Subset Transforms (Dynamic Programming)

:::note references

1. https://codeforces.com/blog/entry/45223
2. https://codeforces.com/blog/entry/72488 :::

Zeta Transform & its inverse

namespace Algorithm {
    template<typename T>
    void zetaTransform(std::span<T> f) {
        int n = std::countr_zero(f.size());
        for (int i = 0; i < n; ++i)
            for (int mask = 0; mask < (1 << n); ++mask)
                if (mask & (1 << i))
                    f[mask] += f[mask ^ (1 << i)];
    }

    /* this is inverse of zetaTransform */
    template<typename T>
    void mobiusTransform(std::span<T> f) {
        int n = std::countr_zero(f.size());
        for (int i = 0; i < n; ++i)
            for (int mask = 0; mask < (1 << n); ++mask)
                if (mask & (1 << i))
                    f[mask] -= f[mask ^ (1 << i)];
    }
}

int main() {
    std::vector<int> a = {3,7,5,6,1,2,2,1};
    Algorithm::zetaTransform<int>(a);
    std::ranges::copy(a, std::ostream_iterator<int>(std::cout, " "));
}

Subset Sum Convolution

$$f \circ g(s) = \sum_{\substack{a, b \subseteq s \\ a \cup b = s \\ |a| + |b| = |s|}} f(a)g(b)$$

Time Complexity: $O(n^2 \times 2^n)$ where $|s| = 2^n$

namespace Algorithm {
    /* unverified */
    template<typename T>
    std::vector<T> subsetSumConvolution(std::span<T> f, std::span<T> g) {
        uint64_t two_power_n = f.size();
        int n = std::countr_zero(two_power_n);

        std::array<std::vector<T>, 32> f_hat, g_hat, h_hat;
        std::ranges::fill(f_hat, std::vector<int>(two_power_n, 0));
        std::ranges::fill(g_hat, std::vector<int>(two_power_n, 0));
        std::ranges::fill(h_hat, std::vector<int>(two_power_n, 0));

        for (int mask = 0; mask < (1 << n); ++mask) {
            int set_bit_count = std::popcount((uint64_t) mask);
            f_hat[set_bit_count][mask] = f[mask];
            g_hat[set_bit_count][mask] = g[mask];
        }

        for (int k = 0; k <= n; ++k) zetaTransform<T>(f_hat[k]);
        for (int k = 0; k <= n; ++k) zetaTransform<T>(g_hat[k]);

        for (int i = 0; i <= n; ++i)
            for (int j = 0; j <= i; ++j)
                for (int mask = 0; mask < (1 << n); ++mask)
                    h_hat[i][mask] += f_hat[j][mask] * g_hat[i - j][mask];

        for (int k = 0; k <= n; ++k) mobiusTransform<T>(h_hat[k]);

        std::vector<T> fog(two_power_n);

        for (int mask = 0; mask < (1 << n); ++mask)
            fog[mask] = h_hat[mask][std::popcount((uint64_t) mask)];

        return fog;
    }
}