<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>

class Solution {
public:

    inline size_t d2(int x1, int y1, int x2, int y2) {
        return (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2);
    }

    string createKey(int centerX, int centerY, size_t ed2) {
        return to_string(centerX) + "," + to_string(centerY) + "," + to_string(ed2);
    }

    double minAreaFreeRect(vector<vector<int>>& ps) {
        unordered_map<string, vector<vector<int>>> m;
        size_t res = numeric_limits<size_t>::max(); // Initialize to max size_t

        for (size_t i = 0; i < ps.size(); ++i) {
            for (size_t j = i + 1; j < ps.size(); ++j) {
                int centerX = ps[i][0] + ps[j][0];
                int centerY = ps[i][1] + ps[j][1];
                size_t len = d2(ps[i][0], ps[i][1], ps[j][0], ps[j][1]);
                string key = createKey(centerX, centerY, len);
                m[key].push_back({ps[i][0], ps[i][1], ps[j][0], ps[j][1]});
            }
        }

        for (const auto& it_c : m) {
            const auto& pairs = it_c.second;
            for (size_t i = 0; i < pairs.size(); ++i) {
                for (size_t j = i + 1; j < pairs.size(); ++j) {
                    const auto& p1 = pairs[i];
                    const auto& p2 = pairs[j];
                    size_t d1 = d2(p1[0], p1[1], p2[0], p2[1]);
                    size_t d2_ = d2(p1[0], p1[1], p2[2], p2[3]);
                    size_t area = d1 * d2_;
                    if (res > area) res = area;
                }
            }
        }

        // Return the square root of the minimum area
        return res == numeric_limits<size_t>::max() ? 0 : sqrt(res);
    }
    
};
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