Black box linear algebra のための行列
(linear_algebra_matrix/blackbox_matrices.hpp)

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• Last update: 2023-12-26 21:26:22+09:00
• Include: #include "linear_algebra_matrix/blackbox_matrices.hpp"

各行列クラスに実装されるべき関数

• int height() : 行列 $\mathbf{M}$ の行数を返す．
• int width() : 行列 $\mathbf{M}$ の列数を返す．
• vector<T> prod(vector<T> v) : 行列 $\mathbf{M}$ に右から列ベクトル $\mathbf{v}$ をかけた結果の列ベクトル $\mathbf{M}\mathbf{v}$ を計算．
• vector<T> prod_left(vector<T> v) : 行列 $\mathbf{M}$ に左から行ベクトル $\mathbf{u}^\top$ をかけた結果の行ベクトル $\mathbf{u}^\top \mathbf{M}$ を計算．
• vector<vector<T>> vecvec() : 行列 $\mathbf{M}$ を二次元のベクトルとして出力．

実装されている具体的な特殊行列

• sparse_matrix<T> : 疎行列．$S$ 個の非零の値追加に対し，prod() / prod_left() の計算量は $O(W + S), O(H + S)$．
• toeplitz_ntt<T> 特に T が $\mathbb{F}_p$ （$p$ は NTT-friendly な素数）の場合の Toeplitz 行列．prod() / prod_left() の計算量は $O((H + W) \log (H + W))$．FFTを用いることで乗算を高速化．
• square_toeplitz_ntt<T> : $N$ 次正方 Toeplitz 行列（不要かもしれない）．prod() / prod_left() の計算量は $O(N \log N)$．

Depends on

• convolution/ntt.hpp
• modint.hpp

Verified with

• linear_algebra_matrix/test/blackbox_matrix_stress.test.cpp
• linear_algebra_matrix/test/det_of_blackbox_matrix.test.cpp

Code

#pragma once
#include "../convolution/ntt.hpp"
#include <algorithm>
#include <cassert>
#include <numeric>
#include <utility>
#include <vector>

// Sparse matrix
template <typename Tp> struct sparse_matrix {
    int H, W;
    std::vector<std::vector<std::pair<int, Tp>>> vals;
    sparse_matrix(int H = 0, int W = 0) : H(H), W(W), vals(H) {}
    int height() const { return H; }
    int width() const { return W; }
    void add_element(int i, int j, Tp val) {
        assert(i >= 0 and i < H);
        assert(j >= 0 and i < W);
        vals[i].emplace_back(j, val);
    }
    std::vector<Tp> prod(const std::vector<Tp> &vec) const {
        assert(W == int(vec.size()));
        std::vector<Tp> ret(H);
        for (int i = 0; i < H; i++) {
            for (const auto &p : vals[i]) ret[i] += p.second * vec[p.first];
        }
        return ret;
    }
    std::vector<Tp> prod_left(const std::vector<Tp> &vec) const {
        assert(H == int(vec.size()));
        std::vector<Tp> ret(W);
        for (int i = 0; i < H; i++) {
            for (const auto &p : vals[i]) ret[p.first] += p.second * vec[i];
        }
        return ret;
    }
    std::vector<std::vector<Tp>> vecvec() const {
        std::vector<std::vector<Tp>> ret(H, std::vector<Tp>(W));
        for (int i = 0; i < H; i++) {
            for (auto p : vals[i]) ret[i][p.first] += p.second;
        }
        return ret;
    }
};

// Toeplitz matrix
// M = [
// [ vw_0 vw_1 vw_2 ... ]
// [ vh_1 ...           ]
// [ vh_2 ...           ]
// [ ...                ] (vw_0 == vh_0)
// prod() / prod_left() : O((H + W) log (H + W))
template <typename NTTModInt> struct toeplitz_ntt {
    int _h, _w;
    int _len_pow2;
    std::vector<NTTModInt> ts, ntt_ts;
    toeplitz_ntt(const std::vector<NTTModInt> &vh, const std::vector<NTTModInt> &vw)
        : _h(vh.size()), _w(vw.size()) {
        assert(vh.size() and vw.size() and vh[0] == vw[0]);
        ts.resize(_h + _w - 1);
        for (int i = 0; i < _w; i++) ts[i] = vw[_w - 1 - i];
        for (int j = 0; j < _h; j++) ts[_w - 1 + j] = vh[j];
        _len_pow2 = 1;
        while (_len_pow2 < int(ts.size()) + std::max(_h, _w) - 1) _len_pow2 *= 2;
        ntt_ts = ts;
        ntt_ts.resize(_len_pow2);
        ntt(ntt_ts, false);
    }
    int height() const { return _h; }
    int width() const { return _w; }
    std::vector<NTTModInt> prod(std::vector<NTTModInt> b) const {
        assert(int(b.size()) == _w);
        b.resize(_len_pow2);
        ntt(b, false);
        for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i];
        ntt(b, true);
        b.erase(b.begin(), b.begin() + _w - 1);
        b.resize(_h);
        return b;
    }
    std::vector<NTTModInt> prod_left(std::vector<NTTModInt> b) const {
        assert(int(b.size()) == _h);
        std::reverse(b.begin(), b.end());
        b.resize(_len_pow2);
        ntt(b, false);
        for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i];
        ntt(b, true);
        b.erase(b.begin(), b.begin() + _h - 1);
        b.resize(_w);
        std::reverse(b.begin(), b.end());
        return b;
    }
    std::vector<std::vector<NTTModInt>> vecvec() const {
        std::vector<std::vector<NTTModInt>> ret(_h, std::vector<NTTModInt>(_w));
        for (int i = 0; i < _h; i++) {
            for (int j = 0; j < _w; j++) ret[i][j] = ts[i - j + _w - 1];
        }
        return ret;
    }
};

// Square Toeplitz matrix
// M = [
// [ t_(N-1) t_(N-2) ... t_1 t_0     ]
// [ t_N     t_(N-1) ... t_2 t_1     ]
// [ ...                             ]
// [ t_(N*2-2)       ...     t_(N-1) ]]
// prod() / prod_left() : O(N log N)
template <typename NTTModInt> struct square_toeplitz_ntt {
    int _dim;
    int _len_pow2;
    std::vector<NTTModInt> ts;
    std::vector<NTTModInt> ntt_ts;
    square_toeplitz_ntt(const std::vector<NTTModInt> &t) : _dim(t.size() / 2 + 1), ts(t) {
        assert(t.size() % 2);
        _len_pow2 = 1;
        while (_len_pow2 < int(ts.size()) + _dim - 1) _len_pow2 *= 2;
        ntt_ts = ts;
        ntt_ts.resize(_len_pow2);
        ntt(ntt_ts, false);
    }
    int height() const { return _dim; }
    int width() const { return _dim; }

    // Calculate A * b
    std::vector<NTTModInt> prod(std::vector<NTTModInt> b) const {
        assert(int(b.size()) == _dim);
        b.resize(_len_pow2);
        ntt(b, false);
        for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i];
        ntt(b, true);
        b.erase(b.begin(), b.begin() + _dim - 1);
        b.resize(_dim);
        return b;
    }
    // Calculate bT * A
    std::vector<NTTModInt> prod_left(std::vector<NTTModInt> b) const {
        std::reverse(b.begin(), b.end());
        b = prod(b);
        std::reverse(b.begin(), b.end());
        return b;
    }
    std::vector<std::vector<NTTModInt>> vecvec() const {
        std::vector<std::vector<NTTModInt>> ret(_dim, std::vector<NTTModInt>(_dim));
        for (int i = 0; i < _dim; i++) {
            for (int j = 0; j < _dim; j++) ret[i][j] = ts[i - j + _dim - 1];
        }
        return ret;
    }
};

#line 2 "modint.hpp"
#include <cassert>
#include <iostream>
#include <set>
#include <vector>

template <int md> struct ModInt {
    using lint = long long;
    constexpr static int mod() { return md; }
    static int get_primitive_root() {
        static int primitive_root = 0;
        if (!primitive_root) {
            primitive_root = [&]() {
                std::set<int> fac;
                int v = md - 1;
                for (lint i = 2; i * i <= v; i++)
                    while (v % i == 0) fac.insert(i), v /= i;
                if (v > 1) fac.insert(v);
                for (int g = 1; g < md; g++) {
                    bool ok = true;
                    for (auto i : fac)
                        if (ModInt(g).pow((md - 1) / i) == 1) {
                            ok = false;
                            break;
                        }
                    if (ok) return g;
                }
                return -1;
            }();
        }
        return primitive_root;
    }
    int val_;
    int val() const noexcept { return val_; }
    constexpr ModInt() : val_(0) {}
    constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; }
    constexpr ModInt(lint v) { _setval(v % md + md); }
    constexpr explicit operator bool() const { return val_ != 0; }
    constexpr ModInt operator+(const ModInt &x) const {
        return ModInt()._setval((lint)val_ + x.val_);
    }
    constexpr ModInt operator-(const ModInt &x) const {
        return ModInt()._setval((lint)val_ - x.val_ + md);
    }
    constexpr ModInt operator*(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.val_ % md);
    }
    constexpr ModInt operator/(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.inv().val() % md);
    }
    constexpr ModInt operator-() const { return ModInt()._setval(md - val_); }
    constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
    constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
    constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
    constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
    friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; }
    friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; }
    friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; }
    friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; }
    constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; }
    constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; }
    constexpr bool operator<(const ModInt &x) const {
        return val_ < x.val_;
    } // To use std::map<ModInt, T>
    friend std::istream &operator>>(std::istream &is, ModInt &x) {
        lint t;
        return is >> t, x = ModInt(t), is;
    }
    constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {
        return os << x.val_;
    }

    constexpr ModInt pow(lint n) const {
        ModInt ans = 1, tmp = *this;
        while (n) {
            if (n & 1) ans *= tmp;
            tmp *= tmp, n >>= 1;
        }
        return ans;
    }

    static constexpr int cache_limit = std::min(md, 1 << 21);
    static std::vector<ModInt> facs, facinvs, invs;

    constexpr static void _precalculation(int N) {
        const int l0 = facs.size();
        if (N > md) N = md;
        if (N <= l0) return;
        facs.resize(N), facinvs.resize(N), invs.resize(N);
        for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
        facinvs[N - 1] = facs.back().pow(md - 2);
        for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
        for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
    }

    constexpr ModInt inv() const {
        if (this->val_ < cache_limit) {
            if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};
            while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
            return invs[this->val_];
        } else {
            return this->pow(md - 2);
        }
    }
    constexpr ModInt fac() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facs[this->val_];
    }
    constexpr ModInt facinv() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facinvs[this->val_];
    }
    constexpr ModInt doublefac() const {
        lint k = (this->val_ + 1) / 2;
        return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
                                : ModInt(k).fac() * ModInt(2).pow(k);
    }

    constexpr ModInt nCr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv() * ModInt(r).facinv();
    }

    constexpr ModInt nPr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv();
    }

    static ModInt binom(int n, int r) {
        static long long bruteforce_times = 0;

        if (r < 0 or n < r) return ModInt(0);
        if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r);

        r = std::min(r, n - r);

        ModInt ret = ModInt(r).facinv();
        for (int i = 0; i < r; ++i) ret *= n - i;
        bruteforce_times += r;

        return ret;
    }

    // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)
    // Complexity: O(sum(ks))
    template <class Vec> static ModInt multinomial(const Vec &ks) {
        ModInt ret{1};
        int sum = 0;
        for (int k : ks) {
            assert(k >= 0);
            ret *= ModInt(k).facinv(), sum += k;
        }
        return ret * ModInt(sum).fac();
    }

    // Catalan number, C_n = binom(2n, n) / (n + 1)
    // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ...
    // https://oeis.org/A000108
    // Complexity: O(n)
    static ModInt catalan(int n) {
        if (n < 0) return ModInt(0);
        return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv();
    }

    ModInt sqrt() const {
        if (val_ == 0) return 0;
        if (md == 2) return val_;
        if (pow((md - 1) / 2) != 1) return 0;
        ModInt b = 1;
        while (b.pow((md - 1) / 2) == 1) b += 1;
        int e = 0, m = md - 1;
        while (m % 2 == 0) m >>= 1, e++;
        ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
        x *= (*this);
        ModInt z = b.pow(m);
        while (y != 1) {
            int j = 0;
            ModInt t = y;
            while (t != 1) j++, t *= t;
            z = z.pow(1LL << (e - j - 1));
            x *= z, z *= z, y *= z;
            e = j;
        }
        return ModInt(std::min(x.val_, md - x.val_));
    }
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};

using ModInt998244353 = ModInt<998244353>;
// using mint = ModInt<998244353>;
// using mint = ModInt<1000000007>;
#line 3 "convolution/ntt.hpp"

#include <algorithm>
#include <array>
#line 7 "convolution/ntt.hpp"
#include <tuple>
#line 9 "convolution/ntt.hpp"

// CUT begin
// Integer convolution for arbitrary mod
// with NTT (and Garner's algorithm) for ModInt / ModIntRuntime class.
// We skip Garner's algorithm if `skip_garner` is true or mod is in `nttprimes`.
// input: a (size: n), b (size: m)
// return: vector (size: n + m - 1)
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner);

constexpr int nttprimes[3] = {998244353, 167772161, 469762049};

// Integer FFT (Fast Fourier Transform) for ModInt class
// (Also known as Number Theoretic Transform, NTT)
// is_inverse: inverse transform
// ** Input size must be 2^n **
template <typename MODINT> void ntt(std::vector<MODINT> &a, bool is_inverse = false) {
    int n = a.size();
    if (n == 1) return;
    static const int mod = MODINT::mod();
    static const MODINT root = MODINT::get_primitive_root();
    assert(__builtin_popcount(n) == 1 and (mod - 1) % n == 0);

    static std::vector<MODINT> w{1}, iw{1};
    for (int m = w.size(); m < n / 2; m *= 2) {
        MODINT dw = root.pow((mod - 1) / (4 * m)), dwinv = 1 / dw;
        w.resize(m * 2), iw.resize(m * 2);
        for (int i = 0; i < m; i++) w[m + i] = w[i] * dw, iw[m + i] = iw[i] * dwinv;
    }

    if (!is_inverse) {
        for (int m = n; m >>= 1;) {
            for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
                for (int i = s; i < s + m; i++) {
                    MODINT x = a[i], y = a[i + m] * w[k];
                    a[i] = x + y, a[i + m] = x - y;
                }
            }
        }
    } else {
        for (int m = 1; m < n; m *= 2) {
            for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
                for (int i = s; i < s + m; i++) {
                    MODINT x = a[i], y = a[i + m];
                    a[i] = x + y, a[i + m] = (x - y) * iw[k];
                }
            }
        }
        int n_inv = MODINT(n).inv().val();
        for (auto &v : a) v *= n_inv;
    }
}
template <int MOD>
std::vector<ModInt<MOD>> nttconv_(const std::vector<int> &a, const std::vector<int> &b) {
    int sz = a.size();
    assert(a.size() == b.size() and __builtin_popcount(sz) == 1);
    std::vector<ModInt<MOD>> ap(sz), bp(sz);
    for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i];
    ntt(ap, false);
    if (a == b)
        bp = ap;
    else
        ntt(bp, false);
    for (int i = 0; i < sz; i++) ap[i] *= bp[i];
    ntt(ap, true);
    return ap;
}
long long garner_ntt_(int r0, int r1, int r2, int mod) {
    using mint2 = ModInt<nttprimes[2]>;
    static const long long m01 = 1LL * nttprimes[0] * nttprimes[1];
    static const long long m0_inv_m1 = ModInt<nttprimes[1]>(nttprimes[0]).inv().val();
    static const long long m01_inv_m2 = mint2(m01).inv().val();

    int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1];
    auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * m01_inv_m2;
    return (r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val()) % mod;
}
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner) {
    if (a.empty() or b.empty()) return {};
    int sz = 1, n = a.size(), m = b.size();
    while (sz < n + m) sz <<= 1;
    if (sz <= 16) {
        std::vector<MODINT> ret(n + m - 1);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) ret[i + j] += a[i] * b[j];
        }
        return ret;
    }
    int mod = MODINT::mod();
    if (skip_garner or
        std::find(std::begin(nttprimes), std::end(nttprimes), mod) != std::end(nttprimes)) {
        a.resize(sz), b.resize(sz);
        if (a == b) {
            ntt(a, false);
            b = a;
        } else {
            ntt(a, false), ntt(b, false);
        }
        for (int i = 0; i < sz; i++) a[i] *= b[i];
        ntt(a, true);
        a.resize(n + m - 1);
    } else {
        std::vector<int> ai(sz), bi(sz);
        for (int i = 0; i < n; i++) ai[i] = a[i].val();
        for (int i = 0; i < m; i++) bi[i] = b[i].val();
        auto ntt0 = nttconv_<nttprimes[0]>(ai, bi);
        auto ntt1 = nttconv_<nttprimes[1]>(ai, bi);
        auto ntt2 = nttconv_<nttprimes[2]>(ai, bi);
        a.resize(n + m - 1);
        for (int i = 0; i < n + m - 1; i++)
            a[i] = garner_ntt_(ntt0[i].val(), ntt1[i].val(), ntt2[i].val(), mod);
    }
    return a;
}

template <typename MODINT>
std::vector<MODINT> nttconv(const std::vector<MODINT> &a, const std::vector<MODINT> &b) {
    return nttconv<MODINT>(a, b, false);
}
#line 5 "linear_algebra_matrix/blackbox_matrices.hpp"
#include <numeric>
#include <utility>
#line 8 "linear_algebra_matrix/blackbox_matrices.hpp"

// Sparse matrix
template <typename Tp> struct sparse_matrix {
    int H, W;
    std::vector<std::vector<std::pair<int, Tp>>> vals;
    sparse_matrix(int H = 0, int W = 0) : H(H), W(W), vals(H) {}
    int height() const { return H; }
    int width() const { return W; }
    void add_element(int i, int j, Tp val) {
        assert(i >= 0 and i < H);
        assert(j >= 0 and i < W);
        vals[i].emplace_back(j, val);
    }
    std::vector<Tp> prod(const std::vector<Tp> &vec) const {
        assert(W == int(vec.size()));
        std::vector<Tp> ret(H);
        for (int i = 0; i < H; i++) {
            for (const auto &p : vals[i]) ret[i] += p.second * vec[p.first];
        }
        return ret;
    }
    std::vector<Tp> prod_left(const std::vector<Tp> &vec) const {
        assert(H == int(vec.size()));
        std::vector<Tp> ret(W);
        for (int i = 0; i < H; i++) {
            for (const auto &p : vals[i]) ret[p.first] += p.second * vec[i];
        }
        return ret;
    }
    std::vector<std::vector<Tp>> vecvec() const {
        std::vector<std::vector<Tp>> ret(H, std::vector<Tp>(W));
        for (int i = 0; i < H; i++) {
            for (auto p : vals[i]) ret[i][p.first] += p.second;
        }
        return ret;
    }
};

// Toeplitz matrix
// M = [
// [ vw_0 vw_1 vw_2 ... ]
// [ vh_1 ...           ]
// [ vh_2 ...           ]
// [ ...                ] (vw_0 == vh_0)
// prod() / prod_left() : O((H + W) log (H + W))
template <typename NTTModInt> struct toeplitz_ntt {
    int _h, _w;
    int _len_pow2;
    std::vector<NTTModInt> ts, ntt_ts;
    toeplitz_ntt(const std::vector<NTTModInt> &vh, const std::vector<NTTModInt> &vw)
        : _h(vh.size()), _w(vw.size()) {
        assert(vh.size() and vw.size() and vh[0] == vw[0]);
        ts.resize(_h + _w - 1);
        for (int i = 0; i < _w; i++) ts[i] = vw[_w - 1 - i];
        for (int j = 0; j < _h; j++) ts[_w - 1 + j] = vh[j];
        _len_pow2 = 1;
        while (_len_pow2 < int(ts.size()) + std::max(_h, _w) - 1) _len_pow2 *= 2;
        ntt_ts = ts;
        ntt_ts.resize(_len_pow2);
        ntt(ntt_ts, false);
    }
    int height() const { return _h; }
    int width() const { return _w; }
    std::vector<NTTModInt> prod(std::vector<NTTModInt> b) const {
        assert(int(b.size()) == _w);
        b.resize(_len_pow2);
        ntt(b, false);
        for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i];
        ntt(b, true);
        b.erase(b.begin(), b.begin() + _w - 1);
        b.resize(_h);
        return b;
    }
    std::vector<NTTModInt> prod_left(std::vector<NTTModInt> b) const {
        assert(int(b.size()) == _h);
        std::reverse(b.begin(), b.end());
        b.resize(_len_pow2);
        ntt(b, false);
        for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i];
        ntt(b, true);
        b.erase(b.begin(), b.begin() + _h - 1);
        b.resize(_w);
        std::reverse(b.begin(), b.end());
        return b;
    }
    std::vector<std::vector<NTTModInt>> vecvec() const {
        std::vector<std::vector<NTTModInt>> ret(_h, std::vector<NTTModInt>(_w));
        for (int i = 0; i < _h; i++) {
            for (int j = 0; j < _w; j++) ret[i][j] = ts[i - j + _w - 1];
        }
        return ret;
    }
};

// Square Toeplitz matrix
// M = [
// [ t_(N-1) t_(N-2) ... t_1 t_0     ]
// [ t_N     t_(N-1) ... t_2 t_1     ]
// [ ...                             ]
// [ t_(N*2-2)       ...     t_(N-1) ]]
// prod() / prod_left() : O(N log N)
template <typename NTTModInt> struct square_toeplitz_ntt {
    int _dim;
    int _len_pow2;
    std::vector<NTTModInt> ts;
    std::vector<NTTModInt> ntt_ts;
    square_toeplitz_ntt(const std::vector<NTTModInt> &t) : _dim(t.size() / 2 + 1), ts(t) {
        assert(t.size() % 2);
        _len_pow2 = 1;
        while (_len_pow2 < int(ts.size()) + _dim - 1) _len_pow2 *= 2;
        ntt_ts = ts;
        ntt_ts.resize(_len_pow2);
        ntt(ntt_ts, false);
    }
    int height() const { return _dim; }
    int width() const { return _dim; }

    // Calculate A * b
    std::vector<NTTModInt> prod(std::vector<NTTModInt> b) const {
        assert(int(b.size()) == _dim);
        b.resize(_len_pow2);
        ntt(b, false);
        for (int i = 0; i < _len_pow2; i++) b[i] *= ntt_ts[i];
        ntt(b, true);
        b.erase(b.begin(), b.begin() + _dim - 1);
        b.resize(_dim);
        return b;
    }
    // Calculate bT * A
    std::vector<NTTModInt> prod_left(std::vector<NTTModInt> b) const {
        std::reverse(b.begin(), b.end());
        b = prod(b);
        std::reverse(b.begin(), b.end());
        return b;
    }
    std::vector<std::vector<NTTModInt>> vecvec() const {
        std::vector<std::vector<NTTModInt>> ret(_dim, std::vector<NTTModInt>(_dim));
        for (int i = 0; i < _dim; i++) {
            for (int j = 0; j < _dim; j++) ret[i][j] = ts[i - j + _dim - 1];
        }
        return ret;
    }
};

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