結果
| 問題 | No.2582 Random Average^K |
| ユーザー |
 heno239
|
| 提出日時 | 2023-12-10 03:48:54 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 440 ms / 2,000 ms |
| コード長 | 17,846 bytes |
| コンパイル時間 | 4,715 ms |
| コンパイル使用メモリ | 209,636 KB |
| 実行使用メモリ | 134,784 KB |
| 最終ジャッジ日時 | 2024-09-27 03:57:54 |
| 合計ジャッジ時間 | 12,149 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / judge2 |
（要ログイン）
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 | 324 ms
134,656 KB |
| testcase_01 | AC | 324 ms
134,656 KB |
| testcase_02 | AC | 323 ms
134,656 KB |
| testcase_03 | AC | 323 ms
134,784 KB |
| testcase_04 | AC | 323 ms
134,656 KB |
| testcase_05 | AC | 324 ms
134,528 KB |
| testcase_06 | AC | 326 ms
134,528 KB |
| testcase_07 | AC | 324 ms
134,784 KB |
| testcase_08 | AC | 324 ms
134,528 KB |
| testcase_09 | AC | 353 ms
134,656 KB |
| testcase_10 | AC | 374 ms
134,656 KB |
| testcase_11 | AC | 372 ms
134,656 KB |
| testcase_12 | AC | 436 ms
134,656 KB |
| testcase_13 | AC | 420 ms
134,656 KB |
| testcase_14 | AC | 326 ms
134,656 KB |
| testcase_15 | AC | 440 ms
134,528 KB |
| testcase_16 | AC | 382 ms
134,656 KB |
| testcase_17 | AC | 384 ms
134,528 KB |
ソースコード
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#include<iostream>
#include<string>
#include<cstdio>
#include<vector>
#include<cmath>
#include<algorithm>
#include<functional>
#include<iomanip>
#include<queue>
#include<ciso646>
#include<random>
#include<map>
#include<set>
#include<bitset>
#include<stack>
#include<unordered_map>
#include<unordered_set>
#include<utility>
#include<cassert>
#include<complex>
#include<numeric>
#include<array>
#include<chrono>
using namespace std;
//#define int long long
typedef long long ll;
typedef unsigned long long ul;
typedef unsigned int ui;
//ll mod = 1;
constexpr ll mod = 998244353;
//constexpr ll mod = 1000000007;
const int mod17 = 1000000007;
const ll INF = (ll)mod17 * mod17;
typedef pair<int, int>P;
#define rep(i,n) for(int i=0;i<n;i++)
#define per(i,n) for(int i=n-1;i>=0;i--)
#define Rep(i,sta,n) for(int i=sta;i<n;i++)
#define rep1(i,n) for(int i=1;i<=n;i++)
#define per1(i,n) for(int i=n;i>=1;i--)
#define Rep1(i,sta,n) for(int i=sta;i<=n;i++)
#define all(v) (v).begin(),(v).end()
typedef pair<ll, ll> LP;
using ld = long double;
typedef pair<ld, ld> LDP;
const ld eps = 1e-10;
const ld pi = acosl(-1.0);
template<typename T>
void chmin(T& a, T b) {
a = min(a, b);
}
template<typename T>
void chmax(T& a, T b) {
a = max(a, b);
}
template<typename T>
vector<T> vmerge(vector<T>& a, vector<T>& b) {
vector<T> res;
int ida = 0, idb = 0;
while (ida < a.size() || idb < b.size()) {
if (idb == b.size()) {
res.push_back(a[ida]); ida++;
}
else if (ida == a.size()) {
res.push_back(b[idb]); idb++;
}
else {
if (a[ida] < b[idb]) {
res.push_back(a[ida]); ida++;
}
else {
res.push_back(b[idb]); idb++;
}
}
}
return res;
}
template<typename T>
void cinarray(vector<T>& v) {
rep(i, v.size())cin >> v[i];
}
template<typename T>
void coutarray(vector<T>& v) {
rep(i, v.size()) {
if (i > 0)cout << " "; cout << v[i];
}
cout << "\n";
}
ll mod_pow(ll x, ll n, ll m = mod) {
if (n < 0) {
ll res = mod_pow(x, -n, m);
return mod_pow(res, m - 2, m);
}
if (abs(x) >= m)x %= m;
if (x < 0)x += m;
//if (x == 0)return 0;
ll res = 1;
while (n) {
if (n & 1)res = res * x % m;
x = x * x % m; n >>= 1;
}
return res;
}
//mod should be <2^31
struct modint {
int n;
modint() :n(0) { ; }
modint(ll m) {
if (m < 0 || mod <= m) {
m %= mod; if (m < 0)m += mod;
}
n = m;
}
operator int() { return n; }
};
bool operator==(modint a, modint b) { return a.n == b.n; }
bool operator<(modint a, modint b) { return a.n < b.n; }
modint operator+=(modint& a, modint b) { a.n += b.n; if (a.n >= mod)a.n -= (int)mod; return a; }
modint operator-=(modint& a, modint b) { a.n -= b.n; if (a.n < 0)a.n += (int)mod; return a; }
modint operator*=(modint& a, modint b) { a.n = ((ll)a.n * b.n) % mod; return a; }
modint operator+(modint a, modint b) { return a += b; }
modint operator-(modint a, modint b) { return a -= b; }
modint operator*(modint a, modint b) { return a *= b; }
modint operator^(modint a, ll n) {
if (n == 0)return modint(1);
modint res = (a * a) ^ (n / 2);
if (n % 2)res = res * a;
return res;
}
ll inv(ll a, ll p) {
return (a == 1 ? 1 : (1 - p * inv(p % a, a)) / a + p);
}
modint operator/(modint a, modint b) { return a * modint(inv(b, mod)); }
modint operator/=(modint& a, modint b) { a = a / b; return a; }
const int max_n = 1 << 24;
modint fact[max_n], factinv[max_n];
void init_f() {
fact[0] = modint(1);
for (int i = 0; i < max_n - 1; i++) {
fact[i + 1] = fact[i] * modint(i + 1);
}
factinv[max_n - 1] = modint(1) / fact[max_n - 1];
for (int i = max_n - 2; i >= 0; i--) {
factinv[i] = factinv[i + 1] * modint(i + 1);
}
}
modint comb(int a, int b) {
if (a < 0 || b < 0 || a < b)return 0;
return fact[a] * factinv[b] * factinv[a - b];
}
modint combP(int a, int b) {
if (a < 0 || b < 0 || a < b)return 0;
return fact[a] * factinv[a - b];
}
ll gcd(ll a, ll b) {
a = abs(a); b = abs(b);
if (a < b)swap(a, b);
while (b) {
ll r = a % b; a = b; b = r;
}
return a;
}
template<typename T>
void addv(vector<T>& v, int loc, T val) {
if (loc >= v.size())v.resize(loc + 1, 0);
v[loc] += val;
}
/*const int mn = 2000005;
bool isp[mn];
vector<int> ps;
void init() {
fill(isp + 2, isp + mn, true);
for (int i = 2; i < mn; i++) {
if (!isp[i])continue;
ps.push_back(i);
for (int j = 2 * i; j < mn; j += i) {
isp[j] = false;
}
}
}*/
//[,val)
template<typename T>
auto prev_itr(set<T>& st, T val) {
auto res = st.lower_bound(val);
if (res == st.begin())return st.end();
res--; return res;
}
//[val,)
template<typename T>
auto next_itr(set<T>& st, T val) {
auto res = st.lower_bound(val);
return res;
}
using mP = pair<modint, modint>;
mP operator+(mP a, mP b) {
return { a.first + b.first,a.second + b.second };
}
mP operator+=(mP& a, mP b) {
a = a + b; return a;
}
mP operator-(mP a, mP b) {
return { a.first - b.first,a.second - b.second };
}
mP operator-=(mP& a, mP b) {
a = a - b; return a;
}
LP operator+(LP a, LP b) {
return { a.first + b.first,a.second + b.second };
}
LP operator+=(LP& a, LP b) {
a = a + b; return a;
}
LP operator-(LP a, LP b) {
return { a.first - b.first,a.second - b.second };
}
LP operator-=(LP& a, LP b) {
a = a - b; return a;
}
mt19937 mt(time(0));
const string drul = "DRUL";
string senw = "SENW";
//DRUL,or SENW
//int dx[4] = { 1,0,-1,0 };
//int dy[4] = { 0,1,0,-1 };
//------------------------------------
int get_premitive_root() {
int primitive_root = 0;
if (!primitive_root) {
primitive_root = [&]() {
set<int> fac;
int v = mod - 1;
for (ll 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 (mod_pow(g, (mod - 1) / i) == 1) { ok = false; break; }
if (ok) return g;
}
return -1;
}();
}
return primitive_root;
}
const int proot = get_premitive_root();
int bsf(int x) {
int res = 0;
while (!(x & 1)) {
res++; x >>= 1;
}
return res;
}
int ceil_pow2(int n) {
int x = 0;
while ((1 << x) < n) x++;
return x;
}
using poly = vector<modint>;
void butterfly(poly& a) {
int n = int(a.size());
int g = proot;
int h = ceil_pow2(n);
static bool first = true;
static modint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
if (first) {
first = false;
modint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
int cnt2 = bsf(mod - 1);
modint e = mod_pow(g, (mod - 1) >> cnt2);
modint ie = (modint)1 / e;
for (int i = cnt2; i >= 2; i--) {
// e^(2^i) == 1
es[i - 2] = e;
ies[i - 2] = ie;
e *= e;
ie *= ie;
}
modint now = 1;
for (int i = 0; i < cnt2 - 2; i++) {
sum_e[i] = es[i] * now;
now *= ies[i];
}
}
for (int ph = 1; ph <= h; ph++) {
int w = 1 << (ph - 1), p = 1 << (h - ph);
modint now = 1;
for (int s = 0; s < w; s++) {
int offset = s << (h - ph + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * now;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
now *= sum_e[bsf(~(unsigned int)(s))];
}
}
}
void butterfly_inv(poly& a) {
int n = int(a.size());
int g = proot;
int h = ceil_pow2(n);
static bool first = true;
static modint sum_ie[30]; // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
if (first) {
first = false;
modint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
int cnt2 = bsf(mod - 1);
modint e = mod_pow(g, (mod - 1) >> cnt2);
modint ie = (modint)1 / e;
for (int i = cnt2; i >= 2; i--) {
// e^(2^i) == 1
es[i - 2] = e;
ies[i - 2] = ie;
e *= e;
ie *= ie;
}
modint now = 1;
for (int i = 0; i < cnt2 - 2; i++) {
sum_ie[i] = ies[i] * now;
now *= es[i];
}
}
for (int ph = h; ph >= 1; ph--) {
int w = 1 << (ph - 1), p = 1 << (h - ph);
modint inow = 1;
for (int s = 0; s < w; s++) {
int offset = s << (h - ph + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] =
(unsigned long long)(mod + (ll)l - (ll)r) *
(ll)inow;
}
inow *= sum_ie[bsf(~(unsigned int)(s))];
}
}
}
poly multiply(poly g, poly h) {
int n = g.size();
int m = h.size();
if (n == 0 || m == 0)return {};
if (min(g.size(), h.size()) < 60) {
poly res(g.size() + h.size() - 1);
rep(i, g.size())rep(j, h.size()) {
res[i + j] += g[i] * h[j];
}
return res;
}
int z = 1 << ceil_pow2(n + m - 1);
g.resize(z);
butterfly(g);
h.resize(z);
butterfly(h);
rep(i, z) {
g[i] *= h[i];
}
butterfly_inv(g);
g.resize(n + m - 1);
modint iz = (modint)1 / (modint)z;
rep(i, n + m - 1) {
g[i] *= iz;
}
return g;
}
struct FormalPowerSeries :vector<modint> {
using vector<modint>::vector;
using fps = FormalPowerSeries;
void shrink() {
while (this->size() && this->back() == (modint)0)this->pop_back();
}
fps operator+(const fps& r)const { return fps(*this) += r; }
fps operator+(const modint& v)const { return fps(*this) += v; }
fps operator-(const fps& r)const { return fps(*this) -= r; }
fps operator-(const modint& v)const { return fps(*this) -= v; }
fps operator*(const fps& r)const { return fps(*this) *= r; }
fps operator*(const modint& v)const { return fps(*this) *= v; }
fps& operator+=(const fps& r) {
if (r.size() > this->size())this->resize(r.size());
rep(i, r.size())(*this)[i] += r[i];
shrink();
return *this;
}
fps& operator+=(const modint& v) {
if (this->empty())this->resize(1);
(*this)[0] += v;
shrink();
return *this;
}
fps& operator-=(const fps& r) {
if (r.size() > this->size())this->resize(r.size());
rep(i, r.size())(*this)[i] -= r[i];
shrink();
return *this;
}
fps& operator-=(const modint& v) {
if (this->empty())this->resize(1);
(*this)[0] -= v;
shrink();
return *this;
}
fps& operator*=(const fps& r) {
if (this->empty() || r.empty())this->clear();
else {
poly ret = multiply(*this, r);
*this = fps(all(ret));
}
shrink();
return *this;
}
fps& operator*=(const modint& v) {
for (auto& x : (*this))x *= v;
shrink();
return *this;
}
fps operator-()const {
fps ret = *this;
for (auto& v : ret)v = -v;
return ret;
}
modint sub(modint x) {
modint t = 1;
modint res = 0;
rep(i, (*this).size()) {
res += t * (*this)[i];
t *= x;
}
return res;
}
fps pre(int sz)const {
fps ret(this->begin(), this->begin() + min((int)this->size(), sz));
ret.shrink();
return ret;
}
fps integral() const {
const int n = (int)this->size();
fps ret(n + 1);
ret[0] = 0;
for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / (modint)(i + 1);
return ret;
}
fps inv(int deg = -1)const {
const int n = this->size();
if (deg == -1)deg = n;
fps ret({ (modint)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;
}
fps diff() const {
const int n = (int)this->size();
fps ret(max(0, n - 1));
for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * (modint)i;
return ret;
}
// F(0) must be 1
fps log(int deg = -1) const {
assert((*this)[0] == 1);
const int n = (int)this->size();
if (deg == -1) deg = n;
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
// F(0) must be 0
fps exp(int deg = -1)const {
assert((*this)[0] == 0);
const int n = (int)this->size();
if (deg == -1)deg = n;
fps ret = { 1 };
for (int i = 1; i < deg; i <<= 1) {
ret = (ret * (pre(i << 1) + 1 - ret.log(i << 1))).pre(i << 1);
}
//cout << "!!!! " << ret.size() << "\n";
return ret.pre(deg);
}
fps div(fps g) {
assert(g.size() && g.back() != (modint)0);
fps f = *this;
if (f.size() < g.size())return {};
int dif = f.size() - g.size();
reverse(all(f));
reverse(all(g));
g = g.inv(dif + 1);
fps fg = f * g;
fps ret(dif + 1);
rep(i, fg.size()) {
int id = i - dif;
if (-dif <= id && id <= 0) {
ret[-id] = fg[i];
}
}
return ret;
}
fps divr(fps g) {
fps ret = (*this) - g * (*this).div(g);
ret.shrink();
return ret;
}
};
using fps = FormalPowerSeries;
//f(r^0),f(r^1),...,f(r^n)
vector<modint> Multipoint_Evaluation(fps c, modint r, int n) {
vector<modint> res(n + 1, 0);
if (c.empty()) {
return res;
}
if (r == (modint)0) {
rep(i, n + 1)res[i] = c[0];
return res;
}
int sz = c.size() + n;
vector<modint> rr(sz);
rr[0] = 1;
rep(i, sz - 1)rr[i + 1] = rr[i] * r;
vector<modint> irr(sz);
modint ir = (modint)1 / r;
irr[0] = 1;
rep(i, sz - 1)irr[i + 1] = irr[i] * ir;
vector<modint> coef(sz);
coef[0] = 1;
rep(i, sz - 1) {
coef[i + 1] = coef[i] * rr[i];
}
vector<modint> icoef(sz);
icoef[0] = 1;
rep(i, sz - 1) {
icoef[i + 1] = icoef[i] * irr[i];
}
fps f(c.size());
rep(i, c.size()) {
f[i] = (modint)c[i] * icoef[i];
}
fps g(sz);
rep(i, sz) {
g[i] = coef[i];
}
reverse(all(f));
f *= g;
rep(i, n + 1) {
modint val = icoef[i];
int loc = i + c.size() - 1;
if (loc < f.size())val *= f[loc];
else val = 0;
res[i] = val;
}
return res;
}
vector<modint> Multipoint_Evaluation(fps c, vector<modint> p) {
int n = p.size();
vector<modint> ret(n);
int sz = 1;
while (sz < n)sz *= 2;
vector<fps> f(2 * sz - 1);
function<void(int, int, int)> dfs = [&](int k, int l, int r) {
if (l + 1 == r) {
f[k] = { -p[l],1 };
}
else {
dfs(2 * k + 1, l, (l + r) / 2);
dfs(2 * k + 2, (l + r) / 2, r);
f[k] = f[2 * k + 1] * f[2 * k + 2];
}
};
dfs(0, 0, n);
vector<fps> g(2 * sz - 1);
function<void(int, int, int)> invdfs = [&](int k, int l, int r) {
if (k == 0) {
g[k] = c.divr(f[k]);
}
else {
g[k] = g[(k - 1) / 2].divr(f[k]);
}
if (r - l <= 100) {
Rep(i, l, r) {
ret[i] = g[k].sub(p[i]);
}
}
else {
invdfs(2 * k + 1, l, (l + r) / 2);
invdfs(2 * k + 2, (l + r) / 2, r);
}
};
invdfs(0, 0, n);
return ret;
}
//reference: https://37zigen.com/berlekamp-massey/
struct berlekamp_massey {
fps a, b;
berlekamp_massey(int n, fps f) {
f.resize(2 * n);
f.shrink();
//deg(a)<n,deg(b)<=n
fps a1 = { 1 }, b1, c1 = f;
fps a2, b2 = { 1 }, c2; c2.resize(2 * n + 1); c2[2 * n] = 1;
while (true) {
if (c1.size() > c2.size()) {
swap(c1, c2);
swap(a1, a2);
swap(b1, b2);
}
if (c1.size() <= n)break;
int dif = c2.size() - c1.size();
modint coef = c2.back() / c1.back();
fps d1, d2, d3;
d1.resize(dif);
d2.resize(dif);
d3.resize(dif);
rep(i, a1.size())d1.push_back(a1[i] * coef);
rep(i, b1.size())d2.push_back(b1[i] * coef);
rep(i, c1.size())d3.push_back(c1[i] * coef);
a2 -= d1;
b2 -= d2;
c2 -= d3;
}
swap(a, c1);
swap(b, a1);
}
//g=x^{-n}*a mod b
ll calc(ll n) {
assert(b[0] != (modint)0);
modint coef = (modint)1 / b[0];
rep(i, a.size())a[i] *= coef;
rep(i, b.size())b[i] *= coef;
fps rx = b;
coef = (modint)-1 / b[0];
rx.erase(rx.begin());
rep(i, rx.size())rx[i] *= coef;
//rx^n
fps z = a;
while (n) {
if (n & 1) {
z *= rx; z = z.divr(b);
}
n >>= 1;
if (n == 0)break;
rx *= rx; rx = rx.divr(b);
}
if (z.empty())return 0;
return z[0] / b[0];
}
};
fps allprod(vector<fps> f) {
while (f.size() > 1) {
vector<fps> nf;
for (int i = 0; i + 1 < f.size(); i += 2) {
nf.push_back(f[i] * f[i + 1]);
}
if (f.size() % 2)nf.push_back(f.back());
swap(f, nf);
}
return f[0];
}
//https://maspypy.com/%e5%a4%9a%e9%a0%85%e5%bc%8f%e3%83%bb%e5%bd%a2%e5%bc%8f%e7%9a%84%e3%81%b9%e3%81%8d%e7%b4%9a%e6%95%b0-%e9%ab%98%e9%80%9f%e3%81%ab%e8%a8%88%e7%ae%97%e3%81%a7%e3%81%8d%e3%82%8b%e3%82%82%e3%81%ae#toc40
//sum_{i}(a_i*exp(b_ix))をm次まで計算
//verified with https://atcoder.jp/contests/arc154/tasks/arc154_f
fps calc_expsum(vector<LP> ps, int m) {
using pfps = pair<fps, fps>;
vector<pfps> vp;
rep(i, ps.size()) {
modint a = ps[i].first;
modint b = ps[i].second;
fps pa = { a };
fps pb = { 1,-b };
vp.push_back({ pa,pb });
}
while (vp.size() > 1) {
vector<pfps> nvp;
for (int i = 0; i + 1 < vp.size(); i += 2) {
pfps& a = vp[i];
pfps& b = vp[i + 1];
fps pa = a.first * b.second + a.second * b.first;
if (pa.size() > m + 1)pa.resize(m + 1);
fps pb = a.second * b.second;
if (pb.size() > m + 1)pb.resize(m + 1);
nvp.push_back({ pa,pb });
}
if (vp.size() & 1)nvp.push_back(vp.back());
swap(vp, nvp);
}
fps las = vp[0].first * vp[0].second.inv(m + 1);
fps res(m + 1);
rep(i, m + 1) {
if (i < las.size())res[i] = las[i];
res[i] *= factinv[i];
}
return res;
}
void solve() {
int n, k; cin >> n >> k;
modint ans = 0;
rep(i, n + 1) {
modint val = comb(n, i);
if ((n - i) % 2)val *= -1;
int loc = n + k;
val *= factinv[loc];
val *= mod_pow(i, loc);
ans += val;
}
ans *= fact[k];
/*fps f(n + 1);
rep1(i, n) {
f[i] = mod_pow(i, k);
f[i] *= comb(n, i);
}
fps g(n + 1);
rep(i, n + 1) {
g[i] = factinv[i];
}
g = g.inv(n + 1);
rep(i, n + 1)f[i] *= fact[n-i];
fps F = f * g;
F.resize(n + 1);
rep(i, n + 1) {
F[i] *= fact[n-i];
}
coutarray(f);
coutarray(g);
coutarray(F);
modint ans = 0;
modint t2 = 1;
rep1(i, n) {
t2 *= (1 + mod) / 2;
modint val = F[i];
val *= t2;
ans += val;
}
ans /= mod_pow(n, k);*/
ans /= mod_pow(n, k);
//ans *= 24;
cout << ans << "\n";
}
signed main() {
ios::sync_with_stdio(false);
cin.tie(0);
//cout << fixed<<setprecision(10);
init_f();
//init();
//while(true)
//expr();
//int t; cin >> t; rep(i, t)
solve();
return 0;
}