从给定的一组 N 个点中平行于坐标轴的正方形计数

给定笛卡尔坐标系中的点数组points[] ，任务是找到平行于坐标轴的正方形的数量。

例子：

Input:points[] = {(0, 0), (0, 2), (2, 0), (2, 2), (1, 1)}
Output: 1
Explanation:
As the points (0, 0), (0, 2), (2, 0), (2, 2) forms square which is parallel to the X-axis and Y-axis, Hence the count of such squares is 1.

Input:points[] = {(2, 0), (0, 2), (2, 2), (0, 0), (-2, 2), (-2, 0)}
Output: 2
Explanation:
As the points (0, 0), (0, 2), (2, 0), (2, 2) forms one square, whereas points (0, 0), (0, 2), (-2, 0), (-2, 2) forms other square which is parallel to the X-axis and Y-axis,
Hence the count of such squares is 2.

方法：思路是从点阵中选择两个点，使得这两个点平行于坐标轴，然后借助点之间的距离找到正方形的其他两个点。如果这些点存在于数组中，那么就有一个这样的可能的正方形。

下面是上述方法的实现：

C++

// C++ implementation to find count of Squares
// that are parallel to the coordinate axis
// from the given set of N points
  
#include 
using namespace std;
  
#define sz(x) int(x.size())
  
// Function to get distance
// between two points
int get_dis(pair p1,
            pair p2)
{
    int a = abs(p1.first - p2.first);
    int b = abs(p1.second - p2.second);
    return ((a * a) + (b * b));
}
  
// Function to check that points
// forms a square and parallel to
// the co-ordinate axis
bool check(pair p1,
           pair p2,
           pair p3,
           pair p4)
{
  
    int d2 = get_dis(p1, p2);
    int d3 = get_dis(p1, p3);
    int d4 = get_dis(p1, p4);
    if (d2 == d3
        && 2 * d2 == d4
        && 2 * get_dis(p2, p4) == get_dis(p2, p3)) {
        return true;
    }
    if (d3 == d4
        && 2 * d3 == d2
        && 2 * get_dis(p3, p2) == get_dis(p3, p4)) {
        return true;
    }
    if (d2 == d4
        && 2 * d2 == d3
        && 2 * get_dis(p2, p3) == get_dis(p2, p4)) {
        return true;
    }
    return false;
}
  
// Function to find all the squares which is
// parallel to co-ordinate axis
int count(map, int> hash,
          vector > v, int n)
{
    int ans = 0;
    map, int> vis;
  
    // Loop to choose two points
    // from the array of points
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            if (i == j)
                continue;
            pair p1
                = make_pair(v[i].first,
                            v[j].second);
            pair p2
                = make_pair(v[j].first,
                            v[i].second);
            set > s;
            s.insert(v[i]);
            s.insert(v[j]);
            s.insert(p1);
            s.insert(p2);
            if (sz(s) != 4)
                continue;
  
            // Condition to check if the
            // other points are present in the map
            if (hash.find(p1) != hash.end()
                && hash.find(p2) != hash.end()) {
                if ((!vis[v[i]] || !vis[v[j]]
                     || !vis[p1] || !vis[p2])
                    && (check(v[i], v[j], p1, p2))) {
  
                    vis[v[i]] = 1;
                    vis[v[j]] = 1;
                    vis[p1] = 1;
                    vis[p2] = 1;
                    ans++;
                }
            }
        }
    }
    cout << ans;
    return ans;
}
  
// Function to Count the number of squares
void countOfSquares(vector > v, int n)
{
    ios_base::sync_with_stdio(0);
    cin.tie(0);
  
    map, int> hash;
  
    // Declaring iterator to a vector
    vector >::iterator ptr;
  
    // Adding the points to hash
    for (ptr = v.begin(); ptr < v.end(); ptr++)
        hash[*ptr] = 1;
  
    // Count the number of squares
    count(hash, v, n);
}
  
// Driver Code
int main()
{
  
    int n = 5;
    vector > v;
    v.push_back(make_pair(0, 0));
    v.push_back(make_pair(0, 2));
    v.push_back(make_pair(2, 0));
    v.push_back(make_pair(2, 2));
    v.push_back(make_pair(0, 1));
  
    // Function call
    countOfSquares(v, n);
    return 0;
}

输出：

1

性能分析：

• 时间复杂度：与上述方法一样，有两个循环需要 O(N ² ) 时间，因此时间复杂度将为O(N ² ) 。
• 辅助空间复杂度：与上述方法一样，使用了额外的空间，因此辅助空间复杂度为O(N) 。

如果您希望与专家一起参加现场课程，请参阅DSA 现场工作专业课程和学生竞争性编程现场课程。