[POJ2079]Triangle（凸包+旋转卡壳）

题目描述

传送门
题意：给出一些点，求顶点在这些点上的面积最大的三角形

题解

其实这不算一个旋转卡壳吧…
很容易得出一个结论，最大的三角形一定是凸包上的边加上离它最远的点
$\color{red}{这个结论是错误的！}$
反例也很容易举出：

所以我们只能固定两个点然后让第三个点单调
所以这道题实际上是$O(n^2)$的，虽然有5w个点，但是凸包上的点远远达不到这个级别

代码

#include<algorithm>
#include<iostream>
#include<cstring>
#include<cstdio>
#include<cmath>
using namespace std;
#define N 100005

const double eps=1e-9;
int dcmp(double x)
{
    if (x<=eps&&x>=-eps) return 0;
    return (x>0)?1:-1;
}
struct Vector
{
    double x,y;
    Vector(double X=0,double Y=0)
    {
        x=X,y=Y;
    }
    bool operator < (Vector const &a) const
    {
        return x<a.x||(x==a.x&&y<a.y);
    }
};
typedef Vector Point;
Vector operator + (Vector a,Vector b) {return Vector(a.x+b.x,a.y+b.y);}
Vector operator - (Vector a,Vector b) {return Vector(a.x-b.x,a.y-b.y);}
Vector operator * (Vector a,double b) {return Vector(a.x*b,a.y*b);}

int n,top;
double x,y,area;
Point p[N],stack[N];

double Cross(Vector a,Vector b)
{
    return a.x*b.y-a.y*b.x;
}
void graham()
{
    sort(p+1,p+n+1);
    top=0;
    for (int i=1;i<=n;++i)
    {
        while (top>1&&dcmp(Cross(stack[top]-stack[top-1],p[i]-stack[top-1]))<=0)
            --top;
        stack[++top]=p[i];
    }
    int k=top;
    for (int i=n-1;i>=1;--i)
    {
        while (top>k&&dcmp(Cross(stack[top]-stack[top-1],p[i]-stack[top-1]))<=0)
            --top;
        stack[++top]=p[i];
    }
    if (n>1) --top;
}
double rotating()
{
    if (top<=2) return 0;
    if (top==3) return fabs(Cross(stack[2]-stack[1],stack[3]-stack[1])/2.0);
    double ans=0;
    for (int i=1;i<=top;++i)
    {
        int j=i%top+1;
        int k=j%top+1;
        while (dcmp(fabs(Cross(stack[i]-stack[j],stack[i]-stack[k]))-fabs(Cross(stack[i]-stack[j],stack[i]-stack[k%top+1])))<0) k=k%top+1;
        while (i!=j&&j!=k&&i!=k)
        {
            ans=max(ans,fabs(Cross(stack[i]-stack[j],stack[i]-stack[k]))/2.0);
            while (dcmp(fabs(Cross(stack[i]-stack[j],stack[i]-stack[k]))-fabs(Cross(stack[i]-stack[j],stack[i]-stack[k%top+1])))<0) k=k%top+1;
            j=j%top+1;
        }
    }
    return ans;
}
int main()
{
    while (~scanf("%d",&n))
    {
        if (n==-1) break;
        for (int i=1;i<=n;++i)
        {
            scanf("%lf%lf",&x,&y);
            p[i]=Point(x,y);
        }
        graham();
        area=rotating();
        printf("%.2lf\n",area);
    }
    return 0;
}