#792031 (C++23) No.1025 Modular Equation

提出ソース
結果

問題	No.1025 Modular Equation
ユーザー	hitonanode
提出日時	2022-08-24 13:24:08
言語	C++23 (gcc 12.3.0 + boost 1.83.0)
結果	TLE
実行時間	-
コード長	14,373 bytes
コンパイル時間	4,495 ms
コンパイル使用メモリ	289,072 KB
実行使用メモリ	13,632 KB
最終ジャッジ日時	2024-10-11 22:42:24
合計ジャッジ時間	12,202 ms
ジャッジサーバーID （参考情報）	judge3 / judge1
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	2 ms 5,248 KB
testcase_01	AC	3 ms 5,248 KB
testcase_02	AC	65 ms 5,248 KB
testcase_03	AC	2 ms 5,248 KB
testcase_04	AC	20 ms 5,248 KB
testcase_05	AC	45 ms 5,248 KB
testcase_06	AC	76 ms 5,248 KB
testcase_07	AC	222 ms 5,248 KB
testcase_08	AC	217 ms 5,248 KB
testcase_09	AC	2 ms 5,248 KB
testcase_10	AC	3 ms 5,248 KB
testcase_11	TLE	-
testcase_12	--	-
testcase_13	--	-
testcase_14	--	-
testcase_15	--	-
testcase_16	--	-
testcase_17	--	-
testcase_18	--	-
testcase_19	--	-
testcase_20	--	-
testcase_21	--	-
testcase_22	--	-
testcase_23	--	-
testcase_24	--	-
testcase_25	--	-
testcase_26	--	-
testcase_27	--	-
testcase_28	--	-
testcase_29	--	-
testcase_30	--	-
testcase_31	--	-
testcase_32	--	-
testcase_33	--	-
testcase_34	--	-
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ソースコード

raw source code
// #pragma GCC optimize("O3", "unroll-loops")
// #pragma GCC target("avx")
#include <bits/stdc++.h>
using namespace std;
using lint = long long int;
using pint = pair<int, int>;
using plint = pair<lint, lint>;
struct fast_ios { fast_ios(){ cin.tie(0); ios::sync_with_stdio(false); cout << fixed << setprecision(20); }; } fast_ios_;
#define ALL(x) (x).begin(), (x).end()
#define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i<i##_end_;i++)
#define IFOR(i, begin, end) for(int i=(end)-1,i##_begin_=(begin);i>=i##_begin_;i--)
#define REP(i, n) FOR(i,0,n)
#define IREP(i, n) IFOR(i,0,n)
template<typename T> void ndarray(vector<T> &vec, int len) { vec.resize(len); }
template<typename T, typename... Args> void ndarray(vector<T> &vec, int len, Args... args) { vec.resize(len); for (auto &v : vec) ndarray(v, args...); }
template<typename T> bool chmax(T &m, const T q) { if (m < q) {m = q; return true;} else return false; }
template<typename T> bool chmin(T &m, const T q) { if (m > q) {m = q; return true;} else return false; }
template<typename T1, typename T2> pair<T1, T2> operator+(const pair<T1, T2> &l, const pair<T1, T2> &r) { return make_pair(l.first + r.first, l.second + r.second); }
template<typename T1, typename T2> pair<T1, T2> operator-(const pair<T1, T2> &l, const pair<T1, T2> &r) { return make_pair(l.first - r.first, l.second - r.second); }
template<typename T> istream &operator>>(istream &is, vector<T> &vec){ for (auto &v : vec) is >> v; return is; }
template<typename T> ostream &operator<<(ostream &os, const vector<T> &vec){ os << "["; for (auto v : vec) os << v << ","; os << "]"; return os; }
template<typename T> ostream &operator<<(ostream &os, const deque<T> &vec){ os << "deq["; for (auto v : vec) os << v << ","; os << "]"; return os; }
template<typename T> ostream &operator<<(ostream &os, const set<T> &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; }
template<typename T> ostream &operator<<(ostream &os, const unordered_set<T> &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; }
template<typename T> ostream &operator<<(ostream &os, const multiset<T> &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; }
template<typename T> ostream &operator<<(ostream &os, const unordered_multiset<T> &vec){ os << "{"; for (auto v : vec) os << v << ","; os << "}"; return os; }
template<typename T1, typename T2> ostream &operator<<(ostream &os, const pair<T1, T2> &pa){ os << "(" << pa.first << "," << pa.second << ")"; return os; }
template<typename TK, typename TV> ostream &operator<<(ostream &os, const map<TK, TV> &mp){ os << "{"; for (auto v : mp) os << v.first << "=>" << v.second << ","; os << "}"; return os; }
template<typename TK, typename TV> ostream &operator<<(ostream &os, const unordered_map<TK, TV> &mp){ os << "{"; for (auto v : mp) os << v.first << "=>" << v.second << ","; os << "}"; return os; }
#define dbg(x) cerr << #x << " = " << (x) << " (L" << __LINE__ << ") " << __FILE__ << endl;


template <int mod>
struct ModInt
{
    using lint = long long;
    static int get_mod() { return mod; }
    static int get_primitive_root() {
        static int primitive_root = 0;
        if (!primitive_root) {
            primitive_root = [&](){
                std::set<int> fac;
                int v = mod - 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 < mod; g++) {
                    bool ok = true;
                    for (auto i : fac) if (ModInt(g).power((mod - 1) / i) == 1) { ok = false; break; }
                    if (ok) return g;
                }
                return -1;
            }();
        }
        return primitive_root;
    }
    int val;
    constexpr ModInt() : val(0) {}
    constexpr ModInt &_setval(lint v) { val = (v >= mod ? v - mod : v); return *this; }
    constexpr ModInt(lint v) { _setval(v % mod + mod); }
    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 + mod); }
    constexpr ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % mod); }
    constexpr ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % mod); }
    constexpr ModInt operator-() const { return ModInt()._setval(mod - 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()._setval(a % mod + x.val); }
    friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % mod - x.val + mod); }
    friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.val % mod); }
    friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.inv() % mod); }
    constexpr bool operator==(const ModInt &x) const { return val == x.val; }
    constexpr bool operator!=(const ModInt &x) const { return val != x.val; }
    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; is >> t; x = ModInt(t); return is; }
    friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { os << x.val;  return os; }
    constexpr lint power(lint n) const {
        lint ans = 1, tmp = this->val;
        while (n) {
            if (n & 1) ans = ans * tmp % mod;
            tmp = tmp * tmp % mod;
            n /= 2;
        }
        return ans;
    }
    constexpr lint inv() const { return this->power(mod - 2); }
    constexpr ModInt operator^(lint n) const { return ModInt(this->power(n)); }
    constexpr ModInt &operator^=(lint n) { return *this = *this ^ n; }

    inline ModInt fac() const {
        static std::vector<ModInt> facs;
        int l0 = facs.size();
        if (l0 > this->val) return facs[this->val];

        facs.resize(this->val + 1);
        for (int i = l0; i <= this->val; i++) facs[i] = (i == 0 ? ModInt(1) : facs[i - 1] * ModInt(i));
        return facs[this->val];
    }

    ModInt doublefac() const {
        lint k = (this->val + 1) / 2;
        if (this->val & 1) return ModInt(k * 2).fac() / ModInt(2).power(k) / ModInt(k).fac();
        else return ModInt(k).fac() * ModInt(2).power(k);
    }

    ModInt nCr(const ModInt &r) const {
        if (this->val < r.val) return ModInt(0);
        return this->fac() / ((*this - r).fac() * r.fac());
    }

    ModInt sqrt() const {
        if (val == 0) return 0;
        if (mod == 2) return val;
        if (power((mod - 1) / 2) != 1) return 0;
        ModInt b = 1;
        while (b.power((mod - 1) / 2) == 1) b += 1;
        int e = 0, m = mod - 1;
        while (m % 2 == 0) m >>= 1, e++;
        ModInt x = power((m - 1) / 2), y = (*this) * x * x;
        x *= (*this);
        ModInt z = b.power(m);
        while (y != 1) {
            int j = 0;
            ModInt t = y;
            while (t != 1) j++, t *= t;
            z = z.power(1LL << (e - j - 1));
            x *= z, z *= z, y *= z;
            e = j;
        }
        return ModInt(std::min(x.val, mod - x.val));
    }
};
using mint = ModInt<1000000007>;

// 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>
vector<MODINT> nttconv(vector<MODINT> a, vector<MODINT> b, bool skip_garner = false);

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(vector<MODINT> &a, bool is_inverse = false)
{
    int n = a.size();
    assert(__builtin_popcount(n) == 1);
    MODINT h = MODINT(MODINT::get_primitive_root()).power((MODINT::get_mod() - 1) / n);
    if (is_inverse) h = 1 / h;

    int i = 0;
    for (int j = 1; j < n - 1; j++) {
        for (int k = n >> 1; k > (i ^= k); k >>= 1);
        if (j < i) swap(a[i], a[j]);
    }

    for (int m = 1; m < n; m *= 2) {
        int m2 = 2 * m;
        long long int base = h.power(n / m2);
        MODINT w(1);
        for(int x = 0; x < m; x++) {
            for (int s = x; s < n; s += m2) {
                MODINT u = a[s], d = a[s + m] * w;
                a[s] = u + d, a[s + m] = u - d;
            }
            w *= base;
        }
    }
    if (is_inverse) {
        long long int n_inv = MODINT(n).inv();
        for (auto &v : a) v *= n_inv;
    }
}
template<int MOD>
vector<ModInt<MOD>> nttconv_(const vector<int> &a, const vector<int> &b) {
    int sz = a.size();
    assert(a.size() == b.size() and __builtin_popcount(sz) == 1);
    vector<ModInt<MOD>> ap(sz), bp(sz);
    for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i];
    if (a == b) {
        ntt(ap, false);
        bp = ap;
    }
    else {
        ntt(ap, false);
        ntt(bp, false);
    }
    for (int i = 0; i < sz; i++) ap[i] *= bp[i];
    ntt(ap, true);
    return ap;
}
long long int extgcd_ntt_(long long int a, long long int b, long long int &x, long long int &y)
{
    long long int d = a;
    if (b != 0) d = extgcd_ntt_(b, a % b, y, x), y -= (a / b) * x;
    else x = 1, y = 0;
    return d;
}
long long int modinv_ntt_(long long int a, long long int m)
{
    long long int x, y;
    extgcd_ntt_(a, m, x, y);
    return (m + x % m) % m;
}
long long int garner_ntt_(int r0, int r1, int r2, int mod)
{
    array<long long int, 4> rs = {r0, r1, r2, 0};
    vector<long long int> coffs(4, 1), constants(4, 0);
    for (int i = 0; i < 3; i++) {
        long long int v = (rs[i] - constants[i]) * modinv_ntt_(coffs[i], nttprimes[i]) % nttprimes[i];
        if (v < 0) v += nttprimes[i];
        for (int j = i + 1; j < 4; j++) {
            (constants[j] += coffs[j] * v) %= (j < 3 ? nttprimes[j] : mod);
            (coffs[j] *= nttprimes[i]) %= (j < 3 ? nttprimes[j] : mod);
        }
    }
    return constants.back();
}
template <typename MODINT>
vector<MODINT> nttconv(vector<MODINT> a, vector<MODINT> b, bool skip_garner)
{
    int sz = 1, n = a.size(), m = b.size();
    while (sz < n + m) sz <<= 1;
    int mod = MODINT::get_mod();
    if (skip_garner or find(begin(nttprimes), end(nttprimes), mod) != 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 {
        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;
}

int P;
vector<mint> prod(const vector<mint> &v1, const vector<mint> &v2)
{
    auto ret = nttconv(v1, v2);
    FOR(i, P, ret.size()) ret[i % P] += ret[i];
    ret.resize(P, 0);
    return ret;
}

vector<mint> vpow(vector<mint> v, int n)
{
    vector<mint> ret(P);
    ret[0] = 1;
    while (n)
    {
        if (n & 1)
        {
            ret = prod(ret, v);
        }
        n >>= 1;
        if (!n) return ret;
        v = prod(v, v);
    }
    return ret;
}

lint power(lint x, lint n, lint MOD)
{
    lint ans = 1;
    while (n>0)
    {
        if (n & 1) (ans *= x) %= MOD;
        (x *= x) %= MOD;
       n >>= 1;
    }
   return ans;
}

int main()
{
    int N, K, B;
    cin >> P >> N >> K >> B;
    if (__gcd(K, P - 1) == 1)
    {
        cout << mint(P).power(N - 1) << "\n";
        return 0;
    }

    vector<int> A(N);
    cin >> A;
    // map<int, int> acnt;
    // for (auto a : A) acnt[a]++;
    vector<mint> f(P);
    REP(x, P) f[power(x, K, P)] += 1;
    vector<int> nonzero_is;
    REP(i, P) if (f[i].val != 0) nonzero_is.push_back(i);

    // int nb_on = 0;
    // REP(i, f.size()) nb_on += (f[i].val > 0);

    vector<int> mini2a(P, -1), minicnt(P);
    for (auto a : A) {
        lint mini = P;
        for (int i : nonzero_is) {
            if (i) chmin(mini, 1LL * i * a % P);
        }
        mini2a.at(mini) = a;
        ++minicnt.at(mini);
    }
    if (P <= 100) dbg(minicnt);
    if (P <= 100) dbg(mini2a);
    if (P <= 100) dbg(f);

    vector<mint> dp(P);
    dp[0] = 1;

    REP(p, P) {
        if (!minicnt.at(p)) continue;
        vector<mint> g(P);
        for (auto i : nonzero_is) g[1LL * i * mini2a[p] % P] += f[i];
        if (P <= 100) dbg(g);
        dp = prod(dp, vpow(g, minicnt[p]));
    }
    if (P <= 100) dbg(dp);

    // if (nb_on < 20)
    // {
    //     for (auto a : A)
    //     {
    //         lint mini = P;
    //         REP(i, P) chmin(mini, 1LL * i * 1 % P);
    //         vector<mint> dpnxt(P);
    //         REP(i, P) if (f[i].val)
    //         {
    //             int ad = 1LL * i * a % P;
    //             REP(j, P) dpnxt[ad + j - (ad + j >= P ? P : 0)] += dp[j] * f[i];
    //         }
    //         dp = dpnxt;
    //     }
    // }
    // else
    // {
    //     map<vector<mint>, int> mp;
    //     for (auto p : acnt)
    //     {
    //         int a = p.first;
    //         vector<mint> g(P);
    //         REP(i, P) g[1LL * i * a % P] += f[i];
    //         dp = prod(dp, vpow(g, p.second));
    //         // mp[g] += p.second;
    //     }
    //     // for (auto p : mp)
    //     // {
    //     //     dp = prod(dp, vpow(p.first, p.second));
    //     // }
    // }
    cout << dp[B] << "\n";
}
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結果

テストケース

ソースコード
