#792987 (C++17) No.2062 Sum of Subset mod 999630629

提出ソース
結果

問題	No.2062 Sum of Subset mod 999630629
コンテスト
ユーザー	Forested
提出日時	2022-08-26 22:33:48
言語	C++17 (gcc 13.3.0 + boost 1.87.0)
結果	WA
実行時間	-
コード長	20,130 bytes
コンパイル時間	2,209 ms
コンパイル使用メモリ	155,644 KB
最終ジャッジ日時	2025-01-31 05:13:03
ジャッジサーバーID （参考情報）	judge4 / judge2
このコードへのチャレンジ
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ファイルパターン	結果
sample	AC * 3
other	AC * 8 WA * 21
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ソースコード

raw source code
#ifndef LOCAL
#define FAST_IO
#endif

// ===== template.hpp =====
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <string>
#include <tuple>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

#define OVERRIDE(a, b, c, d, ...) d
#define REP2(i, n) for (i32 i = 0; i < (i32) (n); ++i)
#define REP3(i, m, n) for (i32 i = (i32) (m); i < (i32) (n); ++i)
#define REP(...) OVERRIDE(__VA_ARGS__, REP3, REP2)(__VA_ARGS__)
#define PER(i, n) for (i32 i = (i32) (n) - 1; i >= 0; --i)
#define ALL(x) begin(x), end(x)

using namespace std;

using u32 = unsigned int;
using u64 = unsigned long long;
using u128 = __uint128_t;
using i32 = signed int;
using i64 = signed long long;
using i128 = __int128_t;
using f64 = double;
using f80 = long double;

template <typename T>
using Vec = vector<T>;

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

istream &operator>>(istream &is, i128 &x) {
    i64 v;
    is >> v;
    x = v;
    return is;
}
ostream &operator<<(ostream &os, i128 x) {
    os << (i64) x;
    return os;
}
istream &operator>>(istream &is, u128 &x) {
    u64 v;
    is >> v;
    x = v;
    return is;
}
ostream &operator<<(ostream &os, u128 x) {
    os << (u64) x;
    return os;
}

[[maybe_unused]] constexpr i32 INF = 1000000100;
[[maybe_unused]] constexpr i64 INF64 = 3000000000000000100;
struct SetUpIO {
    SetUpIO() {
#ifdef FAST_IO
        ios::sync_with_stdio(false);
        cin.tie(nullptr);
#endif
        cout << fixed << setprecision(15);
    }
} set_up_io;
// ===== template.hpp =====

#ifdef DEBUGF
#include "cpl/template/debug.hpp"
#else
#define DBG(x) (void) 0
#endif

// ===== number_theoretic_transform.hpp =====

#include <array>
#include <vector>

// ===== mod_int.hpp =====

#include <cassert>
#include <iostream>
#include <type_traits>

// ===== utils.hpp =====

constexpr bool is_prime(unsigned n) {
    if (n == 0 || n == 1) {
        return false;
    }
    for (unsigned i = 2; i * i <= n; ++i) {
        if (n % i == 0) {
            return false;
        }
    }
    return true;
}

constexpr unsigned mod_pow(unsigned x, unsigned y, unsigned mod) {
    unsigned ret = 1, self = x;
    while (y != 0) {
        if (y & 1) {
            ret = (unsigned) ((unsigned long long) ret * self % mod);
        }
        self = (unsigned) ((unsigned long long) self * self % mod);
        y /= 2;
    }
    return ret;
}

template <unsigned mod>
constexpr unsigned primitive_root() {
    static_assert(is_prime(mod), "`mod` must be a prime number.");
    if (mod == 2) {
        return 1;
    }

    unsigned primes[32] = {};
    int it = 0;
    {
        unsigned m = mod - 1;
        for (unsigned i = 2; i * i <= m; ++i) {
            if (m % i == 0) {
                primes[it++] = i;
                while (m % i == 0) {
                    m /= i;
                }
            }
        }
        if (m != 1) {
            primes[it++] = m;
        }
    }
    for (unsigned i = 2; i < mod; ++i) {
        bool ok = true;
        for (int j = 0; j < it; ++j) {
            if (mod_pow(i, (mod - 1) / primes[j], mod) == 1) {
                ok = false;
                break;
            }
        }
        if (ok)
            return i;
    }
    return 0;
}

// y >= 1
template <typename T>
constexpr T safe_mod(T x, T y) {
    x %= y;
    if (x < 0) {
        x += y;
    }
    return x;
}

// y != 0
template <typename T>
constexpr T floor_div(T x, T y) {
    if (y < 0) {
        x *= -1;
        y *= -1;
    }
    if (x >= 0) {
        return x / y;
    } else {
        return -((-x + y - 1) / y);
    }
}

// y != 0
template <typename T>
constexpr T ceil_div(T x, T y) {
    if (y < 0) {
        x *= -1;
        y *= -1;
    }
    if (x >= 0) {
        return (x + y - 1) / y;
    } else {
        return -(-x / y);
    }
}
// ===== utils.hpp =====

template <unsigned mod>
class ModInt {
    static_assert(mod != 0, "`mod` must not be equal to 0.");
    static_assert(
        mod < (1u << 31),
        "`mod` must be less than (1u << 31) = 2147483648.");

    unsigned val;

public:
    constexpr ModInt() : val(0) {}
    template <typename T, std::enable_if_t<std::is_signed_v<T>> * = nullptr>
    constexpr ModInt(T x) : val((unsigned) ((long long) x % (long long) mod + (x < 0 ? mod : 0))) {}
    template <typename T, std::enable_if_t<std::is_unsigned_v<T>> * = nullptr>
    constexpr ModInt(T x) : val((unsigned) (x % mod)) {}

    static constexpr ModInt raw(unsigned x) {
        ModInt<mod> ret;
        ret.val = x;
        return ret;
    }

    constexpr unsigned get_val() const {
        return val;
    }

    constexpr ModInt operator+() const {
        return *this;
    }
    constexpr ModInt operator-() const {
        return ModInt<mod>(0u) - *this;
    }

    constexpr ModInt &operator+=(const ModInt &rhs) {
        val += rhs.val;
        if (val >= mod)
            val -= mod;
        return *this;
    }
    constexpr ModInt &operator-=(const ModInt &rhs) {
        if (val < rhs.val)
            val += mod;
        val -= rhs.val;
        return *this;
    }
    constexpr ModInt &operator*=(const ModInt &rhs) {
        val = (unsigned long long)val * rhs.val % mod;
        return *this;
    }
    constexpr ModInt &operator/=(const ModInt &rhs) {
        val = (unsigned long long)val * rhs.inv().val % mod;
        return *this;
    }

    friend constexpr ModInt operator+(const ModInt &lhs, const ModInt &rhs) {
        return ModInt<mod>(lhs) += rhs;
    }
    friend constexpr ModInt operator-(const ModInt &lhs, const ModInt &rhs) {
        return ModInt<mod>(lhs) -= rhs;
    }
    friend constexpr ModInt operator*(const ModInt &lhs, const ModInt &rhs) {
        return ModInt<mod>(lhs) *= rhs;
    }
    friend constexpr ModInt operator/(const ModInt &lhs, const ModInt &rhs) {
        return ModInt<mod>(lhs) /= rhs;
    }

    constexpr ModInt pow(unsigned long long x) const {
        ModInt<mod> ret = ModInt<mod>::raw(1);
        ModInt<mod> self = *this;
        while (x != 0) {
            if (x & 1)
                ret *= self;
            self *= self;
            x >>= 1;
        }
        return ret;
    }
    constexpr ModInt inv() const {
        static_assert(is_prime(mod), "`mod` must be a prime number.");
        assert(val != 0);
        return this->pow(mod - 2);
    }

    friend std::istream &operator>>(std::istream &is, ModInt<mod> &x) {
        is >> x.val;
        x.val %= mod;
        return is;
    }

    friend std::ostream &operator<<(std::ostream &os, const ModInt<mod> &x) {
        os << x.val;
        return os;
    }

    friend bool operator==(const ModInt &lhs, const ModInt &rhs) {
        return lhs.val == rhs.val;
    }
    
    friend bool operator!=(const ModInt &lhs, const ModInt &rhs) {
        return lhs.val != rhs.val;
    }
};

[[maybe_unused]] constexpr unsigned mod998244353 = 998244353;
[[maybe_unused]] constexpr unsigned mod1000000007 = 1000000007;

// ===== mod_int.hpp =====
// ===== bitop.hpp =====

template <typename T, typename U>
bool ith_bit(T n, U i) {
    return (n & ((T) 1 << i)) != 0;
}

int popcount(int x) {
    return __builtin_popcount(x);
}
unsigned popcount(unsigned x) {
    return __builtin_popcount(x);
}
long long popcount(long long x) {
    return __builtin_popcountll(x);
}
unsigned long long popcount(unsigned long long x) {
    return __builtin_popcountll(x);
}

// x must not be 0
int clz(int x) {
    return __builtin_clz(x);
}
unsigned clz(unsigned x) {
    return __builtin_clz(x);
}
long long clz(long long x) {
    return __builtin_clzll(x);
}
unsigned long long clz(unsigned long long x) {
    return __builtin_clzll(x);
}

// x must not be 0
int ctz(int x) {
    return __builtin_ctz(x);
}
unsigned ctz(unsigned int x) {
    return __builtin_ctz(x);
}
long long ctz(long long x) {
    return __builtin_ctzll(x);
}
unsigned long long ctz(unsigned long long x) {
    return __builtin_ctzll(x);
}

int floor_log2(int x) {
    return 31 - clz(x);
}
unsigned floor_log2(unsigned x) {
    return 31 - clz(x);
}
long long floor_log2(long long x) {
    return 63 - clz(x);
}
unsigned long long floor_log2(unsigned long long x) {
    return 63 - clz(x);
}

template <typename T>
T mask_n(T x, T n) {
    T mask = ((T) 1 << n) - 1;
    return x & mask;
}
// ===== bitop.hpp =====

template <unsigned mod>
class NumberTheoreticTransform {
    static constexpr int calc_ex() {
        unsigned m = mod - 1;
        int ret = 0;
        while (!(m & 1)) {
            m >>= 1;
            ++ret;
        }
        return ret;
    }

    static constexpr int max_ex = calc_ex();

    std::array<ModInt<mod>, max_ex + 1> root;
    std::array<ModInt<mod>, max_ex + 1> inv_root;
    std::array<ModInt<mod>, max_ex + 2> inc;
    std::array<ModInt<mod>, max_ex + 2> inv_inc;

public:
    void dft(std::vector<ModInt<mod>> &a) const {
        int n = (int) a.size();
        int ex = ctz(n);
        for (int i = 0; i < ex; ++i) {
            int pr = 1 << (ex - 1 - i);
            int len = 1 << i;
            ModInt<mod> zeta(1);
            for (int j = 0; j < len; ++j) {
                int offset = j << (ex - i);
                for (int k = 0; k < pr; ++k) {
                    ModInt<mod> l = a[offset + k];
                    ModInt<mod> r = a[offset + k + pr] * zeta;
                    a[offset + k] = l + r;
                    a[offset + k + pr] = l - r;
                }
                zeta *= inc[ctz(~j)];
            }
        }
    }

    void idft(std::vector<ModInt<mod>> &a) const {
        int n = (int) a.size();
        int ex = ctz(n);
        for (int i = ex - 1; i >= 0; --i) {
            int pr = 1 << (ex - 1 - i);
            int len = 1 << i;
            ModInt<mod> zeta(1);
            for (int j = 0; j < len; ++j) {
                int offset = j << (ex - i);
                for (int k = 0; k < pr; ++k) {
                    ModInt<mod> l = a[offset + k];
                    ModInt<mod> r = a[offset + k + pr];
                    a[offset + k] = l + r;
                    a[offset + k + pr] = (l - r) * zeta;
                }
                zeta *= inv_inc[ctz(~j)];
            }
        }
        ModInt<mod> inv = ModInt<mod>::raw((unsigned) a.size()).inv();
        for (ModInt<mod> &ele : a) {
            ele *= inv;
        }
    }

    constexpr NumberTheoreticTransform() : root(), inv_root() {
        ModInt<mod> g = ModInt<mod>::raw(primitive_root<mod>()).pow((mod - 1) >> max_ex);
        root[max_ex] = g;
        inv_root[max_ex] = g.inv();
        for (int i = max_ex; i > 0; --i) {
            root[i - 1] = root[i] * root[i];
            inv_root[i - 1] = inv_root[i] * inv_root[i];
        }
        ModInt<mod> prod(1);
        for (int i = 2; i <= max_ex; ++i) {
            inc[i - 2] = root[i] * prod;
            prod *= inv_root[i];
        }
        prod = ModInt<mod>(1);
        for (int i = 2; i <= max_ex; ++i) {
            inv_inc[i - 2] = inv_root[i] * prod;
            prod *= root[i];
        }
    }

    std::vector<ModInt<mod>> multiply(
        std::vector<ModInt<mod>> a,
        std::vector<ModInt<mod>> b) const {
        if (a.empty() || b.empty())
            return std::vector<ModInt<mod>>();
        int siz = 1;
        int s = (int) (a.size() + b.size());
        while (siz < s) {
            siz <<= 1;
        }
        a.resize(siz, ModInt<mod>());
        b.resize(siz, ModInt<mod>());
        dft(a);
        dft(b);
        for (int i = 0; i < siz; ++i) {
            a[i] *= b[i];
        }
        idft(a);
        a.resize(s - 1);
        return a;
    }
};

template <unsigned mod>
class NTTMul {
    static constexpr NumberTheoreticTransform<mod> ntt = NumberTheoreticTransform<mod>();

public:
    static void dft(std::vector<ModInt<mod>> &a) {
        ntt.dft(a);
    }

    static void idft(std::vector<ModInt<mod>> &a) {
        ntt.idft(a);
    }

    static std::vector<ModInt<mod>> mul(
        std::vector<ModInt<mod>> lhs,
        std::vector<ModInt<mod>> rhs) {
        return ntt.multiply(std::move(lhs), std::move(rhs));
    }
};

// ===== number_theoretic_transform.hpp =====
// ===== fps_exp.hpp =====

// ===== polynomial.hpp =====

#include <vector>
#include <utility>
#include <cassert>
#include <algorithm>

template <typename T, typename Mul>
class Polynomial {
    std::vector<T> coeff;
    
public:
    using This = Polynomial<T, Mul>;
    
    Polynomial() : coeff() {}
    Polynomial(int n) : coeff(n, T(0)) {}
    Polynomial(std::vector<T> c) : coeff(std::move(c)) {}
    
    const std::vector<T> &vec() const {
        return coeff;
    }
    
    int size() const {
        return (int) coeff.size();
    }
    
    const T &operator[](int idx) const {
        return coeff[idx];
    }
    T &operator[](int idx) {
        return coeff[idx];
    }
    
    T at(int idx) const {
        if (idx < size()) {
            return coeff[idx];
        } else {
            return T(0);
        }
    }
    
    void pre_(int n) {
        assert(n >= 0);
        coeff.resize(n, T(0));
    }
    This pre(int n) const {
        This tmp(*this);
        tmp.pre_(n);
        return tmp;
    }
    
    T operator()(const T &x) const {
        T p(1), sum(0);
        for (const T &ele : coeff) {
            sum += p * ele;
            p *= x;
        }
        return sum;
    }
    
    This &operator+=(const This &rhs) {
        if (coeff.size() < rhs.coeff.size()) {
            coeff.resize(rhs.coeff.size(), T(0));
        }
        for (int i = 0; i < (int) rhs.coeff.size(); ++i) {
            coeff[i] += rhs.coeff[i];
        }
        return *this;
    }
    friend This operator+(This lhs, const This &rhs) {
        lhs += rhs;
        return lhs;
    }
    This &operator-=(const This &rhs) {
        if (coeff.size() < rhs.coeff.size()) {
            coeff.resize(rhs.coeff.size(), T(0));
        }
        for (int i = 0; i < (int) rhs.coeff.size(); ++i) {
            coeff[i] -= rhs.coeff[i];
        }
        return *this;
    }
    friend This operator-(This lhs, const This &rhs) {
        lhs -= rhs;
        return lhs;
    }
    
    This &operator*=(This rhs) {
        coeff = Mul::mul(std::move(coeff), std::move(rhs.coeff));
        return *this;
    }
    friend This operator*(This lhs, This rhs) {
        return This(Mul::mul(std::move(lhs.coeff), std::move(rhs.coeff)));
    }
    
    This diff() const {
        if (coeff.empty()) {
            return This();
        }
        std::vector<T> c(coeff.size() - 1);
        for (int i = 0; i < (int) c.size(); ++i) {
            c[i] = T(i + 1) * coeff[i + 1];
        }
        return This(c);
    }
    This integ() const {
        std::vector<T> c(coeff.size() + 1, T(0));
        for (int i = 0; i < (int) coeff.size(); ++i) {
            c[i + 1] = coeff[i] / T(i + 1);
        }
        return This(c);
    }
};
// ===== polynomial.hpp =====

template <typename T, typename Mul>
Polynomial<T, Mul> fps_exp(const Polynomial<T, Mul> &h, int sz = -1) {
    const std::vector<T> &coeff = h.vec();
    assert(!coeff.empty() && coeff[0] == T(0));
    if (sz == -1) {
        sz = (int) coeff.size();
    }
    assert(sz >= 0);
    std::vector<T> f({T(1)});
    std::vector<T> g({T(1)});
    std::vector<T> dft_f_({T(1), T(1)});
    
    while ((int) f.size() < sz) {
        int n = (int) f.size();
        
        // F_{2n}(g_0)
        std::vector<T> dft_g_2 = g;
        dft_g_2.resize(2 * n, T(0));
        Mul::dft(dft_g_2);
        
        // \delta
        std::vector<T> delta(n, T(0));
        for (int i = 0; i < n; ++i) {
            delta[i] = dft_f_[i] * dft_g_2[i];
        }
        Mul::idft(delta);
        delta.resize(2 * n);
        for (int i = 0; i < n; ++i) {
            std::swap(delta[i], delta[n + i]);
        }
        delta[n] -= T(1);
        
        // F_n(D(f_0))
        std::vector<T> dft_d_f(n, T(0));
        for (int i = 0; i < n - 1; ++i) {
            dft_d_f[i] = T(i + 1) * f[i + 1];
        }
        Mul::dft(dft_d_f);
        
        // D(f_0) g_0
        std::vector<T> d_f_g(n, T(0));
        for (int i = 0; i < n; ++i) {
            d_f_g[i] = dft_d_f[i] * dft_g_2[i];
        }
        Mul::idft(d_f_g);
        d_f_g.resize(2 * n, T(0));
        for (int i = 0; i < n - 1; ++i) {
            T tmp = T(i + 1) * h.at(i + 1);
            d_f_g[n + i] = d_f_g[i] - tmp;
            d_f_g[i] = tmp;
        }
        
        // \delta D(h_0)
        std::vector<T> dft_delta = delta;
        Mul::dft(dft_delta);
        std::vector<T> delta_d_h(2 * n);
        for (int i = 0; i < n - 1; ++i) {
            delta_d_h[i] = T(i + 1) * h.at(i + 1);
        }
        Mul::dft(delta_d_h);
        for (int i = 0; i < 2 * n; ++i) {
            delta_d_h[i] *= dft_delta[i];
        }
        Mul::idft(delta_d_h);
        std::fill(delta_d_h.begin(), delta_d_h.begin() + n, T(0));
        
        // \epsilon
        std::vector<T> eps = std::move(d_f_g);
        for (int i = 0; i < 2 * n; ++i) {
            eps[i] -= T(i + 1) * h.at(i + 1) + delta_d_h[i];
        }
        for (int i = 2 * n - 1; i > 0; --i) {
            eps[i] = eps[i - 1] / T(i);
        }
        eps[0] = T(0);
        
        // \epsilon f_0
        std::vector<T> dft_eps = eps;
        Mul::dft(dft_eps);
        std::vector<T> eps_f(2 * n);
        for (int i = 0; i < 2 * n; ++i) {
            eps_f[i] = dft_eps[i] * dft_f_[i];
        }
        Mul::idft(eps_f);
        std::fill(eps_f.begin(), eps_f.begin() + n - 1, T(0));
        
        // update f
        f.resize(2 * n, T(0));
        for (int i = 0; i < 2 * n; ++i) {
            f[i] -= eps_f[i];
        }
        
        if ((int) f.size() >= sz) {
            break;
        }
        
        // update F_{2n}(f)
        dft_f_ = f;
        dft_f_.resize(4 * n);
        Mul::dft(dft_f_);
        
        // update g
        std::vector<T> fg(dft_f_.begin(), dft_f_.begin() + 2 * n);
        for (int i = 0; i < 2 * n; ++i) {
            fg[i] *= dft_g_2[i];
        }
        Mul::idft(fg);
        std::fill(fg.begin(), fg.begin() + n, T(0));
        Mul::dft(fg);
        for (int i = 0; i < 2 * n; ++i) {
            fg[i] *= dft_g_2[i];
        }
        Mul::idft(fg);
        g.resize(2 * n);
        for (int i = n; i < 2 * n; ++i) {
            g[i] = -fg[i];
        }
    }
    
    f.resize(sz);
    return Polynomial<T, Mul>(f);
}
// ===== fps_exp.hpp =====

constexpr u32 MOD = 998244353;
using Mint = ModInt<MOD>;
using FPS = Polynomial<Mint, NTTMul<MOD>>;

int main() {
    i32 n;
    cin >> n;
    Vec<i32> a(n);
    REP(i, n) {
        cin >> a[i];
    }
    
    Vec<i32> b(n);
    REP(i, n) {
        b[i] = 10000 - a[i];
    }
    
    constexpr i32 TGT = 369371;
    
    Vec<Mint> cnt(TGT + 1);
    REP(i, n) {
        cnt[b[i]] += Mint(1);
    }
    FPS plog(TGT + 1);
    REP(i, 1, TGT + 1) {
        if (cnt[i] == Mint()) {
            continue;
        }
        //cerr << i << ' ' << cnt[i] << '\n';
        for (i32 j = 1; i * j <= TGT; ++j) {
            if (j % 2 == 1) {
                plog[i * j] += cnt[i] / Mint(j);
            } else {
                plog[i * j] -= cnt[i] / Mint(j);
            }
        }
    }
    Mint p = Mint(2).pow(cnt[0].get_val());
    REP(i, TGT + 1) {
        plog[i] *= p;
    }
    FPS pexp = fps_exp(plog);
    REP(i, TGT + 1) {
        if (pexp[i] != Mint()) {
            cerr << i << ' ' << pexp[i] << '\n';
        }
    }
    Mint sum;
    REP(i, TGT + 1) {
        if (10000 * n - i >= 999630629) {
            sum += pexp[i];
        }
    }
    cerr << sum << '\n';
    
    Mint ans;
    REP(i, n) {
        ans += Mint(a[i]) * Mint(2).pow(n - 1);
    }
    ans -= Mint(999630629) * sum;
    cout << ans << '\n';
}
yukicoder

結果

ソースコード
