最近点对问题

Posted 2021-01-31 cwolf9

ZOJ2107
HDU1007
POJ3714

#include <iostream>
#include<cstdio>
#include<string.h>
#include<algorithm>
#include <cmath>
#include<iomanip>
#include <assert.h>
using namespace std;
int n;
struct node{
    double x;
    double y;
};
node s[100001];
node tempC[100001];//用于临时比较的辅助数组
node ta[100001];
double Distance(node a, node b) {
    return sqrt((a.x - b.x)*(a.x - b.x) + (a.y - b.y)*(a.y - b.y));
}
//按x从小到大排序
bool cmp1(node a, node b) {
    return a.x < b.x;
}
//按y从小到大排序
bool cmp2(node a, node b) {
    return a.y < b.y;
}
int merge(int left, int right,int middle,int axis,double delta){
        int i, j, index1, index2;
        int cm=0;
        for (j = left; j <= right; ++j)
            ta[j] = s[j];
        index1 = left;
        index2 = middle + 1;
        i = left;
        while (index1 <= middle && index2 <= right){
            if (ta[index1].y <= ta[index2].y){
                s[i] = ta[index1];
                //将距离AXIS 在delta以内的点放到辅助空间，同时保持y升序
                if (double(abs(s[i].x - axis)) <= delta) {
                    tempC[cm] = s[i];
                    ++cm;
                }
                ++i;
                ++index1;
            }else{
                s[i] = ta[index2];
                if (double(abs(s[i].x - axis)) <= delta){
                    tempC[cm] = s[i];
                    ++cm;
                }
                ++i;
                ++index2;
            }
        }
        while (index1 <= middle){
            s[i] = ta[index1];
            if (double(abs(s[i].x - axis)) <= delta){
                tempC[cm] = s[i];
                ++cm;
            }
            ++i;
            ++index1;
        }
        while (index2 <= right){
            s[i] = ta[index2];
            if (double(abs(s[i].x - axis)) <= delta){
                tempC[cm] = s[i];
                ++cm;
            }
            ++i;
            ++index2;
        }

        return cm - 1;
}
//采用按点数分治，s,e分别为处理坐标的起止
double solveClosest(int start, int end) {
    double tempans = Distance(s[start],s[end]);
    double d1, d2, d3=0;
    if (end - start == 1) {//两个点的情况
        sort(s + start, s + end+1, cmp2); //保持y序列的有序
        return Distance(s[end], s[start]);
    }
    if (end - start == 2) {//三个点的情况
        d1 = Distance(s[start], s[start + 1]);
        d2 = Distance(s[start + 1], s[end]);
        d3 = Distance(s[end], s[start]);
        sort(s + start, s + end+1, cmp2); //保持y序列的有序
        if ((d1<= d2) && (d1<= d3)) {
            return d1;
        }else if (d2<= d3){
            return d2;
        }else return d3;
    }
    int mid = (start + end) / 2;
    d1 =solveClosest(start, mid);
    d2 =solveClosest(mid + 1, end);
    if(d1<=d2) tempans = d1; //开始考察两区域中间的情况 此时左右区间已经内部按y升序
    else tempans = d2;
    for (int i = 0; i < end - start + 1; ++i){
        tempC[i].x = 0;
        tempC[i].y = 0;
    }
    int cnt = 0;
    cnt =merge(start, end, mid, s[mid].x , tempans);//进行归并排序中的合并，并且同时挑出在轴值附近需要比较的所有点（且按照y升序排列）
    //比较每一个tempC[i]所可能产生的最小距离
    for (int i = 0; i < cnt; ++i) {
        int ui = i-1;
        int di = i + 1;
        while (ui >= 0 &&  double(abs(tempC[ui].y - tempC[i].y)) <= tempans){
            if (Distance(tempC[ui], tempC[i]) < tempans)
                tempans = Distance(tempC[ui], tempC[i]);
            --ui;
        }
        while (di <= cnt && double(abs(tempC[di].y - tempC[i].y)) <= tempans){
            if (Distance(tempC[di], tempC[i]) < tempans)
                tempans = Distance(tempC[di], tempC[i]);
            ++di;
        }
    } 
    return tempans;
}
int main(){
    while(~scanf("%d", &n) && n) {
        for(int i = 0; i < n; ++i) {
            scanf("%lf%lf", &s[i].x, &s[i].y);
        }
        sort(s, s + n, cmp1); //先按x从小到大排序
        double ans = solveClosest(0, n - 1);
        printf("%.2f
", ans/2);
    }
    return 0;
}

#include <iostream>
#include <vector>
#include<string.h>
#include<algorithm>
#include <cmath>
#include<iomanip>
#include <assert.h>
#include <cstdio>
using namespace std;

struct point {
    double x;
    double y;
    point(double x, double y) :x(x), y(y) {}
    point() { return; }
};

bool cmp_x(const point & A, const point & B) {
    return A.x < B.x;
}

bool cmp_y(const point & A, const point & B) {
    return A.y < B.y;
}

double distance(const point & A, const point & B){
    return sqrt(pow(A.x - B.x, 2) + pow(A.y - B.y, 2));
}
/*
* function: 合并，同第三区域最近点距离比较
* param: points 点的集合
*        dis 左右两边集合的最近点距离
*        mid x坐标排序后，点集合中中间点的索引值
*/
double merge(vector<point> & points, double dis, int mid){
    vector<point> left, right;
    // 搜集左右两边符合条件的点
    for (int i = 0; i < (int)points.size(); ++i) {
        if (points[i].x - points[mid].x <= 0 && points[i].x - points[mid].x > -dis)
            left.push_back(points[i]);
        else if (points[i].x - points[mid].x > 0 && points[i].x - points[mid].x < dis)
            right.push_back(points[i]);
    }
    sort(right.begin(), right.end(), cmp_y);
    // 遍历左边的点集合，与右边符合条件的计算距离
    for (int i = 0, index; i < left.size(); ++i) {
        for (index = 0; index < right.size() && left[i].y > right[index].y; ++index);
            // 遍历右边坐标上界的6个点
        for (int j = 0; j < 7 && index + j < right.size(); ++j) {
            if (distance(left[i], right[j + index]) < dis)
                dis = distance(left[i], right[j + index]);
        }
        if(index == right.size()) index --;
        // 遍历右边坐标上界的6个点
        for (int j = 0; j < 7; ++j)  {
            if(index-j < 0) break;
            if (distance(left[i], right[index-j]) < dis)
                dis = distance(left[i], right[index-j]);
        }
    }
    return dis;
}


double closest(vector<point> & points){
    if ((int)points.size() == 2) return distance(points[0], points[1]);  // 两个点
    if ((int)points.size() == 3) return min(distance(points[0], points[1]), min(distance(points[0], points[2]), 
        distance(points[1], points[2])));  // 三个点
    int mid = ((int)(points.size()) >> 1) - 1;
    double d1, d2, d;
    vector<point> left(mid + 1), right((int)points.size() - mid - 1);
    copy(points.begin(), points.begin() + mid + 1, left.begin());  // 左边区域点集合
    copy(points.begin() + mid + 1, points.end(), right.begin());  // 右边区域点集合
    d1 = closest(left);
    d2 = closest(right);
    d = min(d1, d2);
    return merge(points, d, mid);
}

int main(){
    int count;
    while(~scanf("%d", &count)){
        if(count == 0) break;
        vector<point> points;
        double x, y;
        for (int i = 0; i < count; ++i){
            scanf("%lf%lf", &x, &y);
            point p(x, y);
            points.push_back(p);
        }
        sort(points.begin(), points.end(), cmp_x);
        printf("%.2f
", closest(points)/2);
    }
    return 0;
}

/**
最近点对问题，时间复杂度为O(n*logn*logn)
*/
#include <iostream>
#include <cstdio>
#include <cstring>
#include <cmath>
#include <algorithm>
using namespace std;
const double INF = 1e20;
const int N = 100005;
 
struct Point{
    double x;
    double y;
}point[N];
int n;
int tmpt[N];
 
bool cmpxy(const Point& a, const Point& b){
    if(a.x != b.x)
        return a.x < b.x;
    return a.y < b.y;
}
 
bool cmpy(const int& a, const int& b){
    return point[a].y < point[b].y;
}
 
double min(double a, double b){
    return a < b ? a : b;
}
 
double dis(int i, int j){
    return sqrt((point[i].x-point[j].x)*(point[i].x-point[j].x)
                + (point[i].y-point[j].y)*(point[i].y-point[j].y));
}
 
double Closest_Pair(int left, int right){
    double d = INF;
    if(left==right)
        return d;
    if(left + 1 == right)
        return dis(left, right);
    int mid = (left+right)>>1;
    double d1 = Closest_Pair(left,mid);
    double d2 = Closest_Pair(mid+1,right);
    d = min(d1,d2);
    int i,j,k=0;
    //分离出宽度为d的区间
    for(i = left; i <= right; i++){
        if(fabs(point[mid].x-point[i].x) <= d)
            tmpt[k++] = i;
    }
    sort(tmpt,tmpt+k,cmpy);
    //线性扫描
    for(i = 0; i < k; i++){
        for(j = i+1; j < k && point[tmpt[j]].y-point[tmpt[i]].y<d; j++){
            double d3 = dis(tmpt[i],tmpt[j]);
            if(d > d3)
                d = d3;
        }
    }
    return d;
}

int main(){
    while(true){
        scanf("%d",&n);
        if(n==0)
            break;
        for(int i = 0; i < n; i++)
            scanf("%lf %lf",&point[i].x,&point[i].y);
        sort(point,point+n,cmpxy);
        printf("%.2lf
",Closest_Pair(0,n-1)/2);
    }
    return 0;
}