結果
| 問題 |
No.8046 yukicoderの過去問
|
| コンテスト | |
| ユーザー |
 HEXAebmr
|
| 提出日時 | 2022-05-02 13:52:20 |
| 言語 | C++14 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 138 ms / 2,000 ms |
| コード長 | 20,293 bytes |
| コンパイル時間 | 4,110 ms |
| コンパイル使用メモリ | 209,936 KB |
| 実行使用メモリ | 21,280 KB |
| 最終ジャッジ日時 | 2024-07-01 16:49:27 |
| 合計ジャッジ時間 | 5,562 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge5 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| other | AC * 9 |
ソースコード
#include <algorithm>
#include <atcoder/all>
#include <cassert>
#include <chrono>
#include <cmath>
#include <complex>
#include <cstdint>
#include <fstream>
#include <functional>
#include <iomanip>
#include <iostream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <queue>
#include <random>
#include <regex>
#include <set>
#include <stack>
#include <string>
#include <tuple>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using namespace atcoder;
using ll = long long;
using ld = long double;
using pii = pair<int, int>;
using pll = pair<ll, ll>;
using vll = vector<ll>;
using vvll = vector<vector<ll>>;
using vvvll = vector<vector<vector<ll>>>;
using vii = vector<int>;
using vvii = vector<vector<int>>;
using vvvii = vector<vector<vector<int>>>;
using vdd = vector<ld>;
using vvdd = vector<vector<ld>>;
using vvvdd = vector<vector<vector<ld>>>;
using vbb = vector<bool>;
using vvbb = vector<vector<bool>>;
using vvvbb = vector<vector<vector<bool>>>;
using vpll = vector<pll>;
using vvpll = vector<vector<pll>>;
using vvvpll = vector<vector<vector<pll>>>;
using vmm1 = vector<modint1000000007>;
using vvmm1 = vector<vector<modint1000000007>>;
using vvvmm1 = vector<vector<vector<modint1000000007>>>;
using vmm2 = vector<modint998244353>;
using vvmm2 = vector<vector<modint998244353>>;
using vvvmm2 = vector<vector<vector<modint998244353>>>;
#define pb push_back
#define mp make_pair
#define sc second
#define fr first
#define endl '\n'
#define stpr std::fixed << setprecision
#define cYES cout << "YES" << endl
#define cNO cout << "NO" << endl
#define cYes cout << "Yes" << endl
#define cNo cout << "No" << endl
#define cerr cout << -1 << endl
#define rep(i, n) for (ll i = 0; i < (n); ++i)
#define drep(i, a, b, d) for (ll i = (a); i <= (b); i += d)
#define Rep(i, a, b) for (ll i = (a); i < (b); ++i)
#define rrep(i, n) for (ll i = n - 1; i >= 0; i--)
#define drrep(i, a, b, d) for (ll i = (a); i >= (b); i -= d)
#define rRep(i, a, b) for (ll i = a; i >= b; i--)
#define crep(i) for (char i = 'a'; i <= 'z'; ++i)
#define Crep(i) for (char i = 'A'; i <= 'Z'; ++i)
#define ALL(x) (x).begin(), (x).end()
#define rALL(x) (x).rbegin(), (x).rend()
#define sort2(A, N) \
sort(A, A + N, \
[](const pii &a, const pii &b) { return a.second < b.second; });
#define debug(v) \
cout << #v << ":"; \
for (auto x : v) { \
cout << x << ' '; \
} \
cout << endl;
int ctoi(const char c) {
if ('0' <= c && c <= '9') return (c - '0');
return -1;
}
ll gcd(ll a, ll b) { return (b == 0 ? a : gcd(b, a % b)); }
ll lcm(ll a, ll b) { return a * b / gcd(a, b); }
ll rup(ll a, ll b) { return a + (b - a % b) % b; }
ll rdc(ll a, ll b) { return a / b * b; }
constexpr ll INF = 1000000011;
constexpr ll MOD = 1000000007;
constexpr ll MOD2 = 998244353;
constexpr ll LINF = 1001002003004005006ll;
constexpr ld EPS = 10e-15;
// using mint = modint1000000007;
// using mint2 = modint998244353;
template <class T, class U>
inline bool chmax(T &lhs, const U &rhs) {
if (lhs < rhs) {
lhs = rhs;
return 1;
}
return 0;
}
template <class T, class U>
inline bool chmin(T &lhs, const U &rhs) {
if (lhs > rhs) {
lhs = rhs;
return 1;
}
return 0;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
for (auto &&x : v) is >> x;
return is;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
is >> p.first;
is >> p.second;
return is;
}
template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
os << p.first << ' ' << p.second;
return os;
}
template <class T>
ostream &operator<<(ostream &os, vector<T> &v) {
for (auto i = begin(v); i != end(v); ++i) {
if (i != begin(v)) os << ' ';
os << *i;
}
return os;
}
using int64 = long long;
// const int mod = 1e9 + 7;
const int mod = 998244353;
const int64 infll = (1LL << 62) - 1;
const int inf = (1 << 30) - 1;
template <typename T1, typename T2>
inline bool chmax(T1 &a, T2 b) {
return a < b && (a = b, true);
}
template <typename T1, typename T2>
inline bool chmin(T1 &a, T2 b) {
return a > b && (a = b, true);
}
template <typename T = int64>
vector<T> make_v(size_t a) {
return vector<T>(a);
}
template <typename T, typename... Ts>
auto make_v(size_t a, Ts... ts) {
return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));
}
template <typename T, typename V>
typename enable_if<is_class<T>::value == 0>::type fill_v(T &t, const V &v) {
t = v;
}
template <typename T, typename V>
typename enable_if<is_class<T>::value != 0>::type fill_v(T &t, const V &v) {
for (auto &e : t) fill_v(e, v);
}
template <typename F>
struct FixPoint : F {
FixPoint(F &&f) : F(forward<F>(f)) {}
template <typename... Args>
decltype(auto) operator()(Args &&...args) const {
return F::operator()(*this, forward<Args>(args)...);
}
};
template <typename F>
inline decltype(auto) MFP(F &&f) {
return FixPoint<F>{forward<F>(f)};
}
#line 1 "math/fft/fast-fourier-transform.cpp"
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
#line 2 "math/fft/arbitrary-mod-convolution.cpp"
/*
* @brief Arbitrary-Mod-Convolution(任意mod畳み込み)
*/
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;
}
};
#line 2 "math/fps/formal-power-series.cpp"
/**
* @brief Formal-Power-Series(形式的冪級数)
*/
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;
for (int i = 0; i < n; i++) {
if ((*this)[i] != T(0)) {
T rev = T(1) / (*this)[i];
P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
if (i * k > deg) return P(deg, T(0));
ret = (ret << (i * k)).pre(deg);
if (ret.size() < deg) ret.resize(deg, T(0));
return ret;
}
}
return *this;
}
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;
}
};
/**
* @brief Linear-Recursion-Formula
*/
template <template <typename> class FPS, typename Mint>
Mint linear_recursion_formula(FPS<Mint> P, FPS<Mint> Q, int64_t k) {
// compute the coefficient [x^k] P/Q of rational power series
Mint ret = 0;
if (P.size() >= Q.size()) {
auto R = P / Q;
P -= R * Q;
P.shrink();
if (k < (int)R.size()) ret += R[k];
}
if (P.empty()) return ret;
P.resize((int)Q.size() - 1);
while (k > 0) {
auto Q2 = Q;
for (int i = 1; i < (int)Q2.size(); i += 2) Q2[i] = -Q2[i];
auto S = P * Q2;
auto T = Q * Q2;
if (k & 1) {
for (int i = 1; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
for (int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
} else {
for (int i = 0; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
for (int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
}
k >>= 1;
}
return ret + P[0];
}
template <typename Mint>
using FPS = FormalPowerSeries<Mint>;
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 modint = ModInt<mod>;
using mint = ModInt<MOD>;
mint Bostan_Mori(FPS<mint> p, FPS<mint> q, ll k) {
if (k < 0) return 0;
p.resize(max(p.size(), q.size()));
q.resize(max(p.size(), q.size()));
while (k) {
FPS<mint> _q = q;
for (int i = 1; i < _q.size(); i += 2) _q[i] = -_q[i];
FPS<mint> v = q, u = p;
v *= _q;
u *= _q;
for (int i = k % 2; i < u.size(); i += 2) p[i / 2] = u[i];
for (int i = 0; i < v.size(); i += 2) q[i / 2] = v[i];
k /= 2;
}
return p[0] / q[0];
}
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
ll K, N;
cin >> K >> N;
vll A(N);
cin >> A;
FPS<mint> X(K + 1);
X[0] = 1;
rep(i, N) { X[A[i]] -= 1; }
X = X.inv();
cout << X[K] << endl;
}