#404330 (C++17) No.940 ワープ ε=ε=ε=ε=ε=│;p>д＜│

• 問題: No.940 ワープ ε=ε=ε=ε=ε=│;p>д＜│
• ユーザー: sigma425
• 提出日時: 2019-12-03 14:57:32
• 言語: C++17 (gcc 12.3.0 + boost 1.83.0)
• 結果: AC
• 実行時間: 2,375 ms / 5,000 ms
• コード長: 15,657 bytes
• コンパイル時間: 2,756 ms
• コンパイル使用メモリ: 225,248 KB
• 実行使用メモリ: 198,288 KB
• 最終ジャッジ日時: 2024-05-06 08:18:23
• 合計ジャッジ時間: 18,835 ms
• ジャッジサーバーID （参考情報）: judge5 / judge1

入力	結果	実行時間; 実行使用メモリ
testcase_00	AC	76 ms; 42,752 KB
testcase_01	AC	76 ms; 42,752 KB
testcase_02	AC	74 ms; 42,752 KB
testcase_03	AC	94 ms; 44,856 KB
testcase_04	AC	73 ms; 42,496 KB
testcase_05	AC	82 ms; 43,824 KB
testcase_06	AC	76 ms; 43,136 KB
testcase_07	AC	77 ms; 43,264 KB
testcase_08	AC	77 ms; 43,264 KB
testcase_09	AC	77 ms; 43,264 KB
testcase_10	AC	77 ms; 43,264 KB
testcase_11	AC	77 ms; 43,008 KB
testcase_12	AC	81 ms; 43,740 KB
testcase_13	AC	76 ms; 43,264 KB
testcase_14	AC	73 ms; 42,752 KB
testcase_15	AC	477 ms; 81,548 KB
testcase_16	AC	2,045 ms; 198,288 KB
testcase_17	AC	659 ms; 81,548 KB
testcase_18	AC	722 ms; 81,548 KB
testcase_19	AC	651 ms; 81,420 KB
testcase_20	AC	795 ms; 81,548 KB
testcase_21	AC	601 ms; 81,548 KB
testcase_22	AC	1,240 ms; 120,332 KB
testcase_23	AC	979 ms; 120,336 KB
testcase_24	AC	1,223 ms; 120,332 KB
testcase_25	AC	2,318 ms; 198,288 KB
testcase_26	AC	2,375 ms; 198,164 KB

ソースコード

#include <bits/stdc++.h>
#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 all(c) c.begin(),c.end()
#define pb push_back
#define fs first
#define sc second
#define chmin(x,y) x=min(x,y)
#define chmax(x,y) x=max(x,y)
using namespace std;
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;
}
using ll = long long;
template<class T> using V = vector<T>;
template<class T> using VV = vector<vector<T>>;
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n-1); }

#ifdef LOCAL
#define show(x) cerr << "LINE" << __LINE__ << " : " << #x << " = " << (x) << endl
#else
#define show(x) true
#endif

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;}
	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));
	}
	ModInt pow(ll p) const {
		ModInt a = 1;
		ModInt x = *this;
		while(p){
			if(p&1) a *= x;
			x *= x;
			p >>= 1;
		}
		return a;
	}
	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<1000000007>;


int bsr(int x) { return 31 - __builtin_clz(x); }
using D = double;
const D pi = acos(-1);
using Pc = complex<D>;

void fft(bool type, vector<Pc> &c){	//multiply : false -> mult -> true
	static vector<Pc> buf[30];
	int N = c.size();
	int s = bsr(N);
	assert(1<<s == N);
	if(buf[s].empty()){
		buf[s]=vector<Pc>(N);
		rep(i,N) buf[s][i] = polar<D>(1,i*2*pi/N);
	}
	vector<Pc> a(N),b(N);
	copy(begin(c),end(c),begin(a));
	rep1(i,s){
		int W = 1<<(s-i);
		for(int y=0;y<N/2;y+=W){
			Pc now = buf[s][y];
			if(type) now = conj(now);
			rep(x,W){
				auto l =       a[y<<1 | x];
				auto r = now * a[y<<1 | x | W];
				b[y | x]        = l+r;
				b[y | x | N>>1] = l-r;
			}
		}
		swap(a,b);
	}
	copy(begin(a),end(a),begin(c));
}
template<class Mint>
vector<Mint> multiply_fft(const vector<Mint>& x,const vector<Mint>& y){
	if(x.empty() || y.empty()) return {};
	const int B = 15;
	const int K = 2;
	int S = x.size()+y.size()-1;
	int N = 1; while(N<S) N*=2;
	vector<Pc> a[K],b[K];
	rep(t,K){
		a[t] = vector<Pc>(N);
		b[t] = vector<Pc>(N);
		rep(i,x.size()) a[t][i] = Pc( (x[i].v >> (t*B)) & ((1<<B)-1) , 0 );
		rep(i,y.size()) b[t][i] = Pc( (y[i].v >> (t*B)) & ((1<<B)-1) , 0 );
		fft(false,a[t]);
		fft(false,b[t]);
	}
	vector<Mint> z(S);
	vector<Pc> c(N);
	Mint base = 1;
	rep(t,K+K-1){
		fill_n(begin(c),N,Pc(0,0));
		rep(xt,K){
			int yt = t-xt;
			if(0<=yt && yt<K){
				rep(i,N) c[i] += a[xt][i] * b[yt][i];
			}
		}
		fft(true,c);
		rep(i,S){
			c[i] *= 1.0/N;
			z[i] += Mint(ll(round(c[i].real()))) * base;
		}
		base *= 1<<B;
	}
	return z;
}
template<class D>
struct Poly{
	vector<D> v;
	int size() const{ return v.size();}	//deg+1
	Poly(){}
	Poly(vector<D> _v) : v(_v){shrink();}

	Poly& shrink(){
		while(!v.empty()&&v.back()==D(0)) v.pop_back();
		return *this;
	}
	D at(int i) const{
		return (i<size())?v[i]:D(0);
	}
	void set(int i,const D& x){		//v[i] := x
		if(i>=size() && !x) return;
		while(i>=size()) v.push_back(D(0));
		v[i]=x;
		shrink();
		return;
	}
	D operator()(D x) const {
		D res = 0;
		int n = size();
		D a = 1;
		rep(i,n){
			res += a*v[i];
			a *= x;
		}
		return res;
	}

	Poly operator+(const Poly &r) const{
		int N=max(size(),r.size());
		vector<D> ret(N);
		rep(i,N) ret[i]=at(i)+r.at(i);
		return Poly(ret);
	}
	Poly operator-(const Poly &r) const{
		int N=max(size(),r.size());
		vector<D> ret(N);
		rep(i,N) ret[i]=at(i)-r.at(i);
		return Poly(ret);
	}
	Poly operator-() const{
		int N=size();
		vector<D> ret(N);
		rep(i,N) ret[i] = -at(i);
		return Poly(ret);
	}
	Poly operator*(const Poly &r) const{
		if(size()==0||r.size()==0) return Poly();
		return mul_fft(r);									// FFT or NTT ?
	}
	Poly operator*(const D &r) const{
		int N=size();
		vector<D> ret(N);
		rep(i,N) ret[i]=v[i]*r;
		return Poly(ret);
	}
	Poly operator/(const D &r) const{
		return *this * r.inv();
	}
	Poly operator/(const Poly &y) const{
		return div_fast(y);
	}
	Poly operator%(const Poly &y) const{
		return rem_fast(y);
//		return rem_naive(y);
	}
	Poly operator<<(const int &n) const{	// *=x^n
		assert(n>=0);
		int N=size();
		vector<D> ret(N+n);
		rep(i,N) ret[i+n]=v[i];
		return Poly(ret);
	}
	Poly operator>>(const int &n) const{	// /=x^n
		assert(n>=0);
		int N=size();
		if(N<=n) return Poly();
		vector<D> ret(N-n);
		rep(i,N-n) ret[i]=v[i+n];
		return Poly(ret);
	}
	bool operator==(const Poly &y) const{
		return v==y.v;
	}
	bool operator!=(const Poly &y) const{
		return v!=y.v;
	}

	Poly& operator+=(const Poly &r) {return *this = *this+r;}
	Poly& operator-=(const Poly &r) {return *this = *this-r;}
	Poly& operator*=(const Poly &r) {return *this = *this*r;}
	Poly& operator*=(const D &r) {return *this = *this*r;}
	Poly& operator/=(const Poly &r) {return *this = *this/r;}
	Poly& operator/=(const D &r) {return *this = *this/r;}
	Poly& operator%=(const Poly &y) {return *this = *this%y;}
	Poly& operator<<=(const int &n) {return *this = *this<<n;}
	Poly& operator>>=(const int &n) {return *this = *this>>n;}

	Poly diff() const {
		int n = size();
		if(n == 0) return Poly();
		V<D> u(n-1);
		rep(i,n-1) u[i] = at(i+1) * (i+1);
		return Poly(u);
	}
	Poly intg() const {
		int n = size();
		V<D> u(n+1);
		rep(i,n) u[i+1] = at(i) / (i+1);
		return Poly(u);
	}

	Poly pow(long long n, int L) const {		// f^n, ignoring x^L,x^{L+1},..
		Poly a({1});
		Poly x = *this;
		while(n){
			if(n&1){
				a *= x;
				a = a.strip(L);
			}
			x *= x;
			x = x.strip(L);
			n /= 2;
		}
		return a;
	}

	/*
		[x^0~n] exp(f) = 1 + f + f^2 / 2 + f^3 / 6 + ..
		f(0) should be 0

		O((N+n) log n)	(N = size())
		NTT, -O3
		- N = n = 100000 : 200 [ms]
		- N = n = 200000 : 400 [ms]
		- N = n = 500000 : 1000 [ms]
	*/
	Poly exp(int n) const {
		assert(at(0) == 0);
		Poly f({1}), g({1});
		for(int i=1;i<=n;i*=2){
			g = (g*2 - f*g*g).strip(i);
			Poly q = (this->diff()).strip(i-1);
			Poly w = (q + g * (f.diff() - f*q)) .strip(2*i-1);
			f = (f + f * (*this - w.intg()).strip(2*i)) .strip(2*i);
		}
		return f.strip(n+1);
	}

	/*
		[x^0~n] log(f) = log(1-(1-f)) = - (1-f) - (1-f)^2 / 2 - (1-f)^3 / 3 - ...
		f(0) should be 1
		O(n log n)

		NTT, -O3
		1e5 : 140 [ms]
		2e5 : 296 [ms]
		5e5 : 640 [ms]
		1e6 : 1343 [ms]
	*/
	Poly log(int n) const {
		assert(at(0) == 1);
		auto f = strip(n+1);
		return (f.diff() * f.inv(n)).strip(n).intg();
	}

	/*
		[x^0~n] sqrt(f)
		f(0) should be 1
		いや平方剰余なら何でもいいと思うけど探すのがめんどくさいので
		+- 2通りだけど 定数項が 1 の方
		O(n log n)

		NTT, -O3
		1e5 : 234 [ms]
		2e5 : 484 [ms]
		5e5 : 1000 [ms]
		1e6 : 2109 [ms]
	*/
	Poly sqrt(int n) const {
		assert(at(0) == 1);
		Poly f = strip(n+1);
		Poly g({1});
		for(int i=1; i<=n; i*=2){
			g = (g + f.strip(2*i)*g.inv(2*i-1)) / 2;
		}
		return g.strip(n+1);
	}

	/*
		[x^0~n] f^-1 = (1-(1-f))^-1 = (1-f) + (1-f)^2 + ...
		f * f.inv(n) = 1 + x^n * poly
		f(0) should be non0
		O(n log n)
	*/
	Poly inv(int n) const {
		assert(at(0) != 0);
		Poly f = strip(n+1);
		Poly g({at(0).inv()});
		for(int i=1; i<=n; i*=2){		//need to strip!!
			g *= (Poly({2}) - f.strip(2*i)*g).strip(2*i);
		}
		return g.strip(n+1);
	}	

	Poly exp_naive(int n) const {
		assert(at(0) == 0);
		Poly res;
		Poly fk({1});
		rep(k,n+1){
			res += fk;
			fk *= *this;
			fk = fk.strip(n+1) / (k+1);
		}
		return res;
	}
	Poly log_naive(int n) const {
		assert(at(0) == 1);
		Poly res;
		Poly g({1});
		rep1(k,n){
			g *= (Poly({1}) - *this);
			g = g.strip(n+1);
			res -= g / k;
		}
		return res;
	}


	Poly mul_naive(const Poly &r) const{
		int N=size(),M=r.size();
		vector<D> ret(N+M-1);
		rep(i,N) rep(j,M) ret[i+j]+=at(i)*r.at(j);
		return Poly(ret);
	}
	Poly mul_ntt(const Poly &r) const{
		return Poly(multiply_ntt(v,r.v));
	}
	Poly mul_fft(const Poly &r) const{
		return Poly(multiply_fft(v,r.v));
	}

	Poly div_fast_with_inv(const Poly &inv, int B) const {
		return (*this * inv)>>(B-1);
	}
	Poly div_fast(const Poly &y) const{
		if(size()<y.size()) return Poly();
		int n = size();
		return div_fast_with_inv(y.inv_div(n-1),n);
	}
	Poly rem_naive(const Poly &y) const{
		Poly x = *this;
		while(y.size()<=x.size()){
			int N=x.size(),M=y.size();
			D coef = x.v[N-1]/y.v[M-1];
			x -= (y<<(N-M))*coef;
		}
		return x;
	}
	Poly rem_fast(const Poly &y) const{
		return *this - y * div_fast(y);
	}
	Poly strip(int n) const {	//ignore x^n , x^n+1,...
		vector<D> res = v;
		res.resize(min(n,size()));
		return Poly(res);
	}
	Poly rev(int n = -1) const {	//ignore x^n ~  ->  return x^(n-1) * f(1/x)
		vector<D> res = v;
		if(n!=-1) res.resize(n);
		reverse(all(res));
		return Poly(res);
	}

	/*
		f.inv_div(n) = x^n / f
		f should be non0
		O((N+n) log n)

		for division
	*/
	Poly inv_div(int n) const {
		n++;
		int d = size() - 1;
		assert(d != -1);
		if(n < d) return Poly();
		Poly a = rev();
		Poly g({at(d).inv()});
		for(int i=1; i+d<=n; i*=2){		//need to strip!!
			g *= (Poly({2})-a.strip(2*i)*g).strip(2*i);
		}
		return g.rev(n-d);
	}


	friend ostream& operator<<(ostream &o,const Poly& x){
		if(x.size()==0) return o<<0;
		rep(i,x.size()) if(x.v[i]!=D(0)){
			o<<x.v[i]<<"x^"<<i;
			if(i!=x.size()-1) o<<" + ";
		}
		return o;
	}
};

V<mint> fact,ifact;
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){
	fact.resize(N);
	ifact.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);
}

vector<mint> extended(int n, const vector< vector<mint> >& coeffs, const vector<mint>& terms) {

	vector<mint> ret(max<int>(n + 1, terms.size()));
	copy(terms.begin(), terms.end(), ret.begin());
	const int order = coeffs.size() - 1;
	const int deg = coeffs[0].size() - 1;
	assert((int) terms.size() >= order);
	for (int m = terms.size(); m <= n; ++m) {
		mint s = 0;
		for (int i = 1; i <= order; ++i) {
			int k = m - i;
			mint t = ret[k];
			for (int d = 0; d <= deg; ++d) {
				s += t * coeffs[i][d];
				t *= k;
			}
		}
		mint denom = 0, mpow = 1;
		for (int d = 0; d <= deg; ++d) {
			denom += mpow * coeffs[0][d];
			mpow *= m;
		}
		ret[m] = -s/denom;
	}
	return ret;
}

vector< vector<mint> > find_recurrence_relation(vector<mint> terms, int deg, int ord = -1, bool verify=true) {

	if(ord != -1){		//given order
		int n = (deg+1)*(ord+1)+ord-1;
		while((int)terms.size()>n) terms.pop_back();
	}

	const int n = terms.size();
	const int B = (n + 2) / (deg + 2); // number of blocks
	const int C = B * (deg + 1); // number of columns
	const int R = n - (B - 1); // number of rows
	assert(B >= 2); assert(R >= C - 1);

	auto error = [] (int order, int deg) {
		fprintf(stderr, 
			"Error: Could not find a recurrence relation "
			"of order <= %d and degree <= %d.\n\n", 
			order, deg);
		assert(0);
	};

	vector< vector<mint> > mat(R, vector<mint>(C));
	for (int y = 0; y < R; ++y) {
		for (int b = 0; b < B; ++b) {
			mint v = terms[y+b];
			for (int d = 0; d <= deg; ++d) {
				mat[y][b * (deg + 1) + d] = v;
				v *= y+b;
			}
		}
	}

	int rank = 0;
	for (int x = 0; x < C; ++x) {
		int pivot = -1;
		for (int y = rank; y < R; ++y) if (mat[y][x] != 0) {
			pivot = y; break;
		}
		if (pivot < 0) break;
		if (pivot != rank) swap(mat[rank], mat[pivot]);
		mint inv = mat[rank][x].inv();
		for (int x2 = x; x2 < C; ++x2) mat[rank][x2] *= inv;
		for (int y = rank + 1; y < R; ++y) if (mat[y][x]) {
			mint c = -mat[y][x];
			for (int x2 = x; x2 < C; ++x2) {
				mat[y][x2] += c * mat[rank][x2];
			}
		}
		++rank;
	}

	if (rank == C) error(B - 1, deg);

	for (int y = rank - 1; y >= 0; --y) if (mat[y][rank]) {
		assert(mat[y][y] == 1);
		mint c = -mat[y][rank];
		for (int y2 = 0; y2 < y; ++y2) {
			mat[y2][rank] += c * mat[y2][y];
		}
	}

	int order = rank / (deg + 1);

	vector< vector<mint> > ret(order + 1, vector<mint>(deg + 1));
	ret[0][rank % (deg + 1)] = 1;
	for (int y = rank - 1; y >= 0; --y) {
		int k = order - y / (deg + 1), d = y % (deg + 1);
		ret[k][d] = -mat[y][rank];
	}

	if (verify) {
		auto extended_terms = extended(n - 1, ret, 
				vector<mint>(terms.begin(), terms.begin() + order));
		for (int i = 0; i < (int) terms.size(); ++i) {
			if (terms[i] != extended_terms[i]) error(B - 1, deg);
		}
	}

	auto verbose = [&] {
		int last = verify ? n - 1 : order + R - 1;
		fprintf(stderr, 
			"[ Found a recurrence relation ]\n"
			"- order %d\n"
			"- degree %d\n"
			"- verified up to a(%d) (number of non-trivial terms: %d)\n",
			order, deg, last, (last + 1) - ((deg + 2) * (order + 1) - 2)
		);
		fprintf(stderr, "{\n");
		for (int k = 0; k <= order; ++k) {
			fprintf(stderr, "  {");
			for (int d = 0; d <= deg; ++d) {
				if (d) fprintf(stderr, ", ");
				fprintf(stderr, "%d", ret[k][d].v);
			}
			fprintf(stderr, "}%s\n", k == order ? "" : ",");
		}
		fprintf(stderr, "}\n\n");
	};
	verbose();

	return ret;
}

void show_extended_sequence(int n, const vector<mint>& terms, int degree, int order = -1) {
	auto coeffs = find_recurrence_relation(terms, degree, order);
	auto extended_terms = extended(n, coeffs, terms);
	for (int i = 0; i < (int) extended_terms.size(); ++i) {
		printf("%d %d\n", i, extended_terms[i].v);
	}
	puts("");
}
V<mint> get_extended_sequence(int n, const vector<mint>& terms, int degree, int order = -1) {
	auto coeffs = find_recurrence_relation(terms, degree, order);
	return extended(n, coeffs, terms);
}
int main(){
	cin.tie(0);
	ios::sync_with_stdio(false);		//DON'T USE scanf/printf/puts !!
	cout << fixed << setprecision(20);
	InitFact(5000010);
	int X,Y,Z; cin >> X >> Y >> Z;
	int ZZ = Z;
	chmin(Z,100000);
	Poly<mint> y({1,-2,1});
	Poly<mint> f({1});
	while((int)f.size() < X+Y+Z+100){
		f *= (y + Poly<mint>({1}));
		y *= y;
	}
	show(f.size());

	mint ans = 0;
	if(X+Y+Z == 0){
		cout << 1 << endl;
		return 0;
	}
	
	V<mint> vals;
	rep(d,20){
		mint ans = 0;
		rep1(k,f.size()){
			ans += Choose(X+k-1,k-1) * Choose(Y+k-1,k-1) * Choose(Z+d+k-1,k-1) * f.at(k-1);
		}
		vals.pb(ans);
	}
	auto v = get_extended_sequence(ZZ-Z,vals,2,3);
	cout << v[ZZ-Z] << endl;
}