BZOJ4083 : [Wf2014]Wire Crossing

WF2014完结撒花~

首先求出所有线段之间的交点，并在交点之间连边，得到一个平面图。

这个平面图不一定连通，故首先添加辅助线使其连通。

然后求出所有域，在相邻域之间连一条代价为$1$的边。

对起点和终点进行定位，然后BFS求最短路即可。

时间复杂度$O(n^2\log n)$。

#include<cstdio>
#include<cmath>
#include<set>
#include<algorithm>
using namespace std;
const double eps=1e-12,inf=110000;
const int N=20010,M=100010;
int n,m,w,q,cnt,cur,i,j,x,idx[N],g[N],v[M],nxt[M],ed,d[N],que[N],h,t,S,T;
inline int sgn(double x){
  if(fabs(x)<eps)return 0;
  return x>0?1:-1;
}
struct P{
  double x,y;
  P(){}
  P(double _x,double _y){x=_x,y=_y;}
  P operator+(P b){return P(x+b.x,y+b.y);}
  P operator-(P b){return P(x-b.x,y-b.y);}
  P operator*(double b){return P(x*b,y*b);}
  P operator/(double b){return P(x/b,y/b);}
  double operator*(P b){return x*b.x+y*b.y;}
}a[N],b[N],st[N],en[N],pool[N];
inline bool cmpP(const P&a,const P&b){return !sgn(a.x-b.x)?a.y<b.y:a.x<b.x;}
inline double cross(P a,P b){return a.x*b.y-a.y*b.x;}
inline bool point_on_segment(P p,P a,P b){
  return sgn(cross(b-a,p-a))==0&&sgn((p-a)*(p-b))<=0;
}
inline int has_intersection(P a,P b,P p,P q){
  int d1=sgn(cross(b-a,p-a)),d2=sgn(cross(b-a,q-a));
  int d3=sgn(cross(q-p,a-p)),d4=sgn(cross(q-p,b-p));
  return d1*d2<0&&d3*d4<0;
}
inline P line_intersection(P a,P b,P p,P q){
  double U=cross(p-a,q-p),D=cross(b-a,q-p);
  return a+(b-a)*(U/D);
}
struct E{
  int x,y;double o;
  E(){}
  E(int _x,int _y){x=_x,y=_y,o=atan2(a[y].x-a[x].x,a[y].y-a[x].y);}
}e[M];
bool del[M],ex[M];int from[M],id[N];
struct EV{
  double x;int y,t;
  EV(){}
  EV(double _x,int _y,int _t){x=_x,y=_y,t=_t;}
}ev[M<<1];
inline bool cmpEV(const EV&a,const EV&b){
  if(sgn(a.x-b.x))return a.x<b.x;
  return a.t<b.t;
}
namespace GetArea{
struct cmp{bool operator()(int a,int b){return e[a].o<e[b].o;}};
set<int,cmp>g[N];set<int,cmp>::iterator k;int i,j,q[M],t;
void work(){
  for(i=0;i<m+m;i++)if(!del[i]&&!ex[i]){
    for(q[t=1]=j=i;;q[++t]=j=*k){
      k=g[e[j].y].find(j^1);k++;
      if(k==g[e[j].y].end())k=g[e[j].y].begin();
      if(*k==i)break;
    }
    double s=0;
    for(j=1;j<=t;j++)s+=cross(a[e[q[j]].x],a[e[q[j]].y]),del[q[j]]=1;
    if(sgn(s)<0)continue;
    for(cnt++,j=1;j<=t;j++)from[q[j]]=cnt;
  }
}
}
namespace ScanLine{
struct cmp{
  bool operator()(int A,int B){
    if(e[A].x==e[B].x)return e[A].o>e[B].o;
    double x=min(a[e[A].x].x,a[e[B].x].x),
           yA=(a[e[A].x].y-a[e[A].y].y)*(x-a[e[A].y].x)/
              (a[e[A].x].x-a[e[A].y].x)+a[e[A].y].y,
           yB=(a[e[B].x].y-a[e[B].y].y)*(x-a[e[B].y].x)/
              (a[e[B].x].x-a[e[B].y].x)+a[e[B].y].y;
    return yA>yB;
  }
};
set<int,cmp>T;
int cnt,i,j,k,g[M],v[M],nxt[M],ed,vis[N],t,tmp[N];
inline bool cmpC(int x,int y){return a[x].x<a[y].x;}
inline void add(int x,int y){v[++ed]=y;nxt[ed]=g[x];g[x]=ed;}
void dfs(int x){
  vis[x]=1;
  if(a[x].y>a[t].y)t=x;
  for(int i=g[x];i;i=nxt[i])if(!vis[v[i]])dfs(v[i]);
}
inline double cal(int A,double x){
  return(a[e[A].x].y-a[e[A].y].y)*(x-a[e[A].y].x)/
        (a[e[A].x].x-a[e[A].y].x)+a[e[A].y].y;
}
void connect(){
  for(i=0;i<m+m;i++)add(e[i].x,e[i].y);
  for(i=1;i<=n;i++)if(!vis[i])dfs(t=i),ev[cnt++]=EV(a[t].x,t,2);
  for(i=0;i<m+m;i++)if(sgn(a[e[i].x].x-a[e[i].y].x)>0){
    ev[cnt++]=EV(a[e[i].y].x,i,1);
    ev[cnt++]=EV(a[e[i].x].x,i,0);
  }
  sort(ev,ev+cnt,cmpEV);
  a[n+1]=P(inf,inf);
  a[n+2]=P(-inf,inf);
  e[m+m]=E(n+1,n+2);
  T.insert(m+m);
  e[m+m+1]=E(n+2,n+1);
  n+=2,m++;
  for(ed=0,i=1;i<=n;i++)g[i]=0;
  for(i=0;i<cnt;i++){
    if(ev[i].t==0)T.erase(ev[i].y);
    if(ev[i].t==1)T.insert(ev[i].y);
    if(ev[i].t==2){
      a[n+1]=P(ev[i].x,a[ev[i].y].y+eps);
      a[n+2]=P(ev[i].x-1,a[ev[i].y].y+eps);
      e[m+m]=E(n+1,n+2);
      T.insert(m+m);
      set<int,cmp>::iterator j=T.find(m+m);
      j--,add(*j,ev[i].y);
      T.erase(m+m);
    }
  }
  int newm=m+m;
  for(i=0;i<m+m;i++){
    for(cnt=0,j=g[i];j;j=nxt[j]){
      if(!sgn(a[v[j]].x-a[e[i].x].x)){
        e[newm++]=E(v[j],e[i].x);
        e[newm++]=E(e[i].x,v[j]);
        continue;
      }
      if(!sgn(a[v[j]].x-a[e[i].y].x)){
        e[newm++]=E(v[j],e[i].y);
        e[newm++]=E(e[i].y,v[j]);
        continue;
      }
      tmp[++cnt]=v[j];
    }
    if(!cnt)continue;
    ex[i]=ex[i^1]=1;
    sort(tmp+1,tmp+cnt+1,cmpC);
    for(k=e[i].y,j=1;j<=cnt;k=n,j++){
      a[++n]=P(a[tmp[j]].x,cal(i,a[tmp[j]].x));
      e[newm++]=E(k,n);
      e[newm++]=E(n,k);
      e[newm++]=E(tmp[j],n);
      e[newm++]=E(n,tmp[j]);
    }
    e[newm++]=E(n,e[i].x);
    e[newm++]=E(e[i].x,n);
  }
  m=newm/2;
}
void location(){
  for(i=cnt=0;i<m+m;i++)if(!ex[i]&&sgn(a[e[i].x].x-a[e[i].y].x)>0){
    ev[cnt++]=EV(a[e[i].y].x,i,1);
    ev[cnt++]=EV(a[e[i].x].x,i,0);
  }
  for(i=0;i<q;i++)ev[cnt++]=EV(b[i].x,i,2);
  sort(ev,ev+cnt,cmpEV);
  T.clear();
  for(i=0;i<cnt;i++){
    if(ev[i].t==0)T.erase(ev[i].y);
    if(ev[i].t==1)T.insert(ev[i].y);
    if(ev[i].t==2){
      a[n+1]=P(ev[i].x,b[ev[i].y].y);
      a[n+2]=P(ev[i].x-1,b[ev[i].y].y);
      e[m+m]=E(n+1,n+2);
      T.insert(m+m);
      set<int,cmp>::iterator j=T.find(m+m);
      if(j!=T.begin())j--,id[ev[i].y]=from[*j];
      T.erase(m+m);
    }
  }
}
}
inline int getid(P o){
  int l=1,r=n,mid;
  while(l<=r){
    mid=(l+r)>>1;
    if(!sgn(o.x-a[mid].x)&&!sgn(o.y-a[mid].y))return mid;
    if(sgn(o.x-a[mid].x)>0||!sgn(o.x-a[mid].x)&&sgn(o.y-a[mid].y)>0)l=mid+1;else r=mid-1;
  }
}
inline void cal0(P a,P b,P c,P d){
  if(!has_intersection(a,b,c,d))return;
  ::a[++n]=line_intersection(a,b,c,d);
}
inline void cal1(P a,P b,P c,P d){
  if(point_on_segment(c,a,b)){pool[++cur]=c;return;}
  if(point_on_segment(d,a,b)){pool[++cur]=d;return;}
  if(!has_intersection(a,b,c,d))return;
  pool[++cur]=line_intersection(a,b,c,d);
}
inline void add(int x,int y){v[++ed]=y;nxt[ed]=g[x];g[x]=ed;}
int main(){
  scanf("%d",&w);
  for(q=2;i<q;i++)scanf("%lf%lf",&b[i].x,&b[i].y);
  for(i=0;i<w;i++){
    scanf("%lf%lf%lf%lf",&st[i].x,&st[i].y,&en[i].x,&en[i].y);
    a[++n]=st[i];
    a[++n]=en[i];
  }
  for(i=0;i<w;i++)for(j=0;j<i;j++)cal0(st[i],en[i],st[j],en[j]);
  sort(a+1,a+n+1,cmpP);
  int _=0;
  for(i=1;i<=n;i++)if(i==1||sgn(a[i].x-a[i-1].x)||sgn(a[i].y-a[i-1].y))a[++_]=a[i];
  n=_;
  for(i=0;i<w;i++){
    pool[1]=st[i];
    pool[cur=2]=en[i];
    for(j=0;j<w;j++)if(i!=j)cal1(st[i],en[i],st[j],en[j]);
    sort(pool+1,pool+cur+1,cmpP);
    for(j=1;j<=cur;j++)idx[j]=getid(pool[j]);
    for(j=1;j<cur;j++)if(idx[j]!=idx[j+1]){
      e[m<<1]=E(idx[j],idx[j+1]);
      e[m<<1|1]=E(idx[j+1],idx[j]);
      m++;
    }
  }
  ScanLine::connect();
  for(i=0;i<m+m;i++)if(!ex[i])GetArea::g[e[i].x].insert(i);
  GetArea::work();
  ScanLine::location();
  for(i=0;i<m+m;i++)if(!ex[i])add(from[i],from[i^1]);
  d[que[h=t=1]=id[0]]=1;
  while(h<=t)for(i=g[x=que[h++]];i;i=nxt[i])if(!d[v[i]])d[que[++t]=v[i]]=d[x]+1;
  return printf("%d",d[id[1]]-1),0;
}