test/judge.yosupo.jp/Kth_Term_of_Linearly_Recurrent_Sequence.0.test.cpp

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• Last update: 2022-10-20 00:25:59+09:00
• Problem: https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence

Depends on

• math/bostan_mori.hpp
• 畳み込み (math/convolution.hpp)
• 形式的冪級数 (math/fps.hpp)
• math/kth_of_lrs.hpp
• modint (math/modint.hpp)
• 数論変換 (math/ntt.hpp)
• template.hpp

Code

#define PROBLEM "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"
#include "../../math/kth_of_lrs.hpp"

int main() {
    using mint = modint998244353;
    ll d, k;
    cin >> d >> k;
    vector<mint> a(d), c(d);
    for (ll i : rep(d)) cin >> a[i];
    for (ll i : rep(d)) cin >> c[i];
    cout << kth_of_lrs(a, c, k) << endl;
}

#line 1 "test/judge.yosupo.jp/Kth_Term_of_Linearly_Recurrent_Sequence.0.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"
#line 2 "math/kth_of_lrs.hpp"

#line 2 "template.hpp"

#include <bits/stdc++.h>
using namespace std;

#define all(a) begin(a), end(a)
#define rall(a) rbegin(a), rend(a)
#define uniq(a) (a).erase(unique(all(a)), (a).end())
#define t0 first
#define t1 second
using ll = long long;
using ull = unsigned long long;
using pll = pair<ll, ll>;
using vll = vector<ll>;
constexpr double pi = 3.14159265358979323846;
constexpr ll dy[9] = {0, 1, 0, -1, 1, 1, -1, -1, 0};
constexpr ll dx[9] = {1, 0, -1, 0, 1, -1, -1, 1, 0};
constexpr ll sign(ll a) { return (a > 0) - (a < 0); }
constexpr ll fdiv(ll a, ll b) { return a / b - ((a ^ b) < 0 && a % b); }
constexpr ll cdiv(ll a, ll b) { return -fdiv(-a, b); }
constexpr ll pw(ll n) { return 1ll << n; }
constexpr ll flg(ll n) { return 63 - __builtin_clzll(n); }
constexpr ll clg(ll n) { return n == 1 ? 0 : flg(n - 1) + 1; }
constexpr ll safemod(ll x, ll mod) { return (x % mod + mod) % mod; }
template <typename T> using priority_queue_rev = priority_queue<T, vector<T>, greater<T>>;
template <typename T> constexpr T sq(const T &a) { return a * a; }
template <typename T, typename U> constexpr bool chmax(T &a, const U &b) { return a < b ? a = b, true : false; }
template <typename T, typename U> constexpr bool chmin(T &a, const U &b) { return a > b ? a = b, true : false; }
template <typename T, typename U> ostream &operator<<(ostream &os, const pair<T, U> &a) {
    os << "(" << a.first << ", " << a.second << ")";
    return os;
}
template <typename T, typename U, typename V> ostream &operator<<(ostream &os, const tuple<T, U, V> &a) {
    os << "(" << get<0>(a) << ", " << get<1>(a) << ", " << get<2>(a) << ")";
    return os;
}
template <typename T> ostream &operator<<(ostream &os, const vector<T> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
template <typename T> ostream &operator<<(ostream &os, const set<T> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
template <typename T> ostream &operator<<(ostream &os, const multiset<T> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
template <typename T, typename U> ostream &operator<<(ostream &os, const map<T, U> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
#ifdef ONLINE_JUDGE
#define dump(...) (void(0))
#else
void debug() { cerr << endl; }
template <typename Head, typename... Tail> void debug(Head &&head, Tail &&... tail) {
    cerr << head;
    if (sizeof...(Tail)) cerr << ", ";
    debug(tail...);
}
#define dump(...) cerr << __LINE__ << ": " << #__VA_ARGS__ << " = ", debug(__VA_ARGS__)
#endif
struct rep {
    struct itr {
        ll v;
        itr(ll v) : v(v) {}
        void operator++() { ++v; }
        ll operator*() const { return v; }
        bool operator!=(itr i) const { return v < *i; }
    };
    ll l, r;
    rep(ll l, ll r) : l(l), r(r) {}
    rep(ll r) : rep(0, r) {}
    itr begin() const { return l; };
    itr end() const { return r; };
};
struct per {
    struct itr {
        ll v;
        itr(ll v) : v(v) {}
        void operator++() { --v; }
        ll operator*() const { return v; }
        bool operator!=(itr i) const { return v > *i; }
    };
    ll l, r;
    per(ll l, ll r) : l(l), r(r) {}
    per(ll r) : per(0, r) {}
    itr begin() const { return r - 1; };
    itr end() const { return l - 1; };
};
struct io_setup {
    static constexpr int PREC = 20;
    io_setup() {
        cout << fixed << setprecision(PREC);
        cerr << fixed << setprecision(PREC);
    };
} iOS;
#line 2 "math/bostan_mori.hpp"

#line 2 "math/fps.hpp"

#line 2 "math/convolution.hpp"

#line 2 "math/ntt.hpp"

#line 4 "math/ntt.hpp"

template <typename mint> void ntt(vector<mint> &a, bool inv = false) {
    ll n = a.size(), m = n >> 1;
    mint root = 2;
    while (root.pow((mint::mod() - 1) >> 1) == 1) root += 1;
    mint wn = root.pow((mint::mod() - 1) / n);
    if (inv) wn = wn.inv();
    vector<mint> b(n);
    for (ll i = 1; i < n; i <<= 1, wn *= wn, swap(a, b)) {
        mint wj = 1;
        for (ll j = 0; j < m; j += i, wj *= wn) {
            for (ll k : rep(i)) {
                b[0 + (j << 1) + k] = (a[0 + j + k] + a[m + j + k]);
                b[i + (j << 1) + k] = (a[0 + j + k] - a[m + j + k]) * wj;
            }
        }
    }
    if (inv) {
        mint ninv = mint(n).inv();
        for (mint &ai : a) ai *= ninv;
    }
}
template <typename mint> void intt(vector<mint> &a) { ntt(a, true); }
#line 5 "math/convolution.hpp"

template <typename V> vector<V> convolution_naive(vector<V> a, vector<V> b) {
    ll na = a.size(), nb = b.size();
    vector<V> c(na + nb - 1);
    if (na < nb) swap(a, b), swap(na, nb);
    for (ll i : rep(na)) {
        for (ll j : rep(nb)) c[i + j] += a[i] * b[j];
    }
    return c;
}

template <typename mint> vector<mint> convolution_ntt(vector<mint> a, vector<mint> b) {
    ll _n = a.size() + b.size() - 1, n;
    for (n = 1; n < _n; n <<= 1) {}
    a.resize(n), b.resize(n);
    ntt(a), ntt(b);
    for (ll i : rep(n)) a[i] *= b[i];
    intt(a);
    a.resize(_n);
    return a;
}

template <typename mint> vector<mint> convolution(const vector<mint> &a, const vector<mint> &b) {
    if (min(a.size(), b.size()) <= 60) {
        return convolution_naive(a, b);
    } else {
        return convolution_ntt(a, b);
    }
}
#line 2 "math/modint.hpp"

#line 4 "math/modint.hpp"

template <typename M> struct modint {
    ll val;
    modint(ll val = 0) : val(val >= 0 ? val % M::mod : (M::mod - (-val) % M::mod) % M::mod) {}
    static ll mod() { return M::mod; }
    modint inv() const {
        ll a = val, b = M::mod, u = 1, v = 0, t;
        while (b > 0) {
            t = a / b;
            swap(a -= t * b, b);
            swap(u -= t * v, v);
        }
        return u;
    }
    modint pow(ll k) const {
        modint ret = 1, mul = val;
        while (k) {
            if (k & 1) ret *= mul;
            mul *= mul;
            k >>= 1;
        }
        return ret;
    }
    modint &operator+=(const modint &a) {
        if ((val += a.val) >= M::mod) val -= M::mod;
        return *this;
    }
    modint &operator-=(const modint &a) {
        if ((val += M::mod - a.val) >= M::mod) val -= M::mod;
        return *this;
    }
    modint &operator*=(const modint &a) {
        (val *= a.val) %= M::mod;
        return *this;
    }
    modint &operator/=(const modint &a) { return *this *= a.inv(); }
    modint operator+() const { return *this; }
    modint operator-() const { return modint(-val); }
    friend bool operator==(const modint &a, const modint &b) { return a.val == b.val; }
    friend bool operator!=(const modint &a, const modint &b) { return rel_ops::operator!=(a, b); }
    friend modint operator+(const modint &a, const modint &b) { return modint(a) += b; }
    friend modint operator-(const modint &a, const modint &b) { return modint(a) -= b; }
    friend modint operator*(const modint &a, const modint &b) { return modint(a) *= b; }
    friend modint operator/(const modint &a, const modint &b) { return modint(a) /= b; }
    friend istream &operator>>(istream &is, modint &a) {
        ll val;
        is >> val;
        a = modint(val);
        return is;
    }
    friend ostream &operator<<(ostream &os, const modint &a) { return os << a.val; }
};

struct _998244353 {
    constexpr static ll mod = 998244353;
};
struct _1000000007 {
    constexpr static ll mod = 1000000007;
};
using modint998244353 = modint<_998244353>;
using modint1000000007 = modint<_1000000007>;

struct arbitrary {
    static ll mod;
};
ll arbitrary::mod;
#line 6 "math/fps.hpp"

template <typename mint> struct fps : vector<mint> {
    using vector<mint>::vector;
    using vector<mint>::operator=;
    fps() : vector<mint>() {}
    fps(const mint &a) : vector<mint>(1, a) {}
    fps(const vector<mint> &a) : vector<mint>(a) {}
    fps(const fps &a) : vector<mint>(a) {}
    fps &operator=(const fps &a) {
        *this = (vector<mint>)a;
        return *this;
    }
    fps &operator+=(const fps &a) {
        if (a.size() > this->size()) this->resize(a.size());
        for (ll i : rep(a.size())) (*this)[i] += a[i];
        return *this;
    }
    fps &operator-=(const fps &a) {
        if (a.size() > this->size()) this->resize(a.size());
        for (ll i : rep(a.size())) (*this)[i] -= a[i];
        return *this;
    }
    fps &operator*=(const fps &a);
    fps &operator/=(const mint &a) {
        for (ll i : rep(this->size())) (*this)[i] /= a;
        return *this;
    };
    fps &operator>>=(ll d) {
        if ((ll)this->size() <= d) {
            *this = {};
        } else {
            this->erase(this->begin(), this->begin() + d);
        }
        return *this;
    }
    fps &operator<<=(ll d) {
        this->insert(this->begin(), d, 0);
        return *this;
    }
    fps &chdot(const fps &a) {
        for (ll i : rep(this->size())) {
            if (i < (ll)a.size()) {
                (*this)[i] *= a[i];
            } else {
                (*this)[i] = 0;
            }
        }
        return *this;
    }
    fps prefix(ll d) const { return fps(this->begin(), this->begin() + min((ll)this->size(), d)); }
    fps differential() const {
        ll n = this->size();
        fps ret(max(0ll, n - 1));
        for (ll i : rep(1, n)) { ret[i - 1] = i * (*this)[i]; }
        return ret;
    }
    fps integral() const {
        ll n = this->size();
        fps ret(n + 1);
        ret[0] = 0;
        if (n > 0) ret[1] = 1;
        for (ll i : rep(2, n + 1)) ret[i] = (-ret[mint::mod() % i]) * (mint::mod() / i);
        for (ll i : rep(n)) ret[i + 1] *= (*this)[i];
        return ret;
    }
    fps inv(ll d) const {
        fps ret{(*this)[0].inv()};
        for (ll m = 1; m < d; m <<= 1) ret = (ret + ret - ret * ret * this->prefix(m << 1)).prefix(m << 1);
        return ret.prefix(d);
    }
    fps log(ll d) const {
        assert((*this)[0] == 1);
        return (this->differential() * this->inv(d)).prefix(d - 1).integral();
    }
    fps exp(ll d) const {
        assert(this->size() == 0 || (*this)[0] == 0);
        fps ret{1};
        for (ll m = 1; m < d; m <<= 1) ret = (ret * (this->prefix(m << 1) + 1 - ret.log(m << 1))).prefix(m << 1);
        return ret.prefix(d);
    }
    fps pow(ll k, ll d) const {
        if (k == 0) {
            fps ret(d);
            if (d) ret[0] = 1;
            return ret;
        }
        for (ll i : rep(this->size())) {
            if ((*this)[i] != 0) {
                if (i > d / k) return fps(d);
                fps ret = (((*this * (*this)[i].inv()) >> i).log(d) * mint(k)).exp(d) * (*this)[i].pow(k);
                ret = (ret << (i * k)).prefix(d);
                ret.resize(d);
                return ret;
            }
        }
        return fps(d);
    }
    friend fps operator+(const fps &a) { return a; }
    friend fps operator-(const fps &a) { return fps() -= a; }
    friend fps operator+(const fps &a, const fps &b) { return fps(a) += b; }
    friend fps operator-(const fps &a, const fps &b) { return fps(a) -= b; }
    friend fps operator*(const fps &a, const fps &b) { return fps(a) *= b; }
    friend fps operator>>(const fps &a, ll d) { return fps(a) >>= d; }
    friend fps operator<<(const fps &a, ll d) { return fps(a) <<= d; }
};

using m9 = modint998244353;

template <> fps<m9> &fps<m9>::operator*=(const fps<m9> &a) {
    *this = convolution(*this, a);
    return *this;
}

template <> fps<m9> fps<m9>::inv(ll d) const {
    fps ret{(*this)[0].inv()};
    for (ll m = 1; m < d; m <<= 1) {
        fps f = this->prefix(m << 1);
        fps g = ret;
        f.resize(m << 1), ntt(f);
        g.resize(m << 1), ntt(g);
        f.chdot(g);
        intt(f);
        f >>= m, f.resize(m << 1), ntt(f);
        f.chdot(g);
        intt(f);
        f = -f;
        ret.insert(ret.end(), f.begin(), f.begin() + m);
    }
    return ret.prefix(d);
}

template <> fps<m9> fps<m9>::exp(ll d) const {
    assert(this->size() == 0 || (*this)[0] == 0);
    fps ret{1}, g{1}, g_freq{1};
    for (ll m = 1; m < d; m <<= 1) {
        fps ret_freq = ret.prefix(m);
        ret_freq.resize(m << 1), ntt(ret_freq);

        fps g_cont = g_freq;
        for (ll i : rep(m)) g_cont[i] *= ret_freq[i << 1];
        intt(g_cont);
        g_cont >>= m >> 1;
        g_cont.resize(m), ntt(g_cont);
        g_cont.chdot(g_freq);
        intt(g_cont);
        g_cont = -g_cont;
        g.insert(g.end(), g_cont.begin(), g_cont.begin() + (m >> 1));

        fps r = this->differential().prefix(m - 1);
        r.resize(m), ntt(r);
        for (ll i : rep(m)) r[i] *= ret_freq[i << 1];
        intt(r);

        fps t = ret.differential() - r;
        t.insert(t.begin(), t.back()), t.pop_back();
        t.resize(m << 1), ntt(t);
        g_freq = g, g_freq.resize(m << 1), ntt(g_freq);
        t.chdot(g_freq);
        intt(t), t.resize(m);

        fps u = (this->prefix(m << 1) - (t << m - 1).integral()) >> m;
        u.resize(m << 1), ntt(u);
        u.chdot(ret_freq);
        intt(u);

        ret += u.prefix(m) << m;
    }
    return ret.prefix(d);
}
#line 5 "math/bostan_mori.hpp"

template <typename V> V bostan_mori(fps<V> p, fps<V> q, ll k) {
    while (k) {
        fps<V> _q(q);
        for (ll i = 1; i < _q.size(); i += 2) _q[i] = -_q[i];
        fps<V> u = p * _q, v = q * _q;
        p.resize((u.size() >> 1) + (u.size() & ~k & 1)), q.resize((v.size() >> 1) + 1);
        for (ll i : rep(p.size())) p[i] = u[i << 1 | (k & 1)];
        for (ll i : rep(q.size())) q[i] = v[i << 1];
        k >>= 1;
    }
    return p[0];
}
#line 5 "math/kth_of_lrs.hpp"

template <typename V> V kth_of_lrs(const vector<V> &a, const vector<V> &c, ll k) {
    fps<V> q = {1};
    q.insert(end(q), begin(c), end(c));
    for (ll i : rep(1, q.size())) q[i] = -q[i];
    fps<V> p = (q * a).prefix(a.size());
    return bostan_mori(p, q, k);
}
#line 3 "test/judge.yosupo.jp/Kth_Term_of_Linearly_Recurrent_Sequence.0.test.cpp"

int main() {
    using mint = modint998244353;
    ll d, k;
    cin >> d >> k;
    vector<mint> a(d), c(d);
    for (ll i : rep(d)) cin >> a[i];
    for (ll i : rep(d)) cin >> c[i];
    cout << kth_of_lrs(a, c, k) << endl;
}

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