結果
| 問題 | No.1145 Sums of Powers |
| ユーザー |
 hitonanode
|
| 提出日時 | 2020-08-09 00:04:57 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 514 ms / 2,000 ms |
| コード長 | 20,639 bytes |
| コンパイル時間 | 3,800 ms |
| コンパイル使用メモリ | 233,604 KB |
| 最終ジャッジ日時 | 2025-01-12 19:08:36 |
|
ジャッジサーバーID （参考情報） |
judge5 / judge5 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| other | AC * 6 |
ソースコード
#include <bits/stdc++.h>
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> 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 V, typename T> void ndfill(V &x, const T &val) { x = val; }
template <typename V, typename T> void ndfill(vector<V> &vec, const T &val) { for (auto &v : vec) ndfill(v, val); }
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> vector<T> srtunq(vector<T> vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; }
template <typename T> istream &operator>>(istream &is, vector<T> &vec) { for (auto &v : vec) is >> v; return is; }
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 vector<T> &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <typename... T> ostream &operator<<(ostream &os, const tuple<T...> &tpl) { std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); 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; }
#ifdef HITONANODE_LOCAL
#define dbg(x) cerr << #x << " = " << (x) << " (L" << __LINE__ << ") " << __FILE__ << endl
#else
#define dbg(x)
#endif
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<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 = 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(std::vector<MODINT> &a, bool is_inverse = false)
{
int n = a.size();
if (n == 1) return;
static const int mod = MODINT::get_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.power((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++) {
#ifdef __clang__
a[i + m] *= w[k];
std::tie(a[i], a[i + m]) = std::make_pair(a[i] + a[i + m], a[i] - a[i + m]);
#else
MODINT x = a[i], y = a[i + m] * w[k];
a[i] = x + y, a[i + m] = x - y;
#endif
}
}
}
}
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++) {
#ifdef __clang__
std::tie(a[i], a[i + m]) = std::make_pair(a[i] + a[i + m], a[i] - a[i + m]);
a[i + m] *= iw[k];
#else
MODINT x = a[i], y = a[i + m];
a[i] = x + y, a[i + m] = (x - y) * iw[k];
#endif
}
}
}
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];
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 extgcd_ntt_(long long a, long long b, long long &x, long long &y)
{
long long 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 modinv_ntt_(long long a, long long m)
{
long long x, y;
extgcd_ntt_(a, m, x, y);
return (m + x % m) % m;
}
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)
{
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::get_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;
}
// 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 int 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);
P D(n - i);
for (int j = i; j < n; j++) D[j - i] = C[j];
D = (D.log(deg) * T(k)).exp(deg) * (*this)[i].power(k);
P E(deg);
if (k * (i > 0) > deg or k * i > deg) return {};
long long int 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;
}
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;
}
};
using fps = FormalPowerSeries<mint>;
int main()
{
int N, M;
cin >> N >> M;
vector<mint> A(N);
cin >> A;
auto rec = [&](auto &&rec, int l, int r) -> pair<fps, fps> {
if (l + 1 == r) return make_pair(fps{1}, fps{1, -A[l]});
else
{
auto a = rec(rec, l, (l + r) / 2);
auto b = rec(rec, (l + r) / 2, r);
return make_pair(a.first * b.second + a.second * b.first, a.second * b.second);
}
};
auto pf = rec(rec, 0, N);
auto f = pf.first * pf.second.inv(M + 1);
FOR(i, 1, M + 1) cout << f.coeff(i) << (i == M ? '\n' : ' ');
}