1 条题解

  • 0
    @ 2022-3-19 7:55:30

    一次减法卷积即可。

    减法卷积就是翻转后再卷积。

    Code

    #include <algorithm>
    #include <stdio.h>
    #include <vector>
    typedef long long llt;
    typedef unsigned uint;typedef unsigned long long ullt;
    typedef bool bol;typedef char chr;typedef void voi;
    typedef double dbl;
    template<typename T>bol _max(T&a,T b){return(a<b)?a=b,true:false;}
    template<typename T>bol _min(T&a,T b){return(b<a)?a=b,true:false;}
    template<typename T>T power(T base,T index,T mod){return((index<=1)?(index?base:1):(power(base*base%mod,index>>1,mod)*power(base,index&1,mod)))%mod;}
    template<typename T>T lowbit(T n){return n&-n;}
    template<typename T>T gcd(T a,T b){return b?gcd(b,a%b):a;}
    template<typename T>T lcm(T a,T b){return(a!=0||b!=0)?a/gcd(a,b)*b:(T)0;}
    template<typename T>T exgcd(T a,T b,T&x,T&y){if(!b)return y=0,x=1,a;T ans=exgcd(b,a%b,y,x);y-=a/b*x;return ans;}
    template<const ullt p=998244353>
    class mod_ullt
    {
        private:
            ullt v;
            inline ullt chg(ullt w){return(w<p)?w:w-p;}
            inline mod_ullt _chg(ullt w){mod_ullt ans;ans.v=(w<p)?w:w-p;return ans;}
        public:
            mod_ullt():v(0){}
            mod_ullt(ullt v):v(v%p){}
            bol empty(){return!v;}
            inline ullt val(){return v;}
            friend bol operator<(mod_ullt a,mod_ullt b){return a.v<b.v;}
            friend bol operator>(mod_ullt a,mod_ullt b){return a.v>b.v;}
            friend bol operator<=(mod_ullt a,mod_ullt b){return a.v<=b.v;}
            friend bol operator>=(mod_ullt a,mod_ullt b){return a.v>=b.v;}
            friend bol operator==(mod_ullt a,mod_ullt b){return a.v==b.v;}
            friend bol operator!=(mod_ullt a,mod_ullt b){return a.v!=b.v;}
            inline friend mod_ullt operator+(mod_ullt a,mod_ullt b){return a._chg(a.v+b.v);}
            inline friend mod_ullt operator-(mod_ullt a,mod_ullt b){return a._chg(a.v+a.chg(p-b.v));}
            inline friend mod_ullt operator*(mod_ullt a,mod_ullt b){return a.v*b.v;}
            friend mod_ullt operator/(mod_ullt a,mod_ullt b){return b._power(p-2)*a.v;}
            inline mod_ullt operator-(){return _chg(p-v);}
            mod_ullt sqrt()
            {
                if(power(v,(p-1)>>1,p)!=1)return 0;
                mod_ullt b=1;do b++;while(b._power((p-1)>>1)==1);
                ullt t=p-1,s=0,k=1;while(!(t&1))s++,t>>=1;
                mod_ullt x=_power((t+1)>>1),e=_power(t);
                while(k<s)
                {
                    if(e._power(1llu<<(s-k-1))!=1)x*=b._power((1llu<<(k-1))*t);
                    e=_power(p-2)*x*x,k++;
                }
                return _min(x,-x),x;
            }
            mod_ullt inv(){return _power(p-2);}
            mod_ullt _power(ullt index){mod_ullt ans(1),w(v);while(index){if(index&1)ans*=w;w*=w,index>>=1;}return ans;}
            voi read(){v=0;chr c;do c=getchar();while(c>'9'||c<'0');do v=(c-'0'+v*10)%p,c=getchar();while(c>='0'&&c<='9');v%=p;}
            voi print()
            {
            	static chr C[20];uint tp=0;
            	ullt w=v;do C[tp++]=w%10+'0',w/=10;while(w);
            	while(tp--)putchar(C[tp]);
            }
            voi println(){print(),putchar('\n');}
            mod_ullt operator++(int){mod_ullt ans=*this;return v=chg(v+1),ans;}
        public:
            inline ullt&operator()(){return v;}
            inline mod_ullt&operator+=(mod_ullt b){return*this=_chg(v+b.v);}
            inline mod_ullt&operator-=(mod_ullt b){return*this=_chg(v+chg(p-b.v));}
            inline mod_ullt&operator*=(mod_ullt b){return*this=v*b.v;}
            mod_ullt&operator/=(mod_ullt b){return*this=b._power(p-2)*v;}
            mod_ullt&operator++(){return v=chg(v+1),*this;}
    };
    template<const ullt p=998244353,const ullt g=3>
    class poly_NTT
    {
        public:
    		typedef mod_ullt<p>modint;
            typedef std::vector<modint>modvec;
    	private:
    		modvec V;
    	public:
    		class NTT
    		{
    			private:
    				uint n;uint*T;modint*G;
    			public:
    				NTT():n(0),T(NULL),G(NULL){}
    				NTT(uint len)
    				{
    					n=1;while(n<len)n<<=1;
                        T=new uint[n],G=new modint[n],T[0]=0,G[0]=1;
    					for(uint i=1;i<n;i++)T[i]=((i&1)?n|T[i>>1]:T[i>>1])>>1;
                        modint w=power(g,(p-1)/n,p),*End=G+n;
                        for(modint*_G=G+1;_G<End;_G++)*_G=_G[-1]*w;
    				}
    				~NTT(){if(T!=NULL)delete[]T,delete[]G,T=NULL,G=NULL;}
    				inline uint size(){return n;}
    				voi bzr(uint len)
    				{
    					n=1;while(n<len)n<<=1;
                        if(T!=NULL)delete[]T,delete[]G;
                        T=new uint[n],G=new modint[n],T[0]=0,G[0]=1;
    					for(uint i=1;i<n;i++)T[i]=((i&1)?n|T[i>>1]:T[i>>1])>>1;
                        modint w=power(g,(p-1)/n,p),*End=G+n;
                        for(modint*_G=G+1;_G<End;_G++)*_G=_G[-1]*w;
    				}
    				voi ntt(modvec &x,bol op)
    				{
    					if(x.size()<n)x.resize(n);
                        for(uint i=0;i<n;i++)if(T[i]<i)std::swap(x[i],x[T[i]]);
    					for(uint i=1;i<n;i<<=1)for(uint j=0;j<n;j+=i<<1)
    					{
    						modint*w=G;
    						for(uint k=0;k<i;k++,w+=n/(2*i))
    						{
    							modint t=x[i+j+k]*(*w);
    							x[i+j+k]=x[j+k]-t,x[j+k]+=t;
    						}
    					}
    					if(op)
    					{
    						for(uint i=1;i*2<n;i++)std::swap(x[i],x[n-i]);
    						modint t=modint(n).inv();for(uint i=0;i<n;i++)x[i]*=t;
    					}
    				}
    				inline modint Omega(uint n){return G[n%size()];}
    				NTT&operator=(NTT b)
    				{
    					if(T!=NULL)delete[]T,delete[]G,T=NULL,G=NULL;
    					if(b.T==NULL)return*this;
    					T=new uint[n],G=new modint[n=b.n];
    					for(uint i=0;i<n;i++)T[i]=b.T[i],G[i]=b.G[i];
    					return*this;
    				}
    		};
    		class DIFDIT
    		{
    			private:
    				uint n;modint*G;
    			public:
    				DIFDIT():n(0),G(NULL){}
    				DIFDIT(uint len)
    				{
    					n=1;while(n<len)n<<=1;
                        G=new modint[n],G[0]=1;
                        modint w=power(g,(p-1)/n,p),*End=G+n;
                        for(modint*_G=G+1;_G<End;_G++)*_G=_G[-1]*w;
    				}
    				~DIFDIT(){if(G!=NULL)delete[]G,G=NULL;}
    				inline uint size(){return n;}
    				voi bzr(uint len)
    				{
    					n=1;while(n<len)n<<=1;
                        if(G!=NULL)delete[]G;
                        G=new modint[n],G[0]=1;
                        modint w=power(g,(p-1)/n,p),*End=G+n;
                        for(modint*_G=G+1;_G<End;_G++)*_G=_G[-1]*w;
    				}
    				voi dif(modvec &x)
    				{
    					if(x.size()<n)x.resize(n);
    					for(uint i=n>>1;i;i>>=1)for(uint j=0;j<n;j+=i<<1) 
    					{
    						modint*w=G;
    						for(uint k=0;k<i;k++,w+=n/(2*i))
    						{
    							modint u=x[j+k],t=x[i+j+k];
    							x[j+k]=u+t,x[i+j+k]=(u-t)*(*w);
    						}
    					}
    				}
    				voi dit(modvec &x)
    				{
    					if(x.size()<n)x.resize(n);
    					for(uint i=1;i<n;i<<=1)for(uint j=0;j<n;j+=i<<1)
    					{
    						modint*w=G;
    						for(uint k=0;k<i;k++,w+=n/(2*i))
    						{
    							modint t=x[i+j+k]*(*w);
    							x[i+j+k]=x[j+k]-t,x[j+k]+=t;
    						}
    					}
    					for(uint i=1;i*2<n;i++)std::swap(x[i],x[n-i]);
    					modint t=modint(n).inv();for(uint i=0;i<n;i++)x[i]*=t;
    				} 
    				DIFDIT&operator=(DIFDIT b)
    				{
    					if(G!=NULL)delete[]G,G=NULL;
    					if(b.G==NULL)return*this;
    					G=new modint[n=b.n];
    					for(uint i=0;i<n;i++)G[i]=b.G[i];
    					return*this;
    				}
    		};
    	public:
    		poly_NTT(){}
    		poly_NTT(uint n){V.resize(n);}
    		poly_NTT(modvec V):V(V){}
    		inline voi bzr(){V.clear();}
    		inline voi push(modint v){V.push_back(v);}
    		inline voi pop(){V.pop_back();}
    		inline voi cut_zero(){while(!V.empty()&&V.back().empty())V.pop_back();}
    		inline voi chg_siz(uint n){V.resize(n);}
    		inline voi chg_deg(uint n){V.resize(n+1);}
    		inline bol empty(){return cut_zero(),V.empty();}
    		inline uint size(){return V.size();}
    		inline uint deg(){return V.size()-1;}
    		inline modint val(uint n){return(n<size())?V[n]:0;}
            inline modvec GET(){return V;}
            poly_NTT reverse()
            {
                poly_NTT ans;for(uint i=size()-1;~i;i--)ans.push(V[i]);
                return ans;
            }
    		friend poly_NTT operator+(poly_NTT a,poly_NTT b)
    		{
    			if(a.size()<b.size())a.chg_siz(b.size());
    			for(uint i=0;i<b.size();i++)a[i]+=b[i];
    			a.cut_zero();return a;
    		}
    		friend poly_NTT operator+(poly_NTT a,modint v)
    		{
    			if(!a.size())a.chg_siz(1);
    			a[0]+=v;return a;
    		}
    		friend poly_NTT operator+(modint v,poly_NTT a)
    		{
    			if(!a.size())a.chg_siz(1);
    			a[0]+=v;return a;
    		}
    		friend poly_NTT operator-(poly_NTT a){return a*modint(p-1);}
    		friend poly_NTT operator-(poly_NTT a,poly_NTT b)
    		{
    			if(a.size()<b.size())a.chg_siz(b.size());
    			for(uint i=0;i<b.size();i++)a[i]-=b[i];
    			a.cut_zero();return a;
    		}
    		friend poly_NTT operator-(poly_NTT a,modint v)
    		{
    			if(!a.size())a.chg_siz(1);
    			a[0]-=v;return a;
    		}
    		friend poly_NTT operator-(modint v,poly_NTT a)
    		{
    			if(!a.size())a.chg_siz(1);
    			a[0]-=v;return-a;
    		}
    		friend poly_NTT operator*(poly_NTT a,poly_NTT b)
    		{
                modvec &A=a.V,&B=b.V;DIFDIT s(A.size()+B.size());
                s.dif(A),s.dif(B);for(uint i=0;i<s.size();i++)A[i]*=B[i];
                s.dit(A),a.cut_zero();return a;
    		}
            friend poly_NTT operator*(poly_NTT A,modint b)
            {
                for(auto&s:A.V)s*=b;
                return A;
            }
            friend poly_NTT operator*(modint b,poly_NTT A)
            {
                for(auto&s:A.V)s*=b;
                return A;
            }
            friend poly_NTT operator<<(poly_NTT a,uint k)
            {
            	poly_NTT ans;ans.chg_siz(k);for(auto v:a.V)ans.push(v);
            	return ans;
            }
            friend poly_NTT operator>>(poly_NTT a,uint k)
            {
            	poly_NTT ans;for(uint i=k;i<a.size();i++)ans.push(a[i]);
            	return ans;
            }
            friend poly_NTT sub_mul(poly_NTT a,poly_NTT b)
            {
                uint len=(a=a.reverse()).size();
                modvec &A=a.V,&B=b.V;
                DIFDIT s(len+B.size());
                s.dif(A),s.dif(B);for(uint i=0;i<s.size();i++)A[i]*=B[i];
                s.dit(A),a.chg_siz(len),a=a.reverse();return a;
            }
            poly_NTT inv(){return inv(size());}
            poly_NTT inv(uint prec)
            {
                modvec ans;DIFDIT s;ans.push_back(V[0].inv());
                for(uint tp=1;tp<prec;tp<<=1)
                {
                    modvec f(tp<<1),t=ans;
                    for(uint i=0;i<(tp<<1);++i)f[i]=val(i);
                    s.bzr(tp<<1),s.dif(f),s.dif(t);
                    for(uint i=0;i<(tp<<1);++i)f[i]=1-f[i]*t[i];
                    s.dit(f);
                    for(uint i=0;i<tp;++i)f[i]=f[i+tp],f[i+tp]=0;
                    s.dif(f);
                    for(uint i=(tp<<1)-1;~i;--i)f[i]*=t[i];
                    s.dit(f),ans.resize(tp<<1);
                	for(uint i=0;i<tp;++i)ans[i+tp]=f[i];
                }
                ans.resize(prec);return ans;
            }
            poly_NTT diff()
            {
                poly_NTT ans;for(uint i=1;i<size();i++)ans.push(i*V[i]);
                return ans;
            }
            poly_NTT inte()
            {
                if(!size())return*this;
                poly_NTT ans(size()+1);ans[1]=1;
                for(uint i=2;i<=size();i++)ans[i]=-ans[p%i]*(p/i);
                for(uint i=1;i<=size();i++)ans[i]*=V[i-1];
                return ans;
            }
            poly_NTT ln(){return ln(size());}
            poly_NTT ln(uint prec)
            {
                poly_NTT a=this->diff()*this->inv(prec);a.chg_siz(prec),a=a.inte(),a.chg_siz(prec);return a;
            }
            poly_NTT exp(){return exp(size());}
            poly_NTT exp(uint prec)
            {
                poly_NTT ans;modvec Inv;ans.push(1),Inv.push_back(1);
                for(uint tp=1;tp<prec;tp<<=1)
                {
                    modvec f,ff=ans.diff().V;
                    for(uint i=0;i<(tp<<1);i++)f.push_back(val(i));
                    f[0]=1;DIFDIT s(tp<<2);s.dif(ans.V),s.dif(Inv);
                    for(uint i=0;i<(tp<<2);i++)Inv[i]*=2-Inv[i]*ans[i];
                    s.dit(Inv),Inv.resize(tp);s.dif(Inv);
                    for(uint i=0;i<(tp<<2);i++)Inv[i]*=2-Inv[i]*ans[i];
                    s.dit(Inv),Inv.resize(tp<<1);s.dif(Inv);s.dif(ff);
                    for(uint i=0;i<(tp<<2);i++)ff[i]*=Inv[i];
                    s.dit(ff);f=(f-poly_NTT(ff).inte()).V;s.dif(f);
                    for(uint i=0;i<(tp<<2);i++)ans[i]*=f[i];
                    s.dit(Inv),s.dit(ans.V),ans.chg_siz(tp<<1);
                }
                ans.chg_siz(prec);return ans;
            }
            friend poly_NTT operator/(poly_NTT a,poly_NTT b)
            {
                a.cut_zero(),b.cut_zero();if(a.size()<b.size())return poly_NTT();
                poly_NTT ans=a.reverse()*b.reverse().inv(a.size()-b.size()+1);
                ans.chg_siz(a.size()-b.size()+1);return ans.reverse();
            }
            friend poly_NTT operator%(poly_NTT a,poly_NTT b){return a-a/b*b;}
    	public:
    		inline modint&operator[](uint n){return V[n];};
            poly_NTT&operator+=(poly_NTT b)
            {
    			if(size()<b.size())chg_siz(b.size());
    			for(uint i=0;i<b.size();i++)V[i]+=b[i];
    			cut_zero();return*this;
            }
            inline poly_NTT&operator+=(modint v)
            {
            	if(!size())chg_siz(1);
            	V[0]+=v;return*this;
            }
            poly_NTT&operator-=(poly_NTT b)
            {
    			if(size()<b.size())chg_siz(b.size());
    			for(uint i=0;i<b.size();i++)V[i]-=b[i];
    			cut_zero();return*this;
            }
            inline poly_NTT&operator-=(modint v)
            {
            	if(!size())chg_siz(1);
            	V[0]-=v;return*this;
            }
            poly_NTT&operator*=(poly_NTT b)
            {
                modvec &A=V,&B=b.V;
                DIFDIT s(A.size()+B.size());
                s.dif(A),s.dif(B);
                for(uint i=0;i<s.size();i++)A[i]*=B[i];
                s.dit(A),cut_zero();return*this;
            }
            poly_NTT&operator*=(modint v)
            {
            	for(auto&t:V)t*=v;
    			return*this;
            }
            poly_NTT&operator/=(poly_NTT b){return*this=*this/b;}
            poly_NTT&operator%=(poly_NTT b){return*this=*this%b;}
            poly_NTT&operator<<=(uint v){return*this=*this<<v;}
            poly_NTT&operator>>=(uint v){return*this=*this>>v;}
    };
    template<const ullt p=998244353,const ullt g=3>
    class poly_eval
    {
        public:
    		typedef mod_ullt<p>modint;
            typedef std::vector<modint>modvec;
            typedef poly_NTT<p,g>poly;
        private:
            std::vector<poly>G,User;modvec X;
            voi mult_eval_dfs1(uint u,uint l,uint r)
            {
                if(l+1==r){G[u].push(1),G[u].push(-X[l]);return;}
                uint mid=(l+r)/2;mult_eval_dfs1(u<<1,l,mid),mult_eval_dfs1(u<<1|1,mid,r),G[u]=G[u<<1]*G[u<<1|1];
            }
            voi mult_eval_dfs2(uint u,uint l,uint r)
            {
                User.back().chg_siz(r-l);
                if(l+1==r){X[l]=User.back().val(0);return;}
                uint mid=(l+r)/2;
                User.push_back(sub_mul(User.back(),G[u<<1|1])),mult_eval_dfs2(u<<1,l,mid);
                User.back()=sub_mul(User[User.size()-2],G[u<<1]),mult_eval_dfs2(u<<1|1,mid,r);
                User.pop_back();
            }
        public:
            voi mult_eval(poly P,modvec &X)
            {
                if(X.empty())return;
                G.resize(X.size()<<2),User.clear(),this->X=X;
                mult_eval_dfs1(1,0,X.size());
                User.push_back(sub_mul(P,G[1].inv(std::max<uint>(P.size(),X.size())+1)));
                mult_eval_dfs2(1,0,X.size());
                G.clear(),User.clear(),X=this->X,this->X.clear();
            }
    };
    template<const ullt p=998244353,const ullt g=3>
    class poly_inter
    {
        public:
    		typedef mod_ullt<p>modint;
            typedef std::vector<modint>modvec;
            typedef poly_NTT<p,g>poly;
            typedef poly_eval<p,g>eval;
        private:
            std::vector<poly>Lim,F,G;eval E;modvec X,H;
            voi dfs(uint l,uint r)
            {
                if(l+1==r)
                {
                    F.push_back(poly()),F.back().push(H[l]),G.push_back(poly()),G.back().push(-X[l]),G.back().push(1);return;
                }
                uint mid=(l+r)>>1;dfs(l,mid),dfs(mid,r);
                F[F.size()-2]=F[F.size()-2]*G.back()+F.back()*G[G.size()-2],F.pop_back(),G[G.size()-2]*=G.back(),G.pop_back();
            }
        public:
            poly fast_inter(modvec X,modvec Y)
            {
                uint n=std::min(X.size(),Y.size());if(!n)return poly();
                X.resize(n),Y.resize(n),this->X=X;poly P;Lim.clear();
                for(uint i=0;i<n;i++)
                {
                    P.bzr(),P.push(-X[i]),P.push(1);
                    uint w=lowbit(i+1);while(w>>=1)P*=Lim.back(),Lim.pop_back();
                    Lim.push_back(P);
                }
                P=Lim.back(),Lim.pop_back();while(Lim.size())P*=Lim.back(),Lim.pop_back();
                E.mult_eval(P.diff(),X),H.resize(n);for(uint i=0;i<n;i++)H[i]=Y[i]/X[i];
                F.clear(),G.clear(),dfs(0,n);
                poly ans=F.back();F.clear(),G.clear(),this->X.clear(),H.clear();return ans;
            }
    };
    template<const ullt p=998244353,const ullt g=3>
    class poly_cpd
    {
        public:
    		typedef mod_ullt<p>modint;
            typedef std::vector<modint>modvec;
            typedef poly_NTT<p,g>poly;
            modint point_eval(poly P,modint x)
            {
                modint ans;
                for(uint i=P.deg();~i;i--)ans=ans*x+P[i];
                return ans;
            }
            poly z_npow(poly P,uint n)
            {
                if(P.empty())return P;
                poly ans(P.deg()*n+1);
                for(uint i=0;i<P.size();i++)ans[i*n]+=P[i];
                return ans;
            }
            poly z_npow(poly P,uint n,uint prec)
            {
                poly ans(prec);
                for(uint i=0;i<P.size()&&i*n<prec;i++)ans[i*n]+=P[i];
                return ans;
            }
            poly z_mul_k(poly P,modint k)
            {
                modint t(1);
                for(uint i=0;i<P.size();i++)P[i]*=t,t*=k;
                return P;
            }
            poly z_add_v(poly P,modint v)
            {
                uint n=P.size();if(!n)return P;
                std::vector<modint>A(n),B(n);
                A[0]=1;for(uint i=1;i<n;i++)A[i]=A[i-1]*i;
                B[n-1]=A[n-1].inv();for(uint i=n-1;i;i--)B[i-1]=B[i]*i;
                poly User(n);modint w(1);
                for(uint i=0;i<n;i++)P[i]*=A[i],User[i]=w*B[i],w*=v;
                P=sub_mul(P,User),P.chg_siz(n);
                for(uint i=0;i<n;i++)P[i]*=B[i];
                return P;
            }
            poly chg_siz(poly P,uint siz){P.chg_siz(siz);return P;}
            poly PolyaInv(poly P,uint prec){return(modint(1)-P).inv(prec);}
            poly PolyaExp(poly P,uint prec)
            {
                modvec inv(prec);
                inv[1]=1;for(uint i=2;i<prec;i++)inv[i]=(p/i)*-inv[p%i];
                poly ans(prec);
                for(uint i=1;i<prec;i++)for(uint j=1;i*j<prec&&j<P.size();j++)ans[i*j]+=P[j]*inv[i];
                return ans.exp(prec);
            }
            poly PolyaInv(poly P){return PolyaInv(P,P.size());}
            poly PolyaExp(poly P){return PolyaExp(P,P.size());}
            voi println(poly P,uint n)
            {
                for(uint i=0;i<n;i++){
                    if(i)putchar(' ');
                    P.val(i).print();
                }
                putchar('\n');
            }
            voi println(poly P){println(P,P.size());}
    };
    const ullt Mod=1004535809,g=3;
    typedef mod_ullt<Mod>modint;
    typedef std::vector<modint>modvec;
    typedef poly_NTT<Mod,g>poly;
    typedef poly_eval<Mod,g>eval;
    typedef poly_inter<Mod,g>inter;
    typedef poly_cpd<Mod,g>cpd;
    int main()
    {
        uint n;scanf("%u",&n);
        poly P(n),Q(n);
        for(uint i=0;i<n;i++)P[i].read(),Q[i].read();
        P=sub_mul(P,Q);
        for(uint i=0;i<n;i++)P.val(i).println();
    	return 0;
    }
    
    
    • 1

    信息

    ID
    2194
    时间
    1000ms
    内存
    256MiB
    难度
    9
    标签
    (无)
    递交数
    9
    已通过
    5
    上传者