<p>You are given an array of points in the <strong>X-Y</strong> plane <code>points</code> where <code>points[i] = [x<sub>i</sub>, y<sub>i</sub>]</code>.</p>

<p>Return <em>the minimum area of any rectangle formed from these points, with sides <strong>not necessarily parallel</strong> to the X and Y axes</em>. If there is not any such rectangle, return <code>0</code>.</p>

<p>Answers within <code>10<sup>-5</sup></code> of the actual answer will be accepted.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/21/1a.png" style="width: 398px; height: 400px;" />
<pre>
<strong>Input:</strong> points = [[1,2],[2,1],[1,0],[0,1]]
<strong>Output:</strong> 2.00000
<strong>Explanation:</strong> The minimum area rectangle occurs at [1,2],[2,1],[1,0],[0,1], with an area of 2.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/22/2.png" style="width: 400px; height: 251px;" />
<pre>
<strong>Input:</strong> points = [[0,1],[2,1],[1,1],[1,0],[2,0]]
<strong>Output:</strong> 1.00000
<strong>Explanation:</strong> The minimum area rectangle occurs at [1,0],[1,1],[2,1],[2,0], with an area of 1.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/22/3.png" style="width: 383px; height: 400px;" />
<pre>
<strong>Input:</strong> points = [[0,3],[1,2],[3,1],[1,3],[2,1]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> There is no possible rectangle to form from these points.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= points.length &lt;= 50</code></li>
	<li><code>points[i].length == 2</code></li>
	<li><code>0 &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 4 * 10<sup>4</sup></code></li>
	<li>All the given points are <strong>unique</strong>.</li>
</ul>

using LL = long long;
using P2I = pair<int, int>;
class Solution {
public:
    double minAreaFreeRect(vector<vector<int>>& points) {
        unordered_map<string, unordered_map<int, vector<P2I>>> center_to_edge_square_to_pairs;
        int n = points.size();
        for(int i=0; i<n; i++){
            int x1 = points[i][0];
            int y1 = points[i][1];
            for(int j=i+1; j<n; j++){
                int x2 = points[j][0];
                int y2 = points[j][1];
                int center_x = (x1+x2);
                int center_y = (y1+y2);
                string center = to_string(center_x) + "_" + to_string(center_y);
                LL edge_square = 1ll*(x2-x1)*(x2-x1) + 1ll*(y2-y1)*(y2-y1);
                center_to_edge_square_to_pairs[center][edge_square].push_back({i, j});
            }
        }

        double min_area = DBL_MAX;
        for(auto& [center, edge_square_to_pairs]: center_to_edge_square_to_pairs){
            for(auto& [edge_square, pairs]: edge_square_to_pairs){
                if(pairs.size()<=1) continue;

                int size = pairs.size();
                for(int i=0; i<size; i++){
                    for(int j=i+1; j<size; j++){
                        auto& [p1, p2] = pairs[i];
                        auto& [p3, p4] = pairs[j];

                        unordered_set<int> distinct_id;
                        distinct_id.insert(p1);
                        distinct_id.insert(p2);
                        distinct_id.insert(p3);
                        distinct_id.insert(p4);
                        if(distinct_id.size()<4) continue;

                        min_area = min(min_area, 1.0*dist(points[p1], points[p3])*dist(points[p1], points[p4]));
                    }
                }
            }
        }
        return min_area==DBL_MAX ?0 : sqrt(min_area);
    }

    long long dist(const vector<int>& a, const vector<int>& b){
        int x = a[0]-b[0];
        int y = a[1]-b[1];
        return 1ll*x*x + 1ll*y*y;
    }
};
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