编程之美-2.11 扩展 寻找距离最远的两个点

一、问题描述

    平面上有n个点，如何寻找距离最远的两个点？

二、解题思路

    第一步，寻找凸包（因为最远距离的两个点一定在凸包上）

    第二步，用旋转卡(qia)壳 寻找距离最大的点

    凸包和旋转卡壳算法参见http://blog.csdn.net/kaytowin/article/details/5140111

三、代码实现

#include<iostream>
#include<vector>
#include<algorithm>
#include<cmath>
#include<stack>
#define INF 100000000
using namespace std;

struct point{
    int x;
    int y;
    point(int x1,int y1):x(x1),y(y1){}
    point(){}
};
struct maxLS{
    point p1;
    point p2;
    int dist;
    maxLS(point pp1,point pp2,int dist1):p1(pp1),p2(pp2),dist(dist1){}
    maxLS(){}
};
int xmult(point p1,point p2,point p0){
    return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
point p0(INF,INF);
vector<point> points;
vector<point> pstack;
void sort(vector<point>& points1){
    for(int j=points.size()-1;j>=1;j--){
        for(int i=1;i<j;i++){
            int xm=xmult(points[i],points[i+1],p0);

            if(xm<0){
                point tmp=points[i];
                points[i]=points[i+1];
                points[i+1]=tmp;
            }
        }
    }
}
void printPoints(vector<point> ps){
    for(int i=0;i<ps.size();i++){
        cout<<"<"<<ps[i].x<<","<<ps[i].y<<">"<<endl;
    }
}
void convexHull(){
    cout<<"打印输入点集:"<<endl;
    printPoints(points);
    sort(points);
    cout<<"打印按极角排序的点集:"<<endl;
    printPoints(points);
    pstack.push_back(points[0]);
    pstack.push_back(points[1]);
    int stack_top=pstack.size()-1;
    for(int i=2;i<points.size();i++){
        stack_top=pstack.size()-1;
        point pt2=pstack[stack_top];
        point pt1=pstack[stack_top-1];
        point ptcur=points[i];
        int xm=(pt2.x-pt1.x)*(ptcur.y-pt2.y)-(ptcur.x-pt2.x)*(pt2.y-pt1.y);
        while(xm<0){
            
            pstack.pop_back();
            stack_top=pstack.size()-1;
            pt2=pstack[stack_top];
            pt1=pstack[stack_top-1];
            xm=(pt2.x-pt1.x)*(ptcur.y-pt2.y)-(ptcur.x-pt2.x)*(pt2.y-pt1.y);
        
        }
        pstack.push_back(points[i]);
    }
    cout<<"打印凸包点集:"<<endl;
    printPoints(pstack);
}
int dist(point p1,point p2){
    return (p2.x-p1.x)*(p2.x-p1.x)+(p2.y-p1.y)*(p2.y-p1.y);
}
maxLS rotatingCalipers(){
    int q=1;
    int maxLength=0;
    maxLS mls;
    for(int i=0;i<pstack.size()-1;i++){
        while(xmult(pstack[i+1],pstack[(q+1)%pstack.size()],pstack[i])>xmult(pstack[i+1],pstack[q],pstack[i])){
            q=(q+1)%pstack.size();
        }
        int d1=dist(pstack[i],pstack[q]);
        int d2=dist(pstack[i+1],pstack[q]);
        if(d1>d2){
            maxLength=d1;
            mls=maxLS(pstack[i],pstack[q],d1);

        }else{
            maxLength=d2;
            mls=maxLS(pstack[i+1],pstack[q],d2);
        }
        
        
    }
    return mls;
}
int main(){
    cout<<"请输入节点数："<<endl;
    int n;
    cin>>n;
    cout<<"请输入"<<n<<"个节点:"<<endl;
    int p0i=0;
    int t=0;
    while(n--){
        int x;
        int y;
        cin>>x>>y;
        if(y<p0.y||(y==p0.y&&x<p0.x)){
            p0.x=x;
            p0.y=y;
            p0i=t;
            
        }
        t++;
        points.push_back(point(x,y));
    }
    point tmp=points[0];
    points[0]=points[p0i];
    points[p0i]=tmp;

    convexHull();
    maxLS maxL= rotatingCalipers();
    cout<<"两点最大距离："<<maxL.dist<<endl;
    cout<<"两点分别是:"<<"<"<<maxL.p1.x<<","<<maxL.p1.y<<">，";
    cout<<"<"<<maxL.p2.x<<","<<maxL.p2.y<<">"<<endl;
    system("pause");
    return 0;
}

四、输入：

9

3 1
4 3
5 2
3 5
6 5
8 4
5 7
2 6
1 4

五、输出：