#896369 (C++17) No.2396 等差二項展開

提出ソース

結果

問題	No.2396 等差二項展開
ユーザー	tokusakurai
提出日時	2023-07-28 23:05:05
言語	C++17 (gcc 12.3.0 + boost 1.83.0)
結果	AC
実行時間	1,647 ms / 6,000 ms
コード長	14,338 bytes
コンパイル時間	2,368 ms
コンパイル使用メモリ	213,940 KB
実行使用メモリ	7,424 KB
最終ジャッジ日時	2024-10-06 21:01:53
合計ジャッジ時間	15,478 ms
ジャッジサーバーID （参考情報）	judge3 / judge4

このコードへのチャレンジ
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テストケース

テストケース表示

入力	結果	実行時間実行使用メモリ
testcase_00	AC	1 ms 5,248 KB
testcase_01	AC	1 ms 5,248 KB
testcase_02	AC	2 ms 5,248 KB
testcase_03	AC	2 ms 5,248 KB
testcase_04	AC	2 ms 5,248 KB
testcase_05	AC	2 ms 5,248 KB
testcase_06	AC	1 ms 5,248 KB
testcase_07	AC	2 ms 5,248 KB
testcase_08	AC	2 ms 5,248 KB
testcase_09	AC	1 ms 5,248 KB
testcase_10	AC	2 ms 5,248 KB
testcase_11	AC	2 ms 5,248 KB
testcase_12	AC	2 ms 5,248 KB
testcase_13	AC	2 ms 5,248 KB
testcase_14	AC	2 ms 5,248 KB
testcase_15	AC	14 ms 5,248 KB
testcase_16	AC	200 ms 5,248 KB
testcase_17	AC	861 ms 5,376 KB
testcase_18	AC	1,375 ms 6,272 KB
testcase_19	AC	1,647 ms 7,424 KB
testcase_20	AC	2 ms 5,248 KB
testcase_21	AC	1 ms 5,248 KB
testcase_22	AC	2 ms 5,248 KB
testcase_23	AC	714 ms 5,888 KB
testcase_24	AC	1,126 ms 6,528 KB
testcase_25	AC	1,336 ms 6,528 KB
testcase_26	AC	1,018 ms 6,144 KB
testcase_27	AC	1,114 ms 6,400 KB
testcase_28	AC	1,217 ms 6,144 KB
testcase_29	AC	1,100 ms 6,528 KB
testcase_30	AC	786 ms 6,144 KB

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ソースコード

raw source code

#include <bits/stdc++.h>
using namespace std;
#define rep(i, n) for (int i = 0; i < (n); i++)
#define per(i, n) for (int i = (n)-1; i >= 0; i--)
#define rep2(i, l, r) for (int i = (l); i < (r); i++)
#define per2(i, l, r) for (int i = (r)-1; i >= (l); i--)
#define each(e, v) for (auto &e : v)
#define MM << " " <<
#define pb push_back
#define eb emplace_back
#define all(x) begin(x), end(x)
#define rall(x) rbegin(x), rend(x)
#define sz(x) (int)x.size()
using ll = long long;
using pii = pair<int, int>;
using pil = pair<int, ll>;
using pli = pair<ll, int>;
using pll = pair<ll, ll>;

template <typename T>
using minheap = priority_queue<T, vector<T>, greater<T>>;

template <typename T>
using maxheap = priority_queue<T>;

template <typename T>
bool chmax(T &x, const T &y) {
    return (x < y) ? (x = y, true) : false;
}

template <typename T>
bool chmin(T &x, const T &y) {
    return (x > y) ? (x = y, true) : false;
}

template <typename T>
int flg(T x, int i) {
    return (x >> i) & 1;
}

int pct(int x) { return __builtin_popcount(x); }
int pct(ll x) { return __builtin_popcountll(x); }
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int botbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int botbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T>
void print(const vector<T> &v, T x = 0) {
    int n = v.size();
    for (int i = 0; i < n; i++) cout << v[i] + x << (i == n - 1 ? '\n' : ' ');
    if (v.empty()) cout << '\n';
}

template <typename T>
void printn(const vector<T> &v, T x = 0) {
    int n = v.size();
    for (int i = 0; i < n; i++) cout << v[i] + x << '\n';
}

template <typename T>
int lb(const vector<T> &v, T x) {
    return lower_bound(begin(v), end(v), x) - begin(v);
}

template <typename T>
int ub(const vector<T> &v, T x) {
    return upper_bound(begin(v), end(v), x) - begin(v);
}

template <typename T>
void rearrange(vector<T> &v) {
    sort(begin(v), end(v));
    v.erase(unique(begin(v), end(v)), end(v));
}

template <typename T>
vector<int> id_sort(const vector<T> &v, bool greater = false) {
    int n = v.size();
    vector<int> ret(n);
    iota(begin(ret), end(ret), 0);
    sort(begin(ret), end(ret), [&](int i, int j) { return greater ? v[i] > v[j] : v[i] < v[j]; });
    return ret;
}

template <typename T>
void reorder(vector<T> &a, const vector<int> &ord) {
    int n = a.size();
    vector<T> b(n);
    for (int i = 0; i < n; i++) b[i] = a[ord[i]];
    swap(a, b);
}

template <typename T>
T floor(T x, T y) {
    assert(y != 0);
    if (y < 0) x = -x, y = -y;
    return (x >= 0 ? x / y : (x - y + 1) / y);
}

template <typename T>
T ceil(T x, T y) {
    assert(y != 0);
    if (y < 0) x = -x, y = -y;
    return (x >= 0 ? (x + y - 1) / y : x / y);
}

template <typename S, typename T>
pair<S, T> operator+(const pair<S, T> &p, const pair<S, T> &q) {
    return make_pair(p.first + q.first, p.second + q.second);
}

template <typename S, typename T>
pair<S, T> operator-(const pair<S, T> &p, const pair<S, T> &q) {
    return make_pair(p.first - q.first, p.second - q.second);
}

template <typename S, typename T>
istream &operator>>(istream &is, pair<S, T> &p) {
    S a;
    T b;
    is >> a >> b;
    p = make_pair(a, b);
    return is;
}

template <typename S, typename T>
ostream &operator<<(ostream &os, const pair<S, T> &p) {
    return os << p.first << ' ' << p.second;
}

struct io_setup {
    io_setup() {
        ios_base::sync_with_stdio(false);
        cin.tie(NULL);
        cout << fixed << setprecision(15);
    }
} io_setup;

constexpr int inf = (1 << 30) - 1;
constexpr ll INF = (1LL << 60) - 1;
// constexpr int MOD = 1000000007;
constexpr int MOD = 998244353;

struct Runtime_Mod_Int {
    int x;

    Runtime_Mod_Int() : x(0) {}

    Runtime_Mod_Int(long long y) {
        x = y % get_mod();
        if (x < 0) x += get_mod();
    }

    static inline int &get_mod() {
        static int mod = 0;
        return mod;
    }

    static void set_mod(int md) { get_mod() = md; }

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

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

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

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

    Runtime_Mod_Int &operator++() { return *this += Runtime_Mod_Int(1); }

    Runtime_Mod_Int operator++(int) {
        Runtime_Mod_Int tmp = *this;
        ++*this;
        return tmp;
    }

    Runtime_Mod_Int &operator--() { return *this -= Runtime_Mod_Int(1); }

    Runtime_Mod_Int operator--(int) {
        Runtime_Mod_Int tmp = *this;
        --*this;
        return tmp;
    }

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

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

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

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

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

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

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

    Runtime_Mod_Int inverse() const {
        assert(*this != Runtime_Mod_Int(0));
        return pow(get_mod() - 2);
    }

    Runtime_Mod_Int pow(long long k) const {
        Runtime_Mod_Int now = *this, ret = 1;
        for (; k > 0; k >>= 1, now *= now) {
            if (k & 1) ret *= now;
        }
        return ret;
    }

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

    friend istream &operator>>(istream &is, Runtime_Mod_Int &p) {
        long long a;
        is >> a;
        p = Runtime_Mod_Int(a);
        return is;
    }
};

using mint = Runtime_Mod_Int;

template <typename T>
struct Matrix {
    vector<vector<T>> A;
    int n, m;

    Matrix(int n, int m) : A(n, vector<T>(m, 0)), n(n), m(m) {}

    inline const vector<T> &operator[](int k) const { return A[k]; }

    inline vector<T> &operator[](int k) { return A[k]; }

    static Matrix I(int l) {
        Matrix ret(l, l);
        for (int i = 0; i < l; i++) ret[i][i] = 1;
        return ret;
    }

    Matrix &operator*=(const Matrix &B) {
        assert(m == B.n);
        Matrix ret(n, B.m);
        for (int i = 0; i < n; i++) {
            for (int k = 0; k < m; k++) {
                for (int j = 0; j < B.m; j++) ret[i][j] += A[i][k] * B[k][j];
            }
        }
        swap(A, ret.A);
        m = B.m;
        return *this;
    }

    Matrix operator*(const Matrix &B) const { return Matrix(*this) *= B; }

    Matrix pow(long long k) const {
        assert(n == m);
        Matrix now = *this, ret = I(n);
        for (; k > 0; k >>= 1, now *= now) {
            if (k & 1) ret *= now;
        }
        return ret;
    }

    bool eq(const T &a, const T &b) const {
        return a == b;
        // return abs(a-b) <= EPS;
    }

    // 行基本変形を用いて簡約化を行い、(rank, det) の組を返す
    pair<int, T> row_reduction(vector<T> &b) {
        assert((int)b.size() == n);
        if (n == 0) return make_pair(0, m > 0 ? 0 : 1);
        int check = 0, rank = 0;
        T det = (n == m ? 1 : 0);
        for (int j = 0; j < m; j++) {
            int pivot = check;
            for (int i = check; i < n; i++) {
                if (A[i][j] != 0) pivot = i;
                // if(abs(A[i][j]) > abs(A[pivot][j])) pivot = i; // T が小数の場合はこちら
            }
            if (check != pivot) det *= T(-1);
            swap(A[check], A[pivot]), swap(b[check], b[pivot]);
            if (eq(A[check][j], T(0))) {
                det = T(0);
                continue;
            }
            rank++;
            det *= A[check][j];
            T r = T(1) / A[check][j];
            for (int k = j + 1; k < m; k++) A[check][k] *= r;
            b[check] *= r;
            A[check][j] = T(1);
            for (int i = 0; i < n; i++) {
                if (i == check) continue;
                if (!eq(A[i][j], 0)) {
                    for (int k = j + 1; k < m; k++) A[i][k] -= A[i][j] * A[check][k];
                    b[i] -= A[i][j] * b[check];
                }
                A[i][j] = T(0);
            }
            if (++check == n) break;
        }
        return make_pair(rank, det);
    }

    pair<int, T> row_reduction() {
        vector<T> b(n, T(0));
        return row_reduction(b);
    }

    // 行基本変形を行い、逆行列を求める
    pair<bool, Matrix> inverse() {
        if (n != m) return make_pair(false, Matrix(0, 0));
        if (n == 0) return make_pair(true, Matrix(0, 0));
        Matrix ret = I(n);
        for (int j = 0; j < n; j++) {
            int pivot = j;
            for (int i = j; i < n; i++) {
                if (A[i][j] != 0) pivot = i;
                // if(abs(A[i][j]) > abs(A[pivot][j])) pivot = i; // T が小数の場合はこちら
            }
            swap(A[j], A[pivot]), swap(ret[j], ret[pivot]);
            if (eq(A[j][j], T(0))) return make_pair(false, Matrix(0, 0));
            T r = T(1) / A[j][j];
            for (int k = j + 1; k < n; k++) A[j][k] *= r;
            for (int k = 0; k < n; k++) ret[j][k] *= r;
            A[j][j] = T(1);
            for (int i = 0; i < n; i++) {
                if (i == j) continue;
                if (!eq(A[i][j], T(0))) {
                    for (int k = j + 1; k < n; k++) A[i][k] -= A[i][j] * A[j][k];
                    for (int k = 0; k < n; k++) ret[i][k] -= A[i][j] * ret[j][k];
                }
                A[i][j] = T(0);
            }
        }
        return make_pair(true, ret);
    }

    // Ax = b の解の 1 つと解空間の基底の組を返す
    vector<vector<T>> Gaussian_elimination(vector<T> b) {
        row_reduction(b);
        vector<vector<T>> ret;
        vector<int> p(n, m);
        vector<bool> is_zero(m, true);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                if (!eq(A[i][j], T(0))) {
                    p[i] = j;
                    break;
                }
            }
            if (p[i] < m) {
                is_zero[p[i]] = false;
            } else if (!eq(b[i], T(0))) {
                return {};
            }
        }
        vector<T> x(m, T(0));
        for (int i = 0; i < n; i++) {
            if (p[i] < m) x[p[i]] = b[i];
        }
        ret.push_back(x);
        for (int j = 0; j < m; j++) {
            if (!is_zero[j]) continue;
            x[j] = T(1);
            for (int i = 0; i < n; i++) {
                if (p[i] < m) x[p[i]] = -A[i][j];
            }
            ret.push_back(x);
            x[j] = T(0);
        }
        return ret;
    }
};

template <typename T>
vector<T> Berlekamp_Massey(const vector<T> &a) {
    int n = a.size();
    vector<T> c = {-1}, c_pre = {0};
    int i_pre = -1;
    T x_pre = 1;
    for (int i = 0; i < n; i++) {
        int d = c.size(), d_pre = c_pre.size();
        T x = 0;
        for (int j = 0; j < d; j++) x += a[i - j] * c[j];
        if (x == 0) continue;
        T coef = -x / x_pre;
        if (d >= d_pre + i - i_pre) {
            for (int j = 0; j < d_pre; j++) c[i - i_pre + j] += coef * c_pre[j];
        } else {
            vector<T> memo = c;
            c.resize(d_pre + i - i_pre);
            for (int j = 0; j < d_pre; j++) { c[i - i_pre + j] += coef * c_pre[j]; }
            c_pre = memo, i_pre = i, x_pre = x;
        }
    }
    return c;
}

template <typename T>
struct Kitamasa {
    // d 項間線形漸化式 a[n] = c[1]*a[n-1]+c[2]*a[n-2]+...+c[d]*a[n-d]
    const vector<T> a, c;
    const int d;
    // f(x) := x^d-c[1]*x^(d-1)-...-c[d-1]*x-c[d]
    vector<T> f;

    Kitamasa(const vector<T> &a, const vector<T> &c) : a(a), c(c), d(a.size()) {
        f.resize(d + 1);
        f[d] = 1;
        for (int i = 1; i <= d; i++) f[d - i] = -c[i];
    }

    // p(x)*q(x) を f(x) で割った余り
    vector<T> mul(const vector<T> &p, const vector<T> &q) const {
        int n = p.size(), m = q.size();
        vector<T> ret(n + m - 1, 0);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) ret[i + j] += p[i] * q[j];
        }
        for (int i = n + m - 2; i >= d; i--) {
            for (int j = 1; j <= d; j++) ret[i - j] -= ret[i] * f[d - j];
        }
        ret.resize(d);
        return ret;
    }

    // p(x)^n を f(x) で割った余り
    vector<T> pow(vector<T> p, long long n) const {
        vector<T> ret(1, 0);
        ret[0] = 1;
        while (n) {
            if (n & 1) ret = mul(ret, p);
            p = mul(p, p), n >>= 1;
        }
        return ret;
    }

    // a[n] (0-indexed)
    T operator[](long long n) const {
        if (n < d) return a[n];
        vector<T> x(d + 1, 0);
        x[1] = 1, x = pow(x, n);
        T ret = 0;
        for (int i = 0; i < d; i++) ret += x[i] * a[i];
        return ret;
    }
};

void solve() {
    ll N, M, L, K, B;
    cin >> N >> M >> L >> K >> B;

    mint::set_mod(B);

    auto mul = [&](vector<mint> x) {
        vector<mint> y(L, 0);
        y[0] = x[0] + x[L - 1] * mint(M);
        rep2(i, 1, L) y[i] = x[i - 1] + x[i];
        return y;
    };

    vector comb(L + 1, vector(L + 1, mint(0)));
    comb[0][0] = 1;
    rep(i, L) {
        rep(j, i + 1) {
            comb[i + 1][j] += comb[i][j];
            comb[i + 1][j + 1] += comb[i][j];
        }
    }

    vector<mint> a(L);
    vector<mint> x(L, 0);
    x[0] = 1;
    rep(i, L) {
        a[i] = x[K];
        x = mul(x);
    }

    vector<mint> c(L + 1, 0);
    rep2(i, 1, L + 1) c[i] = comb[L][i] * (i & 1 ? 1 : -1);
    c[L] += M;

    // vector<mint> c = Berlekamp_Massey(a);
    // print(a), print(c);

    Kitamasa<mint> lr(a, c);
    cout << lr[N] << '\n';
}

int main() {
    int T = 1;
    // cin >> T;
    while (T--) solve();
}

yukicoder

結果

テストケース

ソースコード