結果

問題 No.2688 Cell Proliferation (Hard)
コンテスト
ユーザー momohara
提出日時 2024-03-22 01:54:02
言語 C++23
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 1,903 ms / 4,000 ms
コード長 17,674 bytes
コンパイル時間 6,513 ms
コンパイル使用メモリ 337,756 KB
実行使用メモリ 160,004 KB
最終ジャッジ日時 2024-09-30 10:27:47
合計ジャッジ時間 37,775 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge1
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ファイルパターン 結果
sample AC * 3
other AC * 26
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ソースコード

diff # 

#include <atcoder/all>
#include <bits/stdc++.h>

using namespace std;
using namespace atcoder;

using ll = long long;
using ull = unsigned long long;
using ld = long double;
using P = pair<ll, ll>;
using tp = tuple<ll, ll, ll>;

template <class T>
using vec = vector<T>;
template <class T>
using vvec = vector<vec<T>>;

#define all(hoge) (hoge).begin(), (hoge).end()
#define en '\n'
#define rep(i, m, n) for(ll i = (ll)(m); i < (ll)(n); ++i)
#define rep2(i, m, n) for(ll i = (ll)(n)-1; i >= (ll)(m); --i)
#define REP(i, n) rep(i, 0, n)
#define REP2(i, n) rep2(i, 0, n)

constexpr long long INF = 1LL << 60;
constexpr int INF_INT = 1 << 25;
// constexpr long long MOD = (ll)1e9 + 7;
constexpr long long MOD = 998244353LL;
static const ld pi = 3.141592653589793L;

#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")

template <class T>
inline bool chmin(T &a, T b) {
    if(a > b) {
        a = b;
        return true;
    }
    return false;
}

template <class T>
inline bool chmax(T &a, T b) {
    if(a < b) {
        a = b;
        return true;
    }
    return false;
}

struct Edge {
    int to, rev;
    ll cap;
    Edge(int _to, int _rev, ll _cap) : to(_to), rev(_rev), cap(_cap) {}
};

typedef vector<Edge> Edges;
typedef vector<Edges> Graph;

void add_edge(Graph &G, int from, int to, ll cap, bool revFlag, ll revCap) {
    G[from].push_back(Edge(to, (int)G[to].size(), cap));
    if(revFlag)
        G[to].push_back(Edge(from, (int)G[from].size() - 1, revCap));
}

template <int mod>
struct ModInt {
    int x;

    ModInt() : x(0) {}

    ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}

    ModInt &operator+=(const ModInt &p) {
        if((x += p.x) >= mod)
            x -= mod;
        return *this;
    }

    ModInt &operator-=(const ModInt &p) {
        if((x += mod - p.x) >= mod)
            x -= mod;
        return *this;
    }

    ModInt &operator*=(const ModInt &p) {
        x = (int)(1LL * x * p.x % mod);
        return *this;
    }

    ModInt &operator/=(const ModInt &p) {
        *this *= p.inverse();
        return *this;
    }

    ModInt operator-() const { return ModInt(-x); }

    ModInt operator+(const ModInt &p) const { return ModInt(*this) += p; }

    ModInt operator-(const ModInt &p) const { return ModInt(*this) -= p; }

    ModInt operator*(const ModInt &p) const { return ModInt(*this) *= p; }

    ModInt operator/(const ModInt &p) const { return ModInt(*this) /= p; }

    bool operator==(const ModInt &p) const { return x == p.x; }

    bool operator!=(const ModInt &p) const { return x != p.x; }

    ModInt inverse() const {
        int a = x, b = mod, u = 1, v = 0, t;
        while(b > 0) {
            t = a / b;
            swap(a -= t * b, b);
            swap(u -= t * v, v);
        }
        return ModInt(u);
    }

    ModInt pow(int64_t n) const {
        ModInt ret(1), mul(x);
        while(n > 0) {
            if(n & 1)
                ret *= mul;
            mul *= mul;
            n >>= 1;
        }
        return ret;
    }

    friend ostream &operator<<(ostream &os, const ModInt &p) {
        return os << p.x;
    }

    friend istream &operator>>(istream &is, ModInt &a) {
        int64_t t;
        is >> t;
        a = ModInt<mod>(t);
        return (is);
    }

    static int get_mod() { return mod; }
};

using mint = ModInt<MOD>;

namespace FastFourierTransform {
using real = double;

struct C {
    real x, y;

    C() : x(0), y(0) {}

    C(real x, real y) : x(x), y(y) {}

    inline C operator+(const C &c) const { return C(x + c.x, y + c.y); }

    inline C operator-(const C &c) const { return C(x - c.x, y - c.y); }

    inline C operator*(const C &c) const { return C(x * c.x - y * c.y, x * c.y + y * c.x); }

    inline C conj() const { return C(x, -y); }
};

const real PI = acosl(-1);
int base = 1;
vector<C> rts = {{0, 0},
                 {1, 0}};
vector<int> rev = {0, 1};

void ensure_base(int nbase) {
    if(nbase <= base)
        return;
    rev.resize(1 << nbase);
    rts.resize(1 << nbase);
    for(int i = 0; i < (1 << nbase); i++) {
        rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
    }
    while(base < nbase) {
        real angle = PI * 2.0 / (1 << (base + 1));
        for(int i = 1 << (base - 1); i < (1 << base); i++) {
            rts[i << 1] = rts[i];
            real angle_i = angle * (2 * i + 1 - (1 << base));
            rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
        }
        ++base;
    }
}

void fft(vector<C> &a, int n) {
    assert((n & (n - 1)) == 0);
    int zeros = __builtin_ctz(n);
    ensure_base(zeros);
    int shift = base - zeros;
    for(int i = 0; i < n; i++) {
        if(i < (rev[i] >> shift)) {
            swap(a[i], a[rev[i] >> shift]);
        }
    }
    for(int k = 1; k < n; k <<= 1) {
        for(int i = 0; i < n; i += 2 * k) {
            for(int j = 0; j < k; j++) {
                C z = a[i + j + k] * rts[j + k];
                a[i + j + k] = a[i + j] - z;
                a[i + j] = a[i + j] + z;
            }
        }
    }
}

vector<int64_t> multiply(const vector<int> &a, const vector<int> &b) {
    int need = (int)a.size() + (int)b.size() - 1;
    int nbase = 1;
    while((1 << nbase) < need)
        nbase++;
    ensure_base(nbase);
    int sz = 1 << nbase;
    vector<C> fa(sz);
    for(int i = 0; i < sz; i++) {
        int x = (i < (int)a.size() ? a[i] : 0);
        int y = (i < (int)b.size() ? b[i] : 0);
        fa[i] = C(x, y);
    }
    fft(fa, sz);
    C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
    for(int i = 0; i <= (sz >> 1); i++) {
        int j = (sz - i) & (sz - 1);
        C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
        fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
        fa[i] = z;
    }
    for(int i = 0; i < (sz >> 1); i++) {
        C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
        C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i];
        fa[i] = A0 + A1 * s;
    }
    fft(fa, sz >> 1);
    vector<int64_t> ret(need);
    for(int i = 0; i < need; i++) {
        ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
    }
    return ret;
}
}; // namespace FastFourierTransform

template <typename T>
struct ArbitraryModConvolution {
    using real = FastFourierTransform::real;
    using C = FastFourierTransform::C;

    ArbitraryModConvolution() = default;

    static vector<T> multiply(const vector<T> &a, const vector<T> &b, int need = -1) {
        if(need == -1)
            need = a.size() + b.size() - 1;
        int nbase = 0;
        while((1 << nbase) < need)
            nbase++;
        FastFourierTransform::ensure_base(nbase);
        int sz = 1 << nbase;
        vector<C> fa(sz);
        for(int i = 0; i < a.size(); i++) {
            fa[i] = C(a[i].x & ((1 << 15) - 1), a[i].x >> 15);
        }
        fft(fa, sz);
        vector<C> fb(sz);
        if(a == b) {
            fb = fa;
        } else {
            for(int i = 0; i < b.size(); i++) {
                fb[i] = C(b[i].x & ((1 << 15) - 1), b[i].x >> 15);
            }
            fft(fb, sz);
        }
        real ratio = 0.25 / sz;
        C r2(0, -1), r3(ratio, 0), r4(0, -ratio), r5(0, 1);
        for(int i = 0; i <= (sz >> 1); i++) {
            int j = (sz - i) & (sz - 1);
            C a1 = (fa[i] + fa[j].conj());
            C a2 = (fa[i] - fa[j].conj()) * r2;
            C b1 = (fb[i] + fb[j].conj()) * r3;
            C b2 = (fb[i] - fb[j].conj()) * r4;
            if(i != j) {
                C c1 = (fa[j] + fa[i].conj());
                C c2 = (fa[j] - fa[i].conj()) * r2;
                C d1 = (fb[j] + fb[i].conj()) * r3;
                C d2 = (fb[j] - fb[i].conj()) * r4;
                fa[i] = c1 * d1 + c2 * d2 * r5;
                fb[i] = c1 * d2 + c2 * d1;
            }
            fa[j] = a1 * b1 + a2 * b2 * r5;
            fb[j] = a1 * b2 + a2 * b1;
        }
        fft(fa, sz);
        fft(fb, sz);
        vector<T> ret(need);
        for(int i = 0; i < need; i++) {
            int64_t aa = llround(fa[i].x);
            int64_t bb = llround(fb[i].x);
            int64_t cc = llround(fa[i].y);
            aa = T(aa).x, bb = T(bb).x, cc = T(cc).x;
            ret[i] = aa + (bb << 15) + (cc << 30);
        }
        return ret;
    }
};

template <typename T>
struct FormalPowerSeries : vector<T> {
    using vector<T>::vector;
    using P = FormalPowerSeries;
    using Conv = ArbitraryModConvolution<T>;

    P pre(int deg) const {
        return P(begin(*this), begin(*this) + min((int)this->size(), deg));
    }

    P rev(int deg = -1) const {
        P ret(*this);
        if(deg != -1)
            ret.resize(deg, T(0));
        reverse(begin(ret), end(ret));
        return ret;
    }

    void shrink() {
        while(this->size() && this->back() == T(0))
            this->pop_back();
    }

    P operator+(const P &r) const { return P(*this) += r; }

    P operator+(const T &v) const { return P(*this) += v; }

    P operator-(const P &r) const { return P(*this) -= r; }

    P operator-(const T &v) const { return P(*this) -= v; }

    P operator*(const P &r) const { return P(*this) *= r; }

    P operator*(const T &v) const { return P(*this) *= v; }

    P operator/(const P &r) const { return P(*this) /= r; }

    P operator%(const P &r) const { return P(*this) %= r; }

    P &operator+=(const P &r) {
        if(r.size() > this->size())
            this->resize(r.size());
        for(int i = 0; i < r.size(); i++)
            (*this)[i] += r[i];
        return *this;
    }

    P &operator-=(const P &r) {
        if(r.size() > this->size())
            this->resize(r.size());
        for(int i = 0; i < r.size(); i++)
            (*this)[i] -= r[i];
        return *this;
    }

    // https://judge.yosupo.jp/problem/convolution_mod
    P &operator*=(const P &r) {
        if(this->empty() || r.empty()) {
            this->clear();
            return *this;
        }
        auto ret = Conv::multiply(*this, r);
        return *this = {begin(ret), end(ret)};
    }

    P &operator/=(const P &r) {
        if(this->size() < r.size()) {
            this->clear();
            return *this;
        }
        int n = this->size() - r.size() + 1;
        return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
    }

    P &operator%=(const P &r) {
        return *this -= *this / r * r;
    }

    // https://judge.yosupo.jp/problem/division_of_polynomials
    pair<P, P> div_mod(const P &r) {
        P q = *this / r;
        return make_pair(q, *this - q * r);
    }

    P operator-() const {
        P ret(this->size());
        for(int i = 0; i < this->size(); i++)
            ret[i] = -(*this)[i];
        return ret;
    }

    P &operator+=(const T &r) {
        if(this->empty())
            this->resize(1);
        (*this)[0] += r;
        return *this;
    }

    P &operator-=(const T &r) {
        if(this->empty())
            this->resize(1);
        (*this)[0] -= r;
        return *this;
    }

    P &operator*=(const T &v) {
        for(int i = 0; i < this->size(); i++)
            (*this)[i] *= v;
        return *this;
    }

    P dot(P r) const {
        P ret(min(this->size(), r.size()));
        for(int i = 0; i < ret.size(); i++)
            ret[i] = (*this)[i] * r[i];
        return ret;
    }

    P operator>>(int sz) const {
        if(this->size() <= sz)
            return {};
        P ret(*this);
        ret.erase(ret.begin(), ret.begin() + sz);
        return ret;
    }

    P operator<<(int sz) const {
        P ret(*this);
        ret.insert(ret.begin(), sz, T(0));
        return ret;
    }

    T operator()(T x) const {
        T r = 0, w = 1;
        for(auto &v : *this) {
            r += w * v;
            w *= x;
        }
        return r;
    }

    P diff() const {
        const int n = (int)this->size();
        P ret(max(0, n - 1));
        for(int i = 1; i < n; i++)
            ret[i - 1] = (*this)[i] * T(i);
        return ret;
    }

    P integral() const {
        const int n = (int)this->size();
        P ret(n + 1);
        ret[0] = T(0);
        for(int i = 0; i < n; i++)
            ret[i + 1] = (*this)[i] / T(i + 1);
        return ret;
    }

    // https://judge.yosupo.jp/problem/inv_of_formal_power_series
    // F(0) must not be 0
    P inv(int deg = -1) const {
        assert(((*this)[0]) != T(0));
        const int n = (int)this->size();
        if(deg == -1)
            deg = n;
        P ret({T(1) / (*this)[0]});
        for(int i = 1; i < deg; i <<= 1) {
            ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1);
        }
        return ret.pre(deg);
    }

    // https://judge.yosupo.jp/problem/log_of_formal_power_series
    // F(0) must be 1
    P log(int deg = -1) const {
        assert((*this)[0] == T(1));
        const int n = (int)this->size();
        if(deg == -1)
            deg = n;
        return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
    }

    // https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
    P sqrt(
        int deg = -1, const function<T(T)> &get_sqrt = [](T) { return T(1); }) const {
        const int n = (int)this->size();
        if(deg == -1)
            deg = n;
        if((*this)[0] == T(0)) {
            for(int i = 1; i < n; i++) {
                if((*this)[i] != T(0)) {
                    if(i & 1)
                        return {};
                    if(deg - i / 2 <= 0)
                        break;
                    auto ret = (*this >> i).sqrt(deg - i / 2, get_sqrt);
                    if(ret.empty())
                        return {};
                    ret = ret << (i / 2);
                    if(ret.size() < deg)
                        ret.resize(deg, T(0));
                    return ret;
                }
            }
            return P(deg, 0);
        }
        auto sqr = T(get_sqrt((*this)[0]));
        if(sqr * sqr != (*this)[0])
            return {};
        P ret{sqr};
        T inv2 = T(1) / T(2);
        for(int i = 1; i < deg; i <<= 1) {
            ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
        }
        return ret.pre(deg);
    }

    P sqrt(const function<T(T)> &get_sqrt, int deg = -1) const {
        return sqrt(deg, get_sqrt);
    }

    // https://judge.yosupo.jp/problem/exp_of_formal_power_series
    // F(0) must be 0
    P exp(int deg = -1) const {
        if(deg == -1)
            deg = this->size();
        assert((*this)[0] == T(0));
        const int n = (int)this->size();
        if(deg == -1)
            deg = n;
        P ret({T(1)});
        for(int i = 1; i < deg; i <<= 1) {
            ret = (ret * (pre(i << 1) + T(1) - ret.log(i << 1))).pre(i << 1);
        }
        return ret.pre(deg);
    }

    // https://judge.yosupo.jp/problem/pow_of_formal_power_series
    P pow(int64_t k, int deg = -1) const {
        const int n = (int)this->size();
        if(deg == -1)
            deg = n;
        if(k == 0) {
            P ret(deg, T(0));
            ret[0] = T(1);
            return ret;
        }
        for(int i = 0; i < n; i++) {
            if(i * k > deg)
                return P(deg, T(0));
            if((*this)[i] != T(0)) {
                T rev = T(1) / (*this)[i];
                P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
                ret = (ret << (i * k)).pre(deg);
                if(ret.size() < deg)
                    ret.resize(deg, T(0));
                return ret;
            }
        }
        return *this;
    }

    // https://yukicoder.me/problems/no/215
    P mod_pow(int64_t k, P g) const {
        P modinv = g.rev().inv();
        auto get_div = [&](P base) {
            if(base.size() < g.size()) {
                base.clear();
                return base;
            }
            int n = base.size() - g.size() + 1;
            return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
        };
        P x(*this), ret{1};
        while(k > 0) {
            if(k & 1) {
                ret *= x;
                ret -= get_div(ret) * g;
                ret.shrink();
            }
            x *= x;
            x -= get_div(x) * g;
            x.shrink();
            k >>= 1;
        }
        return ret;
    }

    // https://judge.yosupo.jp/problem/polynomial_taylor_shift
    P taylor_shift(T c) const {
        int n = (int)this->size();
        vector<T> fact(n), rfact(n);
        fact[0] = rfact[0] = T(1);
        for(int i = 1; i < n; i++)
            fact[i] = fact[i - 1] * T(i);
        rfact[n - 1] = T(1) / fact[n - 1];
        for(int i = n - 1; i > 1; i--)
            rfact[i - 1] = rfact[i] * T(i);
        P p(*this);
        for(int i = 0; i < n; i++)
            p[i] *= fact[i];
        p = p.rev();
        P bs(n, T(1));
        for(int i = 1; i < n; i++)
            bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1];
        p = (p * bs).pre(n);
        p = p.rev();
        for(int i = 0; i < n; i++)
            p[i] *= rfact[i];
        return p;
    }
};

void solve() {
    ll p1, p2, q1, q2, t;
    cin >> p1 >> p2 >> q1 >> q2 >> t;
    mint P = mint(p1) / p2;
    mint Q = mint(q1) / q2;
    using FPS = FormalPowerSeries<mint>;

    FPS g(t + 2), h(t + 2);
    mint QQ = 1;
    g[0] = 1;
    h[0] = 1;
    REP(i, t) {
        g[i + 1] = P * QQ * (-1);
        QQ *= Q.pow(i + 1);
        h[i + 1] = QQ;
    }

    auto ginv = g.inv();
    auto f = ginv * h;
    cout << f[t] << en;
}

int main() {
    ios::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);
    // cout << fixed << setprecision(10);

    // ll t;
    // cin >> t;
    // REP(i, t - 1) {
    //     solve();
    // }

    solve();

    return 0;
}
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