假设S在T左边，那么只能往右或者上下走
f[i]表示S到i点的最短路
f[i]=min(f[j]+dis(i,j)(i能看到j))
判断i能看到j就维护一个上凸壳和一个下凸壳
时间复杂度$O(n^2)$
代码写的有点长…
#include<cstdio>
#include<cmath>
#include<algorithm>
#define N 2010
using namespace std;
struct P{int x,y;P(){}P(int _x,int _y){x=_x,y=_y;}P operator-(const P&a){return P(x-a.x,y-a.y);}};
inline int cross(P a,P b){return a.x*b.y-a.y*b.x;}
inline double dis(P a,P b){return sqrt((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y));}
int n,i,X1[N],Y1[N],X2[N],Y2[N],xs,ys,xt,yt,st,en;
double v,f[N][4],ans,inf=1e9;
inline void call(int x,int y,int p,double&t){
t=inf;
P o(x,y),up(x,y+1),down(x,y-1),tmp;
for(int i=p-1;i>st;i--){
tmp=P(X2[i],Y2[i]);
if(cross(up-o,tmp-o)>=0){//在up下边或者重合
up=tmp;
if(cross(down-o,tmp-o)<=0)t=min(t,f[i][3]+dis(o,tmp));//在down上边或者重合
}
tmp=P(X2[i],Y1[i]);
if(cross(down-o,tmp-o)<=0){//在down上边或者重合
down=tmp;
if(cross(up-o,tmp-o)>=0)t=min(t,f[i][2]+dis(o,tmp));//在up下边或者重合
}
tmp=P(X1[i],Y2[i]);
if(cross(up-o,tmp-o)>=0){//在up下边或者重合
up=tmp;
if(cross(down-o,tmp-o)<=0)t=min(t,f[i][1]+dis(o,tmp));//在down上边或者重合
}
tmp=P(X1[i],Y1[i]);
if(cross(down-o,tmp-o)<=0){//在down上边或者重合
down=tmp;
if(cross(up-o,tmp-o)>=0)t=min(t,f[i][0]+dis(o,tmp));//在up下边或者重合
}
}
tmp=P(X2[st],Y2[st]);
if(cross(up-o,tmp-o)>=0){//在up下边或者重合
up=tmp;
if(cross(down-o,tmp-o)<=0)t=min(t,f[st][3]+dis(o,tmp));//在down上边或者重合
}
tmp=P(X2[st],Y1[st]);
if(cross(down-o,tmp-o)<=0){//在down上边或者重合
down=tmp;
if(cross(up-o,tmp-o)>=0)t=min(t,f[st][2]+dis(o,tmp));//在up下边或者重合
}
tmp=P(xs,ys);
if(cross(up-o,tmp-o)>=0&&cross(down-o,tmp-o)<=0)t=dis(o,tmp);//在up下边或者重合 在down上边或者重合
}
inline void calr(int x,int y,int p,double&t){
t=inf;
P o(x,y),up(x,y+1),down(x,y-1),tmp;
tmp=P(X1[p],Y2[p]);
if(cross(up-o,tmp-o)>=0){//在up下边或者重合
up=tmp;
if(cross(down-o,tmp-o)<=0)t=min(t,f[p][1]+dis(o,tmp));//在down上边或者重合
}
tmp=P(X1[p],Y1[p]);
if(cross(down-o,tmp-o)<=0){//在down上边或者重合
down=tmp;
if(cross(up-o,tmp-o)>=0)t=min(t,f[p][0]+dis(o,tmp));//在up下边或者重合
}
for(int i=p-1;i>st;i--){
tmp=P(X2[i],Y2[i]);
if(cross(up-o,tmp-o)>=0){//在up下边或者重合
up=tmp;
if(cross(down-o,tmp-o)<=0)t=min(t,f[i][3]+dis(o,tmp));//在down上边或者重合
}
tmp=P(X2[i],Y1[i]);
if(cross(down-o,tmp-o)<=0){//在down上边或者重合
down=tmp;
if(cross(up-o,tmp-o)>=0)t=min(t,f[i][2]+dis(o,tmp));//在up下边或者重合
}
tmp=P(X1[i],Y2[i]);
if(cross(up-o,tmp-o)>=0){//在up下边或者重合
up=tmp;
if(cross(down-o,tmp-o)<=0)t=min(t,f[i][1]+dis(o,tmp));//在down上边或者重合
}
tmp=P(X1[i],Y1[i]);
if(cross(down-o,tmp-o)<=0){//在down上边或者重合
down=tmp;
if(cross(up-o,tmp-o)>=0)t=min(t,f[i][0]+dis(o,tmp));//在up下边或者重合
}
}
tmp=P(X2[st],Y2[st]);
if(cross(up-o,tmp-o)>=0){//在up下边或者重合
up=tmp;
if(cross(down-o,tmp-o)<=0)t=min(t,f[st][3]+dis(o,tmp));//在down上边或者重合
}
tmp=P(X2[st],Y1[st]);
if(cross(down-o,tmp-o)<=0){//在down上边或者重合
down=tmp;
if(cross(up-o,tmp-o)>=0)t=min(t,f[st][2]+dis(o,tmp));//在up下边或者重合
}
tmp=P(xs,ys);
if(cross(up-o,tmp-o)>=0&&cross(down-o,tmp-o)<=0)t=dis(o,tmp);//在up下边或者重合 在down上边或者重合
}
int main(){
scanf("%d",&n);
for(i=1;i<=n;i++)scanf("%d%d%d%d",&X1[i],&Y1[i],&X2[i],&Y2[i]);
scanf("%d%d%d%d",&xs,&ys,&xt,&yt);
if(xs>xt)swap(xs,xt),swap(ys,yt);
scanf("%lf",&v);
for(i=1;i<=n;i++)if(X1[i]<=xs&&xs<=X2[i]&&Y1[i]<=ys&&ys<=Y2[i]){st=i;break;}
for(i=n;i;i--)if(X1[i]<=xt&&xt<=X2[i]&&Y1[i]<=yt&&yt<=Y2[i]){en=i;break;}
f[st][2]=dis(P(xs,ys),P(X2[st],Y1[st]));
f[st][3]=dis(P(xs,ys),P(X2[st],Y2[st]));
for(i=st+1;i<en;i++){
call(X1[i],Y1[i],i,f[i][0]);
call(X1[i],Y2[i],i,f[i][1]);
calr(X2[i],Y1[i],i,f[i][2]);
calr(X2[i],Y2[i],i,f[i][3]);
}
call(X1[en],Y1[en],en,f[en][0]);
call(X1[en],Y2[en],en,f[en][1]);
calr(xt,yt,en,ans);
printf("%.10f",ans/v);
return 0;
}