Coneology(POJ 2932)
- 原题如下：
Coneology
Time Limit: 5000MS Memory Limit: 65536K Total Submissions: 4937 Accepted: 1086 Description
A student named Round Square loved to play with cones. He would arrange cones with different base radii arbitrarily on the floor and would admire the intrinsic beauty of the arrangement. The student even began theorizing about how some cones dominate other cones: a cone A dominates another cone B when cone B is completely within the cone A. Furthermore, he noted that there are some cones that not only dominate others, but are themselves dominated, thus creating complex domination relations. After studying the intricate relations of the cones in more depth, the student reached an important conclusion: there exist some cones, all-powerful cones, that have unique properties: an all-powerful cone is not dominated by any other cone. The student became so impressed by the mightiness of the all-powerful cones that he decided to worship these all-powerful cones.
Unfortunately, after having arranged a huge number of cones and having worked hard on developing this grandiose cone theory, the student become quite confused with all these cones, and he now fears that he might worship the wrong cones (what if there is an evil cone that tries to trick the student into worshiping it?). You need to help this student by finding the cones he should worship.
Input
The input le specifies an arrangement of the cones. There are in total N cones (1 ≤ N ≤ 40000). Cone i has radius and height equal to Ri, i = 1 … N. Each cone is hollow on the inside and has no base, so it can be placed over another cone with smaller radius. No two cones touch.
The first line of the input contains the integer N. The next N lines each contain three real numbers Ri, xi, yi separated by spaces, where (xi, yi) are the coordinates of the center of the base of cone i.
Output
The first line of the output le should contain the number of cones that the student should worship. The second line contains the indices of the cones that the student should worship in increasing order. Two consecutive numbers should be separated by a single space.
Sample Input
5 1 0 -2 3 0 3 10 0 0 1 0 1.5 10 50 50
Sample Output
2 3 5
- 题解：由于有任意两圆都没有公共点这一条件，要判断一个圆是否在其他圆的内部，只要判断其圆心是否在其他圆内即可。这样判断每个圆是否是最外层的复杂度为O(N)，因此很容易得到O(N2)复杂度的算法。而利用平面扫描技术可以得到更为高效的算法。
在几何问题中，经常利用平面扫描技术来降低算法的复杂度。所谓平面扫描，是指扫描线在平面上按给定轨迹移动的同时，不断根据扫描线扫过部分更新信息，从而得到整体所要求的结果的方法。扫描的方法，既可以从左向右平移与y轴平行的直线，也可以固定射线的端点逆时针转动。
对于这道题而言，我们在从左向右平移与y轴平行的直线的同时，维护与扫描线相交的最外层的圆的集合。从左向右移动的过程中，只有扫描线移动到圆的左右两端时，圆与扫描线的相交关系才会发生变化，因此我们先将所有这样的x坐标枚举出来并排好序。首先，当扫描线移动到某个圆的左端时，我们需要判断该圆是否包含在其他圆中，为此，我们只需从当前与扫描线相交的最外层的圆中，找到上下两侧y坐标方向距离最近的两个圆，并检查它们就足够了，因为，假设该圆被包含与更远的圆中，却不被包含于最近的圆中，就会得出两个圆相交的结论，而这与题目的条件不符，于是，只要用二叉查找树来维护这些圆，就能够在O(logn)时间内取得待检查的圆了。其次，当扫描线移动到某个圆的右端时，如果该圆已经包含于其他圆中就什么也不做，如果是最外层的圆就将它从二叉树中删去。综上，总的复杂度是O(nlogn)。 - 代码：
#include <cstdio> #include <utility> #include <set> #include <vector> #include <algorithm> using namespace std; const int MAX_N=40000; int N; double x[MAX_N], y[MAX_N], r[MAX_N]; bool inside(int i, int j) { double dx=x[i]-x[j], dy=y[i]-y[j]; return dx*dx+dy*dy<=r[j]*r[j]; } void solve() { vector<pair<double, int> > events; for (int i=0; i<N; i++) { events.push_back(make_pair(x[i]-r[i], i)); events.push_back(make_pair(x[i]+r[i], i+N)); } sort(events.begin(), events.end()); set<pair<double, int> > outers; vector<int> res; for (int i=0; i<events.size(); i++) { int id=events[i].second % N; if (events[i].second<N) { set<pair<double, int> >::iterator it=outers.lower_bound(make_pair(y[id], id)); if (it !=outers.end() && inside(id, it->second)) continue; if (it !=outers.begin() && inside(id, (--it)->second)) continue; res.push_back(id); outers.insert(make_pair(y[id], id)); } else { outers.erase(make_pair(y[id], id)); } } sort(res.begin(), res.end()); printf("%d\n", res.size()); for (int i=0; i<res.size(); i++) { printf("%d%c", res[i]+1, i+1==res.size() ? '\n' : ' '); } } int main() { scanf("%d", &N); for (int i=0; i<N; i++) { scanf("%lf%lf%lf", &r[i], &x[i], &y[i]); } solve(); }
