<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) {
        int n = points.size();
        unordered_map<LL, unordered_map<LL, vector<P2I>>> center_to_edge_square_to_pairs;
        LL base = 1e5;
        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 c_x = x1+x2;
                int c_y = y1+y2;
                LL center = c_x*base + c_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_y_pairs]: center_to_edge_square_to_pairs){
            for(auto& [edge_square, pairs]: edge_square_to_y_pairs){
                int size = pairs.size();
                for(int i=0; i<size; i++){
                    auto [a, b] = pairs[i];
                    for(int j=i+1; j<size; j++){
                        auto [c, d] = pairs[j];
                        if(a==c || a==d || b==c || b==d) continue;

                        LL ac = dist(points, a, c);
                        LL ad = dist(points, a, d);
                        min_area = min(min_area, 1.0*ac*ad);
                    }
                }
            }
        }
        return min_area==DBL_MAX ?0 :sqrt(min_area);
    }

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