結果
| 問題 | No.1670 Many Gacha |
| ユーザー |
 hitonanode
|
| 提出日時 | 2021-09-04 22:35:28 |
| 言語 | C++23 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 42 ms / 2,000 ms |
| コード長 | 25,386 bytes |
| コンパイル時間 | 3,306 ms |
| コンパイル使用メモリ | 216,472 KB |
| 実行使用メモリ | 7,812 KB |
| 最終ジャッジ日時 | 2024-06-01 11:25:18 |
| 合計ジャッジ時間 | 5,706 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge4 |
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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
6,812 KB |
| testcase_01 | AC | 2 ms
6,940 KB |
| testcase_02 | AC | 2 ms
6,940 KB |
| testcase_03 | AC | 2 ms
6,944 KB |
| testcase_04 | AC | 2 ms
6,944 KB |
| testcase_05 | AC | 2 ms
6,940 KB |
| testcase_06 | AC | 2 ms
6,940 KB |
| testcase_07 | AC | 2 ms
6,940 KB |
| testcase_08 | AC | 40 ms
7,768 KB |
| testcase_09 | AC | 41 ms
7,792 KB |
| testcase_10 | AC | 40 ms
7,784 KB |
| testcase_11 | AC | 11 ms
7,024 KB |
| testcase_12 | AC | 11 ms
6,944 KB |
| testcase_13 | AC | 11 ms
6,940 KB |
| testcase_14 | AC | 35 ms
7,520 KB |
| testcase_15 | AC | 29 ms
7,212 KB |
| testcase_16 | AC | 32 ms
7,336 KB |
| testcase_17 | AC | 18 ms
7,240 KB |
| testcase_18 | AC | 28 ms
7,172 KB |
| testcase_19 | AC | 19 ms
7,260 KB |
| testcase_20 | AC | 38 ms
7,532 KB |
| testcase_21 | AC | 30 ms
7,404 KB |
| testcase_22 | AC | 31 ms
7,416 KB |
| testcase_23 | AC | 37 ms
7,512 KB |
| testcase_24 | AC | 21 ms
6,940 KB |
| testcase_25 | AC | 32 ms
7,372 KB |
| testcase_26 | AC | 21 ms
7,084 KB |
| testcase_27 | AC | 21 ms
6,952 KB |
| testcase_28 | AC | 19 ms
7,160 KB |
| testcase_29 | AC | 30 ms
7,464 KB |
| testcase_30 | AC | 21 ms
7,120 KB |
| testcase_31 | AC | 25 ms
7,208 KB |
| testcase_32 | AC | 21 ms
7,096 KB |
| testcase_33 | AC | 39 ms
7,720 KB |
| testcase_34 | AC | 2 ms
6,940 KB |
| testcase_35 | AC | 11 ms
6,944 KB |
| testcase_36 | AC | 42 ms
7,812 KB |
ソースコード
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <chrono>
#include <cmath>
#include <complex>
#include <deque>
#include <forward_list>
#include <fstream>
#include <functional>
#include <iomanip>
#include <ios>
#include <iostream>
#include <limits>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <type_traits>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using lint = long long;
using pint = pair<int, int>;
using plint = pair<lint, lint>;
struct fast_ios { fast_ios(){ cin.tie(nullptr), 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, typename V>
void ndarray(vector<T>& vec, const V& val, int len) { vec.assign(len, val); }
template <typename T, typename V, typename... Args> void ndarray(vector<T>& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); }
template <typename T> bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; }
template <typename T> bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; }
int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); }
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> vector<T> sort_unique(vector<T> vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; }
template <typename T> int arglb(const std::vector<T> &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); }
template <typename T> int argub(const std::vector<T> &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); }
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, size_t sz> ostream &operator<<(ostream &os, const array<T, sz> &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; }
#if __cplusplus >= 201703L
template <typename... T> istream &operator>>(istream &is, tuple<T...> &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; }
template <typename... T> ostream &operator<<(ostream &os, const tuple<T...> &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; }
#endif
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, typename TH> ostream &operator<<(ostream &os, const unordered_set<T, TH> &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, typename TH> ostream &operator<<(ostream &os, const unordered_map<TK, TV, TH> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
#ifdef HITONANODE_LOCAL
const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m";
#define dbg(x) cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl
#define dbgif(cond, x) ((cond) ? cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl : cerr)
#else
#define dbg(x) (x)
#define dbgif(cond, x) 0
#endif
#include <iostream>
#include <set>
#include <vector>
template <int md> struct ModInt {
#if __cplusplus >= 201402L
#define MDCONST constexpr
#else
#define MDCONST
#endif
using lint = long long;
MDCONST static int mod() { return md; }
static int get_primitive_root() {
static int primitive_root = 0;
if (!primitive_root) {
primitive_root = [&]() {
std::set<int> fac;
int v = md - 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 < md; g++) {
bool ok = true;
for (auto i : fac)
if (ModInt(g).pow((md - 1) / i) == 1) {
ok = false;
break;
}
if (ok) return g;
}
return -1;
}();
}
return primitive_root;
}
int val;
MDCONST ModInt() : val(0) {}
MDCONST ModInt &_setval(lint v) { return val = (v >= md ? v - md : v), *this; }
MDCONST ModInt(lint v) { _setval(v % md + md); }
MDCONST explicit operator bool() const { return val != 0; }
MDCONST ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); }
MDCONST ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val - x.val + md); }
MDCONST ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % md); }
MDCONST ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % md); }
MDCONST ModInt operator-() const { return ModInt()._setval(md - val); }
MDCONST ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
MDCONST ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
MDCONST ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
MDCONST ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
friend MDCONST ModInt operator+(lint a, const ModInt &x) { return ModInt()._setval(a % md + x.val); }
friend MDCONST ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % md - x.val + md); }
friend MDCONST ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % md * x.val % md); }
friend MDCONST ModInt operator/(lint a, const ModInt &x) {
return ModInt()._setval(a % md * x.inv() % md);
}
MDCONST bool operator==(const ModInt &x) const { return val == x.val; }
MDCONST bool operator!=(const ModInt &x) const { return val != x.val; }
MDCONST 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;
return is >> t, x = ModInt(t), is;
}
MDCONST friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val; }
MDCONST ModInt pow(lint n) const {
ModInt ans = 1, tmp = *this;
while (n) {
if (n & 1) ans *= tmp;
tmp *= tmp, n >>= 1;
}
return ans;
}
static std::vector<ModInt> facs, facinvs, invs;
MDCONST static void _precalculation(int N) {
int l0 = facs.size();
if (N > md) N = md;
if (N <= l0) return;
facs.resize(N), facinvs.resize(N), invs.resize(N);
for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
facinvs[N - 1] = facs.back().pow(md - 2);
for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
}
MDCONST lint inv() const {
if (this->val < std::min(md >> 1, 1 << 21)) {
while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
return invs[this->val].val;
} else {
return this->pow(md - 2).val;
}
}
MDCONST ModInt fac() const {
while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
return facs[this->val];
}
MDCONST ModInt facinv() const {
while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
return facinvs[this->val];
}
MDCONST ModInt doublefac() const {
lint k = (this->val + 1) / 2;
return (this->val & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
: ModInt(k).fac() * ModInt(2).pow(k);
}
MDCONST ModInt nCr(const ModInt &r) const {
return (this->val < r.val) ? 0 : this->fac() * (*this - r).facinv() * r.facinv();
}
MDCONST ModInt nPr(const ModInt &r) const {
return (this->val < r.val) ? 0 : this->fac() * (*this - r).facinv();
}
ModInt sqrt() const {
if (val == 0) return 0;
if (md == 2) return val;
if (pow((md - 1) / 2) != 1) return 0;
ModInt b = 1;
while (b.pow((md - 1) / 2) == 1) b += 1;
int e = 0, m = md - 1;
while (m % 2 == 0) m >>= 1, e++;
ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
x *= (*this);
ModInt z = b.pow(m);
while (y != 1) {
int j = 0;
ModInt t = y;
while (t != 1) j++, t *= t;
z = z.pow(1LL << (e - j - 1));
x *= z, z *= z, y *= z;
e = j;
}
return ModInt(std::min(x.val, md - x.val));
}
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};
using mint = ModInt<998244353>;
// 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>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner);
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(std::vector<MODINT> &a, bool is_inverse = false) {
int n = a.size();
if (n == 1) return;
static const int mod = MODINT::mod();
static const MODINT root = MODINT::get_primitive_root();
assert(__builtin_popcount(n) == 1 and (mod - 1) % n == 0);
static std::vector<MODINT> w{1}, iw{1};
for (int m = w.size(); m < n / 2; m *= 2) {
MODINT dw = root.pow((mod - 1) / (4 * m)), dwinv = 1 / dw;
w.resize(m * 2), iw.resize(m * 2);
for (int i = 0; i < m; i++) w[m + i] = w[i] * dw, iw[m + i] = iw[i] * dwinv;
}
if (!is_inverse) {
for (int m = n; m >>= 1;) {
for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
for (int i = s; i < s + m; i++) {
MODINT x = a[i], y = a[i + m] * w[k];
a[i] = x + y, a[i + m] = x - y;
}
}
}
} else {
for (int m = 1; m < n; m *= 2) {
for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
for (int i = s; i < s + m; i++) {
MODINT x = a[i], y = a[i + m];
a[i] = x + y, a[i + m] = (x - y) * iw[k];
}
}
}
int n_inv = MODINT(n).inv();
for (auto &v : a) v *= n_inv;
}
}
template <int MOD> std::vector<ModInt<MOD>> nttconv_(const std::vector<int> &a, const std::vector<int> &b) {
int sz = a.size();
assert(a.size() == b.size() and __builtin_popcount(sz) == 1);
std::vector<ModInt<MOD>> ap(sz), bp(sz);
for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i];
ntt(ap, false);
if (a == b)
bp = ap;
else
ntt(bp, false);
for (int i = 0; i < sz; i++) ap[i] *= bp[i];
ntt(ap, true);
return ap;
}
long long garner_ntt_(int r0, int r1, int r2, int mod) {
using mint2 = ModInt<nttprimes[2]>;
static const long long m01 = 1LL * nttprimes[0] * nttprimes[1];
static const long long m0_inv_m1 = ModInt<nttprimes[1]>(nttprimes[0]).inv();
static const long long m01_inv_m2 = mint2(m01).inv();
int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1];
auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * m01_inv_m2;
return (r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val) % mod;
}
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner) {
if (a.empty() or b.empty()) return {};
int sz = 1, n = a.size(), m = b.size();
while (sz < n + m) sz <<= 1;
if (sz <= 16) {
std::vector<MODINT> ret(n + m - 1);
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) ret[i + j] += a[i] * b[j];
}
return ret;
}
int mod = MODINT::mod();
if (skip_garner or std::find(std::begin(nttprimes), std::end(nttprimes), mod) != std::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 {
std::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;
}
template <typename MODINT>
std::vector<MODINT> nttconv(const std::vector<MODINT> &a, const std::vector<MODINT> &b) {
return nttconv<MODINT>(a, b, false);
}
// Formal Power Series (形式的冪級数) based on ModInt<mod> / ModIntRuntime
// Reference: https://ei1333.github.io/luzhiled/snippets/math/formal-power-series.html
template <typename T> struct FormalPowerSeries : vector<T> {
using vector<T>::vector;
using P = FormalPowerSeries;
void shrink() {
while (this->size() and 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 T &v) const { return P(*this) /= v; }
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 < (int)r.size(); i++) (*this)[i] += r[i];
shrink();
return *this;
}
P &operator+=(const T &v) {
if (this->empty()) this->resize(1);
(*this)[0] += v;
shrink();
return *this;
}
P &operator-=(const P &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
shrink();
return *this;
}
P &operator-=(const T &v) {
if (this->empty()) this->resize(1);
(*this)[0] -= v;
shrink();
return *this;
}
P &operator*=(const T &v) {
for (auto &x : (*this)) x *= v;
shrink();
return *this;
}
P &operator*=(const P &r) {
if (this->empty() || r.empty())
this->clear();
else {
auto ret = nttconv(*this, r);
*this = P(ret.begin(), ret.end());
}
return *this;
}
P &operator%=(const P &r) {
*this -= *this / r * r;
shrink();
return *this;
}
P operator-() const {
P ret = *this;
for (auto &v : ret) v = -v;
return ret;
}
P &operator/=(const T &v) {
assert(v != T(0));
for (auto &x : (*this)) x /= v;
return *this;
}
P &operator/=(const P &r) {
if (this->size() < r.size()) {
this->clear();
return *this;
}
int n = (int)this->size() - r.size() + 1;
return *this = (reversed().pre(n) * r.reversed().inv(n)).pre(n).reversed(n);
}
P pre(int sz) const {
P ret(this->begin(), this->begin() + min((int)this->size(), sz));
ret.shrink();
return ret;
}
P operator>>(int sz) const {
if ((int)this->size() <= sz) return {};
return P(this->begin() + sz, this->end());
}
P operator<<(int sz) const {
if (this->empty()) return {};
P ret(*this);
ret.insert(ret.begin(), sz, T(0));
return ret;
}
P reversed(int deg = -1) const {
assert(deg >= -1);
P ret(*this);
if (deg != -1) ret.resize(deg, T(0));
reverse(ret.begin(), ret.end());
ret.shrink();
return ret;
}
P differential() const { // formal derivative (differential) of f.p.s.
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;
}
P inv(int deg) const {
assert(deg >= -1);
assert(this->size() and ((*this)[0]) != T(0)); // Requirement: F(0) != 0
const int n = 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); }
ret = ret.pre(deg);
ret.shrink();
return ret;
}
P log(int deg = -1) const {
assert(deg >= -1);
assert(this->size() and ((*this)[0]) == T(1)); // Requirement: F(0) = 1
const int n = (int)this->size();
if (deg == 0) return {};
if (deg == -1) deg = n;
return (this->differential() * this->inv(deg)).pre(deg - 1).integral();
}
P sqrt(int deg = -1) const {
assert(deg >= -1);
const int n = (int)this->size();
if (deg == -1) deg = n;
if (this->empty()) return {};
if ((*this)[0] == T(0)) {
for (int i = 1; i < n; i++)
if ((*this)[i] != T(0)) {
if ((i & 1) or deg - i / 2 <= 0) return {};
return (*this >> i).sqrt(deg - i / 2) << (i / 2);
}
return {};
}
T sqrtf0 = (*this)[0].sqrt();
if (sqrtf0 == T(0)) return {};
P y = (*this) / (*this)[0], ret({T(1)});
T inv2 = T(1) / T(2);
for (int i = 1; i < deg; i <<= 1) { ret = (ret + y.pre(i << 1) * ret.inv(i << 1)) * inv2; }
return ret.pre(deg) * sqrtf0;
}
P exp(int deg = -1) const {
assert(deg >= -1);
assert(this->empty() or ((*this)[0]) == T(0)); // Requirement: F(0) = 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);
}
P pow(long long k, int deg = -1) const {
assert(deg >= -1);
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 C = (*this) * rev, D(n - i);
for (int j = i; j < n; j++) D[j - i] = C.coeff(j);
D = (D.log(deg) * T(k)).exp(deg) * (*this)[i].pow(k);
if (k * (i > 0) > deg or k * i > deg) return {};
P E(deg);
long long S = i * k;
for (int j = 0; j + S < deg and j < (int)D.size(); j++) E[j + S] = D[j];
E.shrink();
return E;
}
}
return *this;
}
// Calculate f(X + c) from f(X), O(NlogN)
P shift(T c) const {
const int n = (int)this->size();
P ret = *this;
for (int i = 0; i < n; i++) { ret[i] *= T(i).fac(); }
reverse(ret.begin(), ret.end());
P exp_cx(n, 1);
for (int i = 1; i < n; i++) { exp_cx[i] = exp_cx[i - 1] * c / i; }
ret = (ret * exp_cx), ret.resize(n);
reverse(ret.begin(), ret.end());
for (int i = 0; i < n; i++) { ret[i] /= T(i).fac(); }
return ret;
}
T coeff(int i) const {
if ((int)this->size() <= i or i < 0) return T(0);
return (*this)[i];
}
T eval(T x) const {
T ret = 0, w = 1;
for (auto &v : *this) ret += w * v, w *= x;
return ret;
}
};
// multipoint polynomial evaluation
// input: xs = [x_0, ..., x_{N - 1}]: points to evaluate
// f = \sum_i^M f_i x^i
// Complexity: O(N (lgN)^2) building, O(N (lgN)^2 + M lg M) evaluation
template <typename Tfield> struct MultipointEvaluation {
int nx;
using polynomial = FormalPowerSeries<Tfield>;
std::vector<polynomial> segtree;
MultipointEvaluation(const std::vector<Tfield> &xs) : nx(xs.size()) {
segtree.resize(nx * 2 - 1);
for (int i = 0; i < nx; i++) { segtree[nx - 1 + i] = {-xs[i], 1}; }
for (int i = nx - 2; i >= 0; i--) { segtree[i] = segtree[2 * i + 1] * segtree[2 * i + 2]; }
}
std::vector<Tfield> ret;
void _eval_rec(polynomial f, int now) {
f %= segtree[now];
if (now - (nx - 1) >= 0) {
ret[now - (nx - 1)] = f.coeff(0);
return;
}
_eval_rec(f, 2 * now + 1);
_eval_rec(f, 2 * now + 2);
}
std::vector<Tfield> evaluate_polynomial(const polynomial &f) {
ret.resize(nx);
_eval_rec(f, 0);
return ret;
}
std::vector<Tfield> evaluate_polynomial(const std::vector<Tfield> &f) {
return evaluate_polynomial(polynomial(f.begin(), f.end()));
}
std::vector<Tfield> _interpolate_coeffs;
polynomial _rec_interpolation(int now, const std::vector<Tfield> &ys) const {
int i = now - (nx - 1);
if (i >= 0) return {ys[i]};
auto retl = _rec_interpolation(2 * now + 1, ys);
auto retr = _rec_interpolation(2 * now + 2, ys);
return retl * segtree[2 * now + 2] + retr * segtree[2 * now + 1];
}
std::vector<Tfield> polynomial_interpolation(std::vector<Tfield> ys) {
assert(nx == int(ys.size()));
if (_interpolate_coeffs.empty()) {
_interpolate_coeffs = evaluate_polynomial(segtree[0].differential());
for (auto &x : _interpolate_coeffs) x = x.inv();
}
for (int i = 0; i < nx; i++) ys[i] *= _interpolate_coeffs[i];
return _rec_interpolation(0, ys);
}
};
using fps = FormalPowerSeries<mint>;
// x(1 + (x - 1)(1 + (x - 2)(... (x - (n - 1)(f))))) = gf + h
pair<fps, fps> rec(int n) {
if (n == 1) return {{0, 1}, {1}};
int h = n / 2;
auto [g0, h0] = rec(h);
g0.resize(h + 2);
h0.resize(h + 2);
fps g1 = g0.shift(-(n - h)), h1 = h0.shift(-(n - h));
if (n % 2) {
fps z{-h, 1};
g1 = g1 * z;
h1 = h1 * z + mint(1);
}
auto g = g1 * g0;
auto hret = g0 * h1 + h0;
return {g, hret};
}
int main() {
int N, M;
cin >> N >> M;
vector<int> A(M);
cin >> A;
mint ret = 0;
FOR(k, 1, N + 1) {
int a = *lower_bound(A.begin(), A.end(), k);
ret += mint(N - k + 1).inv() * a;
}
cout << ret << '\n';
}