CF 553E Kyoya and Train

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给定n个点m条边的有向图，起点为1，终点为n，如果到达时间>t要罚款x,通过第i条边的代价是ci，以时间k经过的概率为pi,k(1≤k≤t)，∑pi,k=1。求最优期望代价。
n≤50,m≤100,t≤2∗105,ci≤106,x≤106.

fft+概率dp+Bellman Ford

ShinFeb的水分(6801 ms)

#include<set>#include<cmath>#include<queue>#include<cstdio>#include<cstring>#include<iostream>#include<algorithm>#define inf (1<<30)#define INF (1ll<<62)#define prt(x) cout<<#x<<":"<<x<<" "#define prtn(x) cout<<#x<<":"<<x<<endl#define huh(x) printf("--------------case(%d)--------------\n",x)#define st_ huh(1234)#define en_ huh(4321)#define ONLINE_JUDGEusing namespace std;typedef long long ll;typedef pair<int,int> ii;///////////////////template<class T>inline void Max(T &x,T y){if(x<y)x=y;};template<class T>inline void Min(T &x,T y){if(x>y)x=y;};template<class T>void sc(T &x){    x=0;char c;    while(c=getchar(),c<48);    do x=x*10+(c^48);    while(c=getchar(),c>47);}template<class T>void sim(T x){    if(!x)return;    sim(x/10);    putchar('0'+x%10);}template<class T>void pt(T x){    if(!x)putchar('0');    else sim(x);    putchar('\n');}///////////////////const int maxt=20005;const int maxn=55;const int maxm=105;const int maxs=65536;const double pi=acos(-1);struct comp{    double x,y;    comp(){}    comp(double x,double y):x(x),y(y){}    comp operator+(const comp &a)const{        return comp(x+a.x,y+a.y);    }    comp operator-(const comp &a)const{        return comp(x-a.x,y-a.y);    }    comp operator*(const comp &a)const{        return comp(x*a.x-y*a.y,y*a.x+x*a.y);    }}A[maxs],B[maxs],wm[maxs];int N,rev[maxs];void init(int len){    int l=0;    for(N=1;N<=len;N<<=1)l++;    for(int i=1;i<N;i++)rev[i]=(rev[i>>1]>>1)|((i&1)<<l-1);}void dft(comp *a,int f){    for(int i=0;i<N;i++)if(rev[i]<i)swap(a[i],a[rev[i]]);    for(int m=1;m<N;m<<=1){        wm[0]=comp(1,0);comp wn(cos(pi/m),f*sin(pi/m));        for(int j=1;j<m;j++)wm[j]=wm[j-1]*wn;        for(int j=0;j<N;j+=(m<<1)){            for(int k=0;k<m;k++){                comp x=a[j+k],y=a[j+k+m]*wm[k];                a[j+k]=x+y;a[j+k+m]=x-y;            }        }    }if(f==-1)for(int i=0;i<N;i++)a[i].x/=N,a[i].y/=N;}int n,m,t,x;int u[maxm],v[maxm],w[maxm];int dis[maxn];double p[maxm][maxt];double psum[maxm][maxt];double g[maxm][maxt];double f[maxn][maxt];int main(){    #ifndef ONLINE_JUDGE    freopen("data.in","r",stdin);    freopen("data.out","w",stdout);    #endif    sc(n);sc(m);sc(t);sc(x);    for(int i=0;i<m;i++){        sc(u[i]);sc(v[i]);sc(w[i]);        for(int j=1;j<=t;j++){            sc(p[i][j]);            p[i][j]/=100000.0;        }        psum[i][t]=p[i][t];        for(int j=t-1;j>=1;j--)            psum[i][j]=psum[i][j+1]+p[i][j];    }    memset(dis,-1,sizeof(dis));    dis[n]=0;    for(int i=1;i<=n;i++){        bool update=0;        for(int j=0;j<m;j++){            int u=::u[j],v=::v[j],w=::w[j];            if(dis[v]==-1)continue;            if(dis[u]==-1||dis[u]>dis[v]+w){                dis[u]=dis[v]+w;                update=1;            }        }        if(!update)break;    }    for(int i=1;i<n;i++){        for(int j=0;j<=t;j++)            f[i][j]=dis[i]+x;    }    init(t<<1);//    for(int i=1;;i++){        bool flag=0;        for(int j=0;j<m;j++){            int u=::u[j],v=::v[j],w=::w[j];            for(int k=0;k<=t;k++){                A[k]=comp(p[j][k],0);                B[k]=comp(f[v][k],0);            }               for(int k=t+1;k<N;k++)                A[k]=B[k]=comp(0,0);            dft(A,1);dft(B,1);            for(int k=0;k<N;k++)A[k]=A[k]*B[k];            dft(A,-1);            for(int k=0;k<=t;k++){                g[j][k]=A[k].x+w;                if(k!=t)g[j][k]+=psum[j][k+1]*(dis[v]+x);            }            for(int k=1;k<=t;k++){                if(f[u][k]>g[j][k]){                    f[u][k]=g[j][k];                    flag=true;                }            }        }        if(!flag)break;    }    printf("%f\n",f[1][t]);    return 0;}

fft+概率dp+spfa
yts1999学长的spfa(5303 ms)

以上是乱搞。

fft+概率dp+分治
PS:让人联想到计蒜客百度地图的实时路况，官方题解不知道在搞什么。。。

PPFish学长的分治(3493 ms)
ShinFeb的分治(4539 ms)

#include<set>#include<cmath>#include<queue>#include<cstdio>#include<cstring>#include<iostream>#include<algorithm>#define inf (1<<30)#define INF (1ll<<62)#define prt(x) cout<<#x<<":"<<x<<" "#define prtn(x) cout<<#x<<":"<<x<<endl#define huh(x) printf("--------------case(%d)--------------\n",x)#define st_ huh(1234)#define en_ huh(4321)#define ONLINE_JUDGEusing namespace std;typedef long long ll;typedef pair<int,int> ii;///////////////////template<class T>inline void Max(T &x,T y){if(x<y)x=y;};template<class T>inline void Min(T &x,T y){if(x>y)x=y;};template<class T>void sc(T &x){    x=0;char c;    while(c=getchar(),c<48);    do x=x*10+(c^48);    while(c=getchar(),c>47);}template<class T>void sim(T x){    if(!x)return;    sim(x/10);    putchar('0'+x%10);}template<class T>void pt(T x){    if(!x)putchar('0');    else sim(x);    putchar('\n');}///////////////////const int maxt=20005;const int maxn=55;const int maxm=105;const int maxs=65536;const double pi=acos(-1);struct comp{    double x,y;    comp(){}    comp(double x,double y):x(x),y(y){}    comp operator+(const comp &a)const{        return comp(x+a.x,y+a.y);    }    comp operator-(const comp &a)const{        return comp(x-a.x,y-a.y);    }    comp operator*(const comp &a)const{        return comp(x*a.x-y*a.y,y*a.x+x*a.y);    }}A[maxs],B[maxs],wm[maxs];int N,rev[maxs];void init(int len){    int l=0;    for(N=1;N<=len;N<<=1)l++;    for(int i=1;i<N;i++)rev[i]=(rev[i>>1]>>1)|((i&1)<<l-1);}void dft(comp *a,int f){    for(int i=0;i<N;i++)if(rev[i]<i)swap(a[i],a[rev[i]]);    for(int m=1;m<N;m<<=1){        wm[0]=comp(1,0);comp wn(cos(pi/m),f*sin(pi/m));        for(int j=1;j<m;j++)wm[j]=wm[j-1]*wn;        for(int j=0;j<N;j+=(m<<1)){            for(int k=0;k<m;k++){                comp x=a[j+k],y=a[j+k+m]*wm[k];                a[j+k]=x+y;a[j+k+m]=x-y;            }        }    }if(f==-1)for(int i=0;i<N;i++)a[i].x/=N;}int n,m,t,x;int u[maxm],v[maxm],w[maxm];int dis[maxn];double p[maxm][maxt];double psum[maxm][maxt];double g[maxm][maxt];double f[maxn][maxt];void calc(int l,int r,int idx){    int v=::v[idx];    int mid=l+r>>1;    int na=r-l,nb=mid-l;    //p[][],f[][]    int nn=na+nb;    init(nn);    for(int i=0;i<=na;i++)        A[i]=comp(p[idx][i],0);    for(int i=na+1;i<N;i++)        A[i]=comp(0,0);    //[0,r-l]    for(int i=0;i<=nb;i++)        B[i]=comp(f[v][i+l],0);    for(int i=nb+1;i<N;i++)        B[i]=comp(0,0);    //[l,mid]    dft(A,1);dft(B,1);    for(int i=0;i<N;i++)A[i]=A[i]*B[i];    dft(A,-1);    //[l,r-l+mid]->[0,nn]    //[mid,r-l+mid]->[nb,nn]    for(int i=nb+1,j=mid+1;i<=nn&&j<=r;i++,j++)        g[idx][j]+=A[i].x;}void divide(int l,int r){    if(l==r){        for(int i=0;i<m;i++){            int u=::u[i],v=::v[i],w=::w[i];            Min(f[u][l],g[i][l]+w+psum[i][l+1]*dis[v]);        }        return;    }    int mid=l+r>>1;    divide(l,mid);    for(int i=0;i<m;i++)calc(l,r,i);    divide(mid+1,r);}int main(){    #ifndef ONLINE_JUDGE    freopen("data.in","r",stdin);    freopen("data.out","w",stdout);    #endif    sc(n);sc(m);sc(t);sc(x);    for(int i=0;i<m;i++){        sc(u[i]);sc(v[i]);sc(w[i]);        for(int j=1;j<=t;j++){            sc(p[i][j]);            p[i][j]/=100000.0;        }        psum[i][t+1]=0;        for(int j=t;j>=1;j--)            psum[i][j]=psum[i][j+1]+p[i][j];    }    memset(dis,-1,sizeof(dis));    dis[n]=0;    for(int i=1;i<=n;i++){        bool update=0;        for(int j=0;j<m;j++){            int u=::u[j],v=::v[j],w=::w[j];            if(dis[v]==-1)continue;            if(dis[u]==-1||dis[u]>dis[v]+w){                dis[u]=dis[v]+w;                update=1;            }        }        if(!update)break;    }    for(int i=1;i<=n;i++){        dis[i]+=x;        if(i!=n)for(int j=0;j<=t;j++)            f[i][j]=dis[i];    }    divide(0,t);//[0,t]    printf("%f\n",f[1][t]);    return 0;}

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