#935948 (C++14) No.2556 Increasing Matrix

• 問題: No.2556 Increasing Matrix
• ユーザー: sigma425
• 提出日時: 2023-12-03 18:04:07
• 言語: C++14 (gcc 12.3.0 + boost 1.83.0)
• 結果: AC
• 実行時間: 4,760 ms / 6,000 ms
• コード長: 15,569 bytes
• コンパイル時間: 3,139 ms
• コンパイル使用メモリ: 196,340 KB
• 実行使用メモリ: 48,744 KB
• 最終ジャッジ日時: 2023-12-03 18:04:34
• 合計ジャッジ時間: 26,112 ms
• ジャッジサーバーID （参考情報）: judge14 / judge15

入力	結果	実行時間; 実行使用メモリ
testcase_00	AC	18 ms; 15,044 KB
testcase_01	AC	18 ms; 15,044 KB
testcase_02	AC	17 ms; 15,044 KB
testcase_03	AC	17 ms; 15,044 KB
testcase_04	AC	17 ms; 15,044 KB
testcase_05	AC	18 ms; 15,044 KB
testcase_06	AC	18 ms; 15,172 KB
testcase_07	AC	18 ms; 15,172 KB
testcase_08	AC	19 ms; 15,172 KB
testcase_09	AC	34 ms; 15,300 KB
testcase_10	AC	22 ms; 15,172 KB
testcase_11	AC	27 ms; 15,300 KB
testcase_12	AC	23 ms; 15,172 KB
testcase_13	AC	60 ms; 15,556 KB
testcase_14	AC	120 ms; 16,068 KB
testcase_15	AC	108 ms; 16,068 KB
testcase_16	AC	90 ms; 15,812 KB
testcase_17	AC	148 ms; 16,580 KB
testcase_18	AC	2,870 ms; 33,476 KB
testcase_19	AC	3,243 ms; 41,384 KB
testcase_20	AC	3,526 ms; 41,816 KB
testcase_21	AC	2,296 ms; 31,964 KB
testcase_22	AC	202 ms; 16,964 KB
testcase_23	AC	4,736 ms; 48,744 KB
testcase_24	AC	4,760 ms; 48,744 KB

ソースコード

// #pragma GCC target("avx,avx2")
// #pragma GCC optimize("Ofast")
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
#define rep(i,n) for(int i=0;i<int(n);i++)
#define rep1(i,n) for(int i=1;i<=int(n);i++)
#define per(i,n) for(int i=int(n)-1;i>=0;i--)
#define per1(i,n) for(int i=int(n);i>0;i--)
#define all(c) c.begin(),c.end()
#define si(x) int(x.size())
#define pb push_back
#define eb emplace_back
#define fs first
#define sc second
template<class T> using V = vector<T>;
template<class T> using VV = vector<vector<T>>;
template<class T,class U> bool chmax(T& x, U y){
	if(x<y){ x=y; return true; }
	return false;
}
template<class T,class U> bool chmin(T& x, U y){
	if(y<x){ x=y; return true; }
	return false;
}
template<class T> void mkuni(V<T>& v){sort(all(v));v.erase(unique(all(v)),v.end());}
template<class T> int lwb(const V<T>& v, const T& a){return lower_bound(all(v),a) - v.begin();}
template<class T>
V<T> Vec(size_t a) {
    return V<T>(a);
}
template<class T, class... Ts>
auto Vec(size_t a, Ts... ts) {
  return V<decltype(Vec<T>(ts...))>(a, Vec<T>(ts...));
}
template<class S,class T> ostream& operator<<(ostream& o,const pair<S,T> &p){
	return o<<"("<<p.fs<<","<<p.sc<<")";
}
template<class T> ostream& operator<<(ostream& o,const vector<T> &vc){
	o<<"{";
	for(const T& v:vc) o<<v<<",";
	o<<"}";
	return o;
}
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n-1); }

#ifdef LOCAL
#define show(x) cerr << "LINE" << __LINE__ << " : " << #x << " = " << (x) << endl
void dmpr(ostream& os){os<<endl;}
template<class T,class... Args>
void dmpr(ostream&os,const T&t,const Args&... args){
	os<<t<<" ~ ";
	dmpr(os,args...);
}
#define shows(...) cerr << "LINE" << __LINE__ << " : ";dmpr(cerr,##__VA_ARGS__)
#define dump(x) cerr << "LINE" << __LINE__ << " : " << #x << " = {";  \
	for(auto v: x) cerr << v << ","; cerr << "}" << endl;
#else
#define show(x) void(0)
#define dump(x) void(0)
#define shows(...) void(0)
#endif

template<class D> D divFloor(D a, D b){
	return a / b - (((a ^ b) < 0 && a % b != 0) ? 1 : 0);
}
template<class D> D divCeil(D a, D b) {
	return a / b + (((a ^ b) > 0 && a % b != 0) ? 1 : 0);
}
template<class T>
T rnd(T l,T r){	//[l,r)
	using D = uniform_int_distribution<T>;
	static random_device rd;
	static mt19937 gen(rd());
	return D(l,r-1)(gen);
}
template<class T>
T rnd(T n){	//[0,n)
	return rnd(T(0),n);
}

template<unsigned int mod_>
struct ModInt{
	using uint = unsigned int;
	using ll = long long;
	using ull = unsigned long long;

	constexpr static uint mod = mod_;

	uint v;
	ModInt():v(0){}
	ModInt(ll _v):v(normS(_v%mod+mod)){}
	explicit operator bool() const {return v!=0;}
	static uint normS(const uint &x){return (x<mod)?x:x-mod;}		// [0 , 2*mod-1] -> [0 , mod-1]
	static ModInt make(const uint &x){ModInt m; m.v=x; return m;}
	ModInt operator+(const ModInt& b) const { return make(normS(v+b.v));}
	ModInt operator-(const ModInt& b) const { return make(normS(v+mod-b.v));}
	ModInt operator-() const { return make(normS(mod-v)); }
	ModInt operator*(const ModInt& b) const { return make((ull)v*b.v%mod);}
	ModInt operator/(const ModInt& b) const { return *this*b.inv();}
	ModInt& operator+=(const ModInt& b){ return *this=*this+b;}
	ModInt& operator-=(const ModInt& b){ return *this=*this-b;}
	ModInt& operator*=(const ModInt& b){ return *this=*this*b;}
	ModInt& operator/=(const ModInt& b){ return *this=*this/b;}
	ModInt& operator++(int){ return *this=*this+1;}
	ModInt& operator--(int){ return *this=*this-1;}
	template<class T> friend ModInt operator+(T a, const ModInt& b){ return (ModInt(a) += b);}
	template<class T> friend ModInt operator-(T a, const ModInt& b){ return (ModInt(a) -= b);}
	template<class T> friend ModInt operator*(T a, const ModInt& b){ return (ModInt(a) *= b);}
	template<class T> friend ModInt operator/(T a, const ModInt& b){ return (ModInt(a) /= b);}
	ModInt pow(ll p) const {
		if(p<0) return inv().pow(-p);
		ModInt a = 1;
		ModInt x = *this;
		while(p){
			if(p&1) a *= x;
			x *= x;
			p >>= 1;
		}
		return a;
	}
	ModInt inv() const {		// should be prime
		return pow(mod-2);
	}
	// ll extgcd(ll a,ll b,ll &x,ll &y) const{
	// 	ll p[]={a,1,0},q[]={b,0,1};
	// 	while(*q){
	// 		ll t=*p/ *q;
	// 		rep(i,3) swap(p[i]-=t*q[i],q[i]);
	// 	}
	// 	if(p[0]<0) rep(i,3) p[i]=-p[i];
	// 	x=p[1],y=p[2];
	// 	return p[0];
	// }
	// ModInt inv() const {
	// 	ll x,y;
	// 	extgcd(v,mod,x,y);
	// 	return make(normS(x+mod));
	// }

	bool operator==(const ModInt& b) const { return v==b.v;}
	bool operator!=(const ModInt& b) const { return v!=b.v;}
	bool operator<(const ModInt& b) const { return v<b.v;}
	friend istream& operator>>(istream &o,ModInt& x){
		ll tmp;
		o>>tmp;
		x=ModInt(tmp);
		return o;
	}
	friend ostream& operator<<(ostream &o,const ModInt& x){ return o<<x.v;}
};
using mint = ModInt<998244353>;
//using mint = ModInt<1000000007>;

V<mint> fact,ifact,invs;
mint Choose(int a,int b){
	if(b<0 || a<b) return 0;
	return fact[a] * ifact[b] * ifact[a-b];
}
void InitFact(int N){	//[0,N]
	N++;
	fact.resize(N);
	ifact.resize(N);
	invs.resize(N);
	fact[0] = 1;
	rep1(i,N-1) fact[i] = fact[i-1] * i;
	ifact[N-1] = fact[N-1].inv();
	for(int i=N-2;i>=0;i--) ifact[i] = ifact[i+1] * (i+1);
	rep1(i,N-1) invs[i] = fact[i-1] * ifact[i];
}

// inplace_fmt (without bit rearranging)
// fft:
// 		a[rev(i)] <- \sum_j \zeta^{ij} a[j]
// invfft:
//		a[i] <- (1/n) \sum_j \zeta^{-ij} a[rev(j)]
// These two are inversions.


// !!! CHANGE IF MOD is unusual !!!
const int ORDER_2_MOD_MINUS_1 = 23;	// ord_2 (mod-1)
const mint PRIMITIVE_ROOT = 3; // primitive root of (Z/pZ)*

void fft(V<mint>& a){
	static constexpr uint mod = mint::mod;
	static constexpr uint mod2 = mod + mod;
	static const int H = ORDER_2_MOD_MINUS_1;
	static const mint root = PRIMITIVE_ROOT;
	static mint magic[H-1];

	int n = si(a);
	assert(!(n & (n-1))); assert(n >= 1); assert(n <= 1<<H);	// n should be power of 2

	if(!magic[0]){		// precalc
		rep(i,H-1){
			mint w = -root.pow(((mod-1)>>(i+2))*3);
			magic[i] = w;
		}
	}
	int m = n;
	if(m >>= 1){
		rep(i,m){
			uint v = a[i+m].v;					// < M
			a[i+m].v = a[i].v + mod - v;		// < 2M
			a[i].v += v;						// < 2M
		}
	}
	if(m >>= 1){
		mint p = 1;
		for(int h=0,s=0; s<n; s += m*2){
			for(int i=s;i<s+m;i++){
				uint v = (a[i+m] * p).v;		// < M
				a[i+m].v = a[i].v + mod - v;	// < 3M
				a[i].v += v;					// < 3M
			}
			p *= magic[__builtin_ctz(++h)];
		}
	}
	while(m){
		if(m >>= 1){
			mint p = 1;
			for(int h=0,s=0; s<n; s += m*2){
				for(int i=s;i<s+m;i++){
					uint v = (a[i+m] * p).v;		// < M
					a[i+m].v = a[i].v + mod - v;	// < 4M
					a[i].v += v;					// < 4M
				}
				p *= magic[__builtin_ctz(++h)];
			}
		}
		if(m >>= 1){
			mint p = 1;
			for(int h=0,s=0; s<n; s += m*2){
				for(int i=s;i<s+m;i++){
					uint v = (a[i+m] * p).v;								// < M
					a[i].v = (a[i].v >= mod2) ? a[i].v - mod2 : a[i].v;	// < 2M
					a[i+m].v = a[i].v + mod - v;							// < 3M
					a[i].v += v;											// < 3M
				}
				p *= magic[__builtin_ctz(++h)];
			}
		}
	}
	rep(i,n){
		a[i].v = (a[i].v >= mod2) ? a[i].v - mod2 : a[i].v;		// < 2M
		a[i].v = (a[i].v >= mod) ? a[i].v - mod : a[i].v;		// < M
	}
	// finally < mod !!
}
void invfft(V<mint>& a){
	static constexpr uint mod = mint::mod;
	static constexpr uint mod2 = mod + mod;
	static const int H = ORDER_2_MOD_MINUS_1;
	static const mint root = PRIMITIVE_ROOT;
	static mint magic[H-1];

	int n = si(a);
	assert(!(n & (n-1))); assert(n >= 1); assert(n <= 1<<H);	// n should be power of 2

	if(!magic[0]){		// precalc
		rep(i,H-1){
			mint w = -root.pow(((mod-1)>>(i+2))*3);
			magic[i] = w.inv();
		}
	}
	int m = 1;
	if(m < n>>1){
		mint p = 1;
		for(int h=0,s=0; s<n; s += m*2){
			for(int i=s;i<s+m;i++){
				ull x = a[i].v + mod - a[i+m].v;	// < 2M
				a[i].v += a[i+m].v;					// < 2M
				a[i+m].v = (p.v * x) % mod;			// < M
			}
			p *= magic[__builtin_ctz(++h)];
		}
		m <<= 1;
	}
	for(;m < n>>1; m <<= 1){
		mint p = 1;
		for(int h=0,s=0; s<n; s+= m*2){
			for(int i=s;i<s+(m>>1);i++){
				ull x = a[i].v + mod2 - a[i+m].v;	// < 4M
				a[i].v += a[i+m].v;					// < 4M
				a[i].v = (a[i].v >= mod2) ? a[i].v - mod2 : a[i].v;	// < 2M
				a[i+m].v = (p.v * x) % mod;		// < M
			}
			for(int i=s+(m>>1); i<s+m; i++){
				ull x = a[i].v + mod - a[i+m].v;	// < 2M
				a[i].v += a[i+m].v;	// < 2M
				a[i+m].v = (p.v * x) % mod;	// < M
			}
			p *= magic[__builtin_ctz(++h)];
		}
	}
	if(m < n){
		rep(i,m){
			uint x = a[i].v + mod2 - a[i+m].v;	// < 4M
			a[i].v += a[i+m].v;	// < 4M
			a[i+m].v = x;	// < 4M
		}
	}
	const mint in = mint(n).inv();
	rep(i,n) a[i] *= in;	// < M
	// finally < mod !!
}

// A,B = 500000 -> 70ms
// verify https://judge.yosupo.jp/submission/44937
V<mint> multiply(V<mint> a, V<mint> b) {
	int A = si(a), B = si(b);
	if (!A || !B) return {};
	int n = A+B-1;
	int s = 1; while(s<n) s*=2;
	if(a == b){			// # of fft call : 3 -> 2
		a.resize(s); fft(a);
		rep(i,s) a[i] *= a[i];
	}else{
		a.resize(s); fft(a);
		b.resize(s); fft(b);
		rep(i,s) a[i] *= b[i];
	}
	invfft(a); a.resize(n);
	return a;
}

/*
	係数アクセス
		f[i] でいいが、 配列外参照する可能性があるなら at/set
	
*/

template<class mint>
struct Poly: public V<mint>{
	using vector<mint>::vector;
	Poly() {}
	explicit Poly(int n) : V<mint>(n){}		// poly<mint> a; a = 2; shouldn't be [0,0]
	Poly(int n, mint c) : V<mint>(n,c){}
	Poly(const V<mint>& a) : V<mint>(a){}
	Poly(initializer_list<mint> li) : V<mint>(li){}

	int size() const { return V<mint>::size(); }
	mint at(int i) const {
		return i<size() ? (*this)[i] : 0;
	}
	void set(int i, mint x){
		if(i>=size() && !x) return;
		while(i>=size()) this->pb(0);
		(*this)[i] = x;
		return;
	}
	mint operator()(mint x) const {		// eval
		mint res = 0;
		int n = size();
		mint a = 1;
		rep(i,n){
			res += a * (*this)[i];
			a *= x;
		}
		return res;
	}
	Poly low(int n) const {		// ignore x^n (take first n), but not empty
		return Poly(this->begin(), this->begin()+min(max(n,1),size()));
	}
	Poly rev() const {
		return Poly(this->rbegin(), this->rend());
	}
	friend ostream& operator<<(ostream &o,const Poly& f){
		o << "[";
		rep(i,f.size()){
			o << f[i];
			if(i != f.size()-1) o << ",";
		}
		o << "]";
		return o;
	}

	Poly operator-() const {
		Poly res = *this;
		for(auto& v: res) v = -v;
		return res;
	}
	Poly& operator+=(const mint& c){
		(*this)[0] += c;
		return *this;
	}
	Poly& operator-=(const mint& c){
		(*this)[0] -= c;
		return *this;
	}
	Poly& operator*=(const mint& c){
		for(auto& v: *this) v *= c;
		return *this;
	}
	Poly& operator/=(const mint& c){
		return *this *= mint(1)/mint(c);
	}
	Poly& operator+=(const Poly& r){
		if(size() < r.size()) this->resize(r.size(),0);
		rep(i,r.size()) (*this)[i] += r[i];
		return *this;
	}
	Poly& operator-=(const Poly& r){
		if(size() < r.size()) this->resize(r.size(),0);
		rep(i,r.size()) (*this)[i] -= r[i];
		return *this;
	}
	Poly& operator*=(const Poly& r){
		return *this = multiply(*this,r);
	}

	// 何回も同じrで割り算するなら毎回rinvを計算するのは無駄なので、呼び出し側で一回計算した後直接こっちを呼ぶと良い
	// 取るべきinvの長さに注意
	// 例えば mod r で色々計算したい時は、基本的に deg(r) * 2 長さの多項式を r で割ることになる
	// とはいえいったん rinv を長く計算したらより短い場合はprefix見るだけだし、 rinv としてムダに長いものを渡しても問題ないので
	// 割られる多項式として最大の次数を取ればよい

	Poly quotient(const Poly& r, const Poly& rinv){
		int m = r.size(); assert(r[m-1].v);
		int n = size();
		int s = n-m+1;
		if(s <= 0) return {0};
		return (rev().low(s)*rinv.low(s)).low(s).rev();
	}
	Poly& operator/=(const Poly& r){
		return *this = quotient(r,r.rev().inv(max(size()-r.size(),0)+1));
	}
	Poly& operator%=(const Poly& r){
		*this -= *this/r * r;
		return *this = low(r.size()-1);
	}

	Poly operator+(const mint& c) const {return Poly(*this) += c; }
	Poly operator-(const mint& c) const {return Poly(*this) -= c; }
	Poly operator*(const mint& c) const {return Poly(*this) *= c; }
	Poly operator/(const mint& c) const {return Poly(*this) /= c; }
	Poly operator+(const Poly& r) const {return Poly(*this) += r; }
	Poly operator-(const Poly& r) const {return Poly(*this) -= r; }
	Poly operator*(const Poly& r) const {return Poly(*this) *= r; }
	Poly operator/(const Poly& r) const {return Poly(*this) /= r; }
	Poly operator%(const Poly& r) const {return Poly(*this) %= r; }

	Poly diff() const {
		Poly g(max(size()-1,0));
		rep(i,g.size()) g[i] = (*this)[i+1] * (i+1);
		return g;
	}
	Poly intg() const {
		assert(si(invs) > size());
		Poly g(size()+1);
		rep(i,size()) g[i+1] = (*this)[i] * invs[i+1];
		return g;
	}
	Poly square() const {
		return multiply(*this,*this);
	}

	// 1/f(x) mod x^s
	// N = s = 500000 -> 90ms
	// inv は 5 回 fft(2n) を呼んでいるので、multiply が 3 回 fft(2n) を呼ぶのと比べると
	// だいたい multiply の 5/3 倍の時間がかかる
	// 導出: Newton
	// 		fg = 1 mod x^m
	// 		(fg-1)^2 = 0 mod x^2m
	// 		f(2g-fg^2) = 1 mod x^2m
	// verify: https://judge.yosupo.jp/submission/44938
	Poly inv(int s) const {
		Poly r(s);
		r[0] = mint(1)/at(0);
		for(int n=1;n<s;n*=2){			// 5 times fft : length 2n
			V<mint> f = low(2*n); f.resize(2*n);
			fft(f);
			V<mint> g = r.low(2*n); g.resize(2*n);
			fft(g);
			rep(i,2*n) f[i] *= g[i];
			invfft(f);
			rep(i,n) f[i] = 0;
			fft(f);
			rep(i,2*n) f[i] *= g[i];
			invfft(f);
			for(int i=n;i<min(2*n,s);i++) r[i] -= f[i];
		}
		return r;
	}
};
template<class mint>
V<mint> MultipointEval(const Poly<mint>& f, V<mint> a){
	int Q = a.size();
	int s = 1; while(s < Q) s *= 2;
	V<Poly<mint>> g(s+s,{1});
	rep(i,Q) g[s+i] = {-a[i],1};
	for(int i=s-1;i>0;i--) g[i] = g[i*2] * g[i*2+1];
	g[1] = f % g[1];
	for(int i=2;i<s+Q;i++) g[i] = g[i>>1] % g[i];
	V<mint> res(Q);
	rep(i,Q) res[i] = g[s+i][0];
	return res;
}
/*
	差積 \prod_{i<j} (A_j-A_i)
	f_0 := 1, f_1 := (x-A_0), .. , f_N := (x-A_0)..(x-A_{N-1}) として
	ans = \prod_{i} f_i(A_i)
*/
mint diffProd(V<mint> A){
	int N = si(A);
	int s = 1, h = 0; while(s < N) {s *= 2; h++;}
	V<Poly<mint>> g(s+s,{1});
	rep(i,N) g[s+i] = {-A[i],1};
	for(int i=s-1;i>0;i--) g[i] = g[i*2] * g[i*2+1];
	mint res = 1;
	rep(k,h+1){
		rep(i,1<<k) if(!(i&1)){
			int y = (i+1)<<(h-k), z = (i+2)<<(h-k);
			V<mint> ps;
			for(int p=y;p<min(z,N);p++) ps.eb(A[p]);
			int id = (1<<k)+i;
			auto qs = MultipointEval<mint>(g[id],ps);
			for(auto q: qs) res *= q;
		}
	}
	return res;
}

/*
	SSYT of shape A with values [1,M]
	A: decreasing
	O(M log^2 M)
	多分 O(N log^2 N) にできる (N > M なら ans = 0 に注意)
*/
mint CountSSYT(V<ll> A, ll M){
	if(M >= 1000000) assert(false);
	while(!A.empty() && A.back() == 0) A.pop_back();
	int N = si(A);
	if(N > M) return 0;
	rep(i,M-N) A.eb(0);
	N = M;
	V<mint> B(N); rep(i,N) B[i] = A[i]+N-i;
	reverse(all(B));
	mint numer = diffProd(B);
	mint denom = 1; rep(i,N) denom *= fact[i];
	return numer/denom;
}



/*
	左下と右上で同じなので ans = ()^2 
	() は Gelfand-Tsetlin pattern そのものなので、
	() = # of {SSYT of shape λ := (A_N,..,A_1) で値が [1,N] }

	s_λ(1^n) = \prod_{u \in λ} (n+c(u)) / h(u)
	どちらも差積の形でかける
*/


int main(){
	cin.tie(0);
	ios::sync_with_stdio(false);		//DON'T USE scanf/printf/puts !!
	cout << fixed << setprecision(20);
	InitFact(1000000);

	int N; cin >> N;
	V<ll> A(N); rep(i,N) cin >> A[i]; reverse(all(A));
	cout << CountSSYT(A,N).pow(2) << endl;
}