carry = 0;
        for (register int j = 0; j < b.size(); j++)
        {
            res[i + j] += a[i] * b[j] + carry;
            carry = res[i + j] / BASE;
            res[i + j] %= BASE;
        }
        if (carry)
        {
            res[i + b.size()] += carry;
        }
    }
    while (res.size() > 1 && res.back() == 0)
    {
        res.pop_back();
    }
    return res;
}

inline vector<long long> karatsuba_opt(const vector<long long> &a, const vector<long long> &b, int l, int r)
{
    if (r - l <= CUTOFF)
    {
        return multiply_simple(vector<long long>(a.begin() + l, a.begin() + r), vector<long long>(b.begin() + l, b.begin() + r));
    }
    int mid = (r - l) >> 1, shift = mid << 1;
    vector<long long> lo = karatsuba_opt(a, b, l, l + mid), hi = karatsuba_opt(a, b, l + mid, r), a_sum(mid + 1), b_sum(mid + 1);
    for (register int i = 0; i < mid; ++i)
    {
        a_sum[i] = (l + mid + i < a.size() ? a[l + mid + i] : 0) + a[l + i];
        b_sum[i] = (l + mid + i < b.size() ? b[l + mid + i] : 0) + b[l + i];
    }
    vector<long long> mid_term = karatsuba_opt(a_sum, b_sum, 0, mid);
    mid_term.resize(mid_term.size() + shift, 0);
    rotate(mid_term.rbegin(), mid_term.rbegin() + shift, mid_term.rend());
    vector<long long> res(lo.size() + shift, 0);
    copy(lo.begin(), lo.end(), res.begin());
    add_inplace(res, hi);
    add_inplace(res, mid_term);
    sub_inplace(res, lo);
    sub_inplace(res, hi);
    return res;
}

inline vector<long long> karatsuba(const vector<long long> &a, const vector<long long> &b)
{
    int size = max(a.size(), b.size());
    vector<long long> a_pad = a, b_pad = b;
    a_pad.resize(size, 0);
    b_pad.resize(size, 0);
    return karatsuba_opt(a_pad, b_pad, 0, size);
}

int main()
{
    ios::sync_with_stdio(false), cin.tie(nullptr), cout.tie(nullptr);
    string a, b;
    cin >> a >> b;
    vector<long long> num_a = str_to_num(a), num_b = str_to_num(b), product = karatsuba(num_a, num_b);
    cout << num_to_str(product) << '\n';
    return 0;
}

时间复杂度 $O(\frac{3n^{\log(3)}}{w})$ （其中 $w$ 为压位位数 $8$ ），简单计算得 $O(1.2 \times 10^9)$ ，无可救药了吗？

使用 __int128 的话， $w$ 约为 $40$ 可能勉强 卡不过 ！！！