bzoj 2823: [AHOI2012]信号塔 最小圆覆盖

题目大意:

给定n个点,求面积最小的园覆盖所有点.其中\(n \leq 10^6\)

题解:

恩。。。

刚拿到这道题的时候...

什么？？？最小圆覆盖不是\(O(n^3)\)的随机增量算法吗？？？？？

\(10^6\)又是个什么鬼？？？？？？？？？

然后去%了popoqqq大爷的题解...原来这道题数据是随机的啊。。。

随机数据有一个性质,在凸包上的点不超过\(logn\)

所以我们求凸包然后在上面跑随机增量算法即可

#include <cmath>
#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;
typedef long long ll;
inline void read(int &x){
	x=0;char ch;bool flag = false;
	while(ch=getchar(),ch<'!');if(ch == '-') ch=getchar(),flag = true;
	while(x=10*x+ch-'0',ch=getchar(),ch>'!');if(flag) x=-x;
}
inline int cat_max(const int &a,const int &b){return a>b ? a:b;}
inline int cat_min(const int &a,const int &b){return a<b ? a:b;}
const int maxn = 1000010;
const double eps = 1e-9;
inline int dcmp(const double &x){
	if(x < eps && x > -eps) return 0;
	return x > 0 ? 1 : -1;
}
struct Point{
	double x,y;
	Point(const double &a = 0,const double &b = 0){x=a;y=b;}
	void print(){
		printf("Point : (%lf,%lf)\n",x,y);
	}
};
typedef Point Vector;
inline Vector operator + (const Vector &a,const Vector &b){
	return Vector(a.x+b.x,a.y+b.y);
}
inline Vector operator - (const Vector &a,const Vector &b){
	return Vector(a.x-b.x,a.y-b.y);
}
inline Vector operator / (const Vector &a,const double &b){
	return Vector(a.x/b,a.y/b);
}
inline bool operator < (const Point &a,const Point &b){
	return a.x == b.x ? a.y < b.y : a.x < b.x;
}
inline double operator * (const Vector &a,const Vector &b){
	return a.x*b.x + a.y*b.y;
}
inline double cross(const Vector &a,const Vector &b){
	return a.x*b.y - a.y*b.x;
}
inline double sqr(const double &x){
	return x*x;
}
inline double dis(const Point &a,const Point &b){
	return sqrt(sqr(a.x - b.x) + sqr(a.y - b.y));
}
inline Point getPoint(const Point &p0,const Point &p1,const Point &p2){
	double a1 = p1.x - p0.x,b1 = p1.y - p0.y,c1 = (a1*a1 + b1*b1)/2.0;
	double a2 = p2.x - p0.x,b2 = p2.y - p0.y,c2 = (a2*a2 + b2*b2)/2.0;
	double d = a1*b2 - a2*b1;
	return Point(p0.x+(c1*b2-c2*b1)/d,p0.y+(a1*c2-a2*c1)/d);
}
Point o;double r;
inline bool in_cir(const Point &p){
	return dcmp(dis(p,o) - r) <= 0;
}
Point p[maxn];int n;
inline void getMinCir(){
	o = Point(0,0);r = 0;
	for(int i=2;i<=n;++i){
		if(!in_cir(p[i])){ o = p[i];r = 0;
			for(int j=1;j<i;++j){
				if(!in_cir(p[j])){o = (p[i]+p[j])/2.0;r = dis(p[i],o);
					for(int k=1;k<j;++k){
						if(!in_cir(p[k])){
							o = getPoint(p[i],p[j],p[k]);
							r = dis(p[i],o);
						}
					}
				}
			}
		}
	}return;
}
Point ch[maxn];int m;
inline void convex(){
	sort(p+1,p+n+1);m = 0;
	for(int i=1;i<=n;++i){
		while(m > 1 && cross(ch[m] - ch[m-1],p[i] - ch[m]) <= 0) -- m;
		ch[++m] = p[i];
	}int k = m;
	for(int i=n-1;i>=1;--i){
		while(m > k && cross(ch[m] - ch[m-1],p[i] - ch[m]) <= 0) -- m;
		ch[++m] = p[i];
	}if(n > 1) -- m;
	swap(n,m);swap(p,ch);
}
int main(){
	read(n);
	for(int i=1;i<=n;++i){
		scanf("%lf%lf",&p[i].x,&p[i].y);
	}
	convex();getMinCir();
	printf("%.2lf %.2lf %.2lf\n",o.x,o.y,r);
	getchar();getchar();
	return 0;
}