POJ1912 A highway and the seven dwarfs (判断凸包与直线相交 logn)

POJ1912 给定n个点 和若干条直线，判断对于一条直线，是否存在两个点在直线的两侧。

显然原命题等价于 凸包与直线是否相交。

O（n）的算法是显而易见的 但是直线数量太多 就会复杂到O（n^2）由于n<=100000 会TLE

凸包有个很好的性质，我们没有利用， 那就是凸包的边相对于要判断的直线是极角有序的！

于是得到算法：

构造好凸包后，二分找凸包上第一个与正向直线夹角大于0的线段和第一个与反向直线夹角大于0的线段

然后判断两线段的起点是否在直线两侧即可。

代码实现有一点注意的细节:不要用上下界的方法实现二分，会很困难，用另外一种调整跳转距离的方法来实现就会简单很多。

#include<iostream>
#include<stdio.h>
#include<stdlib.h>
#include<string.h>
#include<math.h>
#include<algorithm>
#include<queue>
#include<vector>
using namespace std;
const double eps=1e-9;

int cmp(double x)
{
 if(fabs(x)<eps)return 0;
 if(x>0)return 1;
 	else return -1;
}

const double pi=acos(-1.0);

inline double sqr(double x)
{
 return x*x;
}

struct point
{
 double x,y;
 point (){}
 point (double a,double b):x(a),y(b){}
 bool input()
 	{
 	 return scanf("%lf%lf",&x,&y)!=EOF;
	}
 friend point operator +(const point &a,const point &b)
 	{
 	 return point(a.x+b.x,a.y+b.y);
	}
 friend point operator -(const point &a,const point &b)
 	{
 	 return point(a.x-b.x,a.y-b.y);
	}
 friend bool operator ==(const point &a,const point &b)
 	{
 	 return cmp(a.x-b.x)==0&&cmp(a.y-b.y)==0;
	}
 friend point operator *(const point &a,const double &b)
 	{
 	 return point(a.x*b,a.y*b);
	}
 friend point operator*(const double &a,const point &b)
 	{
 	 return point(a*b.x,a*b.y);
	}
 friend point operator /(const point &a,const double &b)
 	{
 	 return point(a.x/b,a.y/b);
	}
 double norm()
 	{
 	 return sqr(x)+sqr(y);
	}
};

struct line
{
 point a,b;
 line(){};
 line(point x,point y):a(x),b(y)
 {

 }
};
double det(const point &a,const point &b)
{
 return a.x*b.y-a.y*b.x;
}

double dot(const point &a,const point &b)
{
 return a.x*b.x+a.y*b.y;
}

double dist(const point &a,const point &b)
{
 return (a-b).norm();
}

point rotate_point(const point &p,double A)
{
 double tx=p.x,ty=p.y;
 return point(tx*cos(A)-ty*sin(A),tx*sin(A)+ty*cos(A));
}

bool parallel(line a,line b)
{
 return !cmp(det(a.a-a.b,b.a-b.b));
}

bool line_joined(line a,line b,point &res)
{
 if(parallel(a,b))return false;
 double s1=det(a.a-b.a,b.b-b.a);
 double s2=det(a.b-b.a,b.b-b.a);
 res=(s1*a.b-s2*a.a)/(s1-s2);
 return true;
}

bool pointonSegment(point p,point s,point t)
{
 return cmp(det(p-s,t-s))==0&&cmp(dot(p-s,p-t))<=0;
}

void PointProjLine(const point p,const point s,const point t,point &cp)
{
 double r=dot((t-s),(p-s))/dot(t-s,t-s);
 cp=s+r*(t-s);
}

struct polygon_convex
{
 vector<point>P;
 polygon_convex(int Size=0)
 	{
 	 P.resize(Size);
	}
};

struct halfPlane
{
 double a,b,c;
 halfPlane(point p,point q)
 	{
 	 a=p.y-q.y;
 	 b=q.x-p.x;
 	 c=det(p,q);
	}
 halfPlane(double aa,double bb,double cc)
 	{
 	 a=aa;b=bb;c=cc;
	}

};

double calc(halfPlane &L,point &a)
{
 return a.x*L.a +a.y*L.b+L.c;
}

point Intersect(point &a,point &b,halfPlane &L)
{
 point res;
 double t1=calc(L,a),t2=calc(L,b);
 res.x=(t2*a.x-t1*b.x)/(t2-t1);
 res.y=(t2*a.y-t1*b.y)/(t2-t1);
 //cout<<res.x<<" "<<res.y<<endl;
 return res;
}

polygon_convex cut(polygon_convex &a,halfPlane &L)
{
 int n=a.P.size();
 polygon_convex res;
 for(int i=0;i<n;i++)
 	{
 	 if(calc(L,a.P[i])>-eps)res.P.push_back(a.P[i]);
 	 	else
 	 		{
 	 		 int j;
 	 		 j=i-1;
 	 		 if(j<0)j=n-1;
 	 		 if(calc(L,a.P[j])>-eps)
			   	  res.P.push_back(Intersect(a.P[j],a.P[i],L));
 	 		 j=i+1;
 	 		 if(j==n)j=0;
 	 		 if(calc(L,a.P[j])>-eps)
 	 		 	res.P.push_back(Intersect(a.P[i],a.P[j],L));
			}
	}
 return res;
}
double INF=10000;
polygon_convex core(vector<point> &a)
{
 polygon_convex res;
 res.P.push_back(point(0,0));
 res.P.push_back(point(INF,0));
 res.P.push_back(point(INF,INF));
 res.P.push_back(point(0,INF));
 int n=a.size();
 for(int i=0;i<n-1;i+=2)
 	{
 	 halfPlane L(a[i],a[(i+1)]);
 	 res=cut(res,L);
	}
 return res;
}
bool comp_less(const point &a,const point &b)
{
 return cmp(a.x-b.x)<0||cmp(a.x-b.x)==0&&cmp(a.y-b.y)<0;

}

polygon_convex convex_hull(vector<point> a)
{
 polygon_convex res(2*a.size()+5);
 sort(a.begin(),a.end(),comp_less);
 a.erase(unique(a.begin(),a.end()),a.end());//删去重复点
 int m=0;
 for(int i=0;i<a.size();i++)
 	{
 	 while(m>1&&cmp(det(res.P[m-1]-res.P[m-2],a[i]-res.P[m-2]))<=0)--m;
 	 res.P[m++]=a[i];
	}
 int k=m;
 for(int i=int(a.size())-2;i>=0;--i)
 	{
 	 while(m>k&&cmp(det(res.P[m-1]-res.P[m-2],a[i]-res.P[m-2]))<=0)--m;
 	 res.P[m++]=a[i];
	}
 res.P.resize(m);
 if(a.size()>1)res.P.resize(m-1);
 return res;
}

bool is_convex(vector<point> &a)
{
 for(int i=0;i<a.size();i++)
 	{
 	 int i1=(i+1)%int(a.size());
 	 int i2=(i+2)%int(a.size());
 	 int i3=(i+3)%int(a.size());
 	 if((cmp(det(a[i1]-a[i],a[i2]-a[i1]))*cmp(det(a[i2]-a[i1],a[i3]-a[i2])))<0)
	  	return false;
	}
 return true;
}
int containO(const polygon_convex &a,const point &b)
{
 int n=a.P.size();
 point g=(a.P[0]+a.P[n/3]+a.P[2*n/3])/3.0;
 int l=0,r=n;
 while(l+1<r)
 	{
 	 int mid=(l+r)/2;
 	 if(cmp(det(a.P[l]-g,a.P[mid]-g))>0)
 	 	{
 	 	 if(cmp(det(a.P[l]-g,b-g))>=0&&cmp(det(a.P[mid]-g,b-g))<0)r=mid;
 	 	 	else l=mid;
		}else
			{
			 if(cmp(det(a.P[l]-g,b-g))<0&&cmp(det(a.P[mid]-g,b-g))>=0)l=mid;
 	 	 		else r=mid;
			}
	}
 r%=n;
 int z=cmp(det(a.P[r]-b,a.P[l]-b))-1;
 if(z==-2)return 1;
 return z;
}
long long int distant(point a,point b)
{
 return  (int(b.x)-int(a.x))*(int(b.x)-int(a.x))+(int(b.y)-int(a.y))*(int(b.y)-int(a.y));
}
double  convex_diameter(polygon_convex &a,int &First,int &Second)
{
 vector<point> &p=a.P;
 int n=p.size();
 double maxd=0;
 if(n==1)
 	{
 	 First=Second=0;
 	 return maxd;
	}
 #define next(i)((i+1)%n)
 for(int i=0,j=1;i<n;++i)
 	{
 	 while(cmp(det(p[next(i)]-p[i],p[j]-p[i])-det(p[next(i)]-p[i],p[next(j)]-p[i]))<0)
 	 	j=next(j);
 	 double d=dist(p[i],p[j]);
 	 if(d>maxd)
 	 	{
 	 	 maxd=d;
 	 	 First=i,Second=j;
		}
	 d=dist(p[next(i)],p[next(j)]);
	 if(d>maxd)
	 	{
	 	 maxd=d;
 	 	 First=next(i),Second=next(j);
		}

	}
 return maxd;
}

double area(vector<point>a)
{
 double sum=0;
 for(int i=0;i<a.size();i++)
 	sum+=det(a[(i+1)%(a.size())],a[i]);
 	return sum/2.;
}
int sumn;
int nex(int a,int b)
{
 a=a+b;
 while(a<0)a+=sumn;
 while(a>=sumn)a-=sumn;
 return a;
}
bool Convex_cross_Segment(point a,point b,polygon_convex &pc)
{
 sumn=pc.P.size();
 if(pc.P.size()<2)return true;
 if(pc.P.size()==2)
 	{
 	 if(cmp(det(a-b,pc.P[0]-a)*det(a-b,pc.P[1]-a))<0)return false;
 	 	else return true;
	}
 int len=pc.P.size()/2,loc1=-1,loc2=-1,locn=pc.P.size()/2;
 while(true)
 	{
 	 if(cmp(det(a-b,pc.P[nex(locn,1)]-pc.P[locn]))>0&&cmp(det(a-b,pc.P[locn]-pc.P[nex(locn,-1)]))<=0)
	  	{loc1=locn;break;}
	 if(cmp(det(a-b,pc.P[nex(locn,1)]-pc.P[locn]))>0)
	 	{locn=nex(locn,-len);if(len>1)len/=2;continue;}
	 	else{locn=nex(locn,len);if(len>1)len/=2;continue;}
	}
 len=pc.P.size()/2;
  while(true)
 	{
 	 if(cmp(det(a-b,pc.P[nex(locn,1)]-pc.P[locn]))<0&&cmp(det(a-b,pc.P[locn]-pc.P[nex(locn,-1)]))>=0)
	  	{loc2=locn;break;}
	 if(cmp(det(a-b,pc.P[nex(locn,1)]-pc.P[locn]))<0)
	 	{locn=nex(locn,-len);if(len>1)len/=2;continue;}
	 	else{locn=nex(locn,len);if(len>1)len/=2;continue;}
	}
 if(cmp(det(a-b,pc.P[loc1]-a)*det(a-b,pc.P[loc2]-a))<0)return false;
 	 	else return true;
}
vector<point>pp;
int main()
{freopen("t.txt","r",stdin);
 int N;
 scanf("%d",&N);
 pp.resize(N);
 for(int i=0;i<N;i++)
 	 pp[i].input();
 polygon_convex pc=convex_hull(pp);
 point a,b;
 while(a.input()&&b.input())
 	 if(Convex_cross_Segment(a,b,pc))printf("GOOD\n");
 	 	else printf("BAD\n");
 return 0;
}