<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:    
    double minAreaFreeRect(vector<vector<int>>& points) {
        long ans = LONG_MAX;
        // map<mid_point, map<length, vector>>
        map<vector<long>, multimap<long, vector<int>>> db;
        for (auto i = points.begin(); i != points.end(); i++) {
            for (auto j = i+1; j != points.end(); j++) {
                if (i == j) continue;
                
                long x1 = j->at(0) - i->at(0);
                long y1 = j->at(1) - i->at(1);
                
                long length = x1 * x1 + y1 * y1;

                vector<long> mid = {i->at(0) + j->at(0), i->at(1) + j->at(1)};
                
                if (db.find(mid) == db.end()) {
                    db.insert(std::make_pair(mid, multimap<long, vector<int>>()));
                }
                
                auto &m = (db.find(mid))->second;
                if (m.find(length) != m.end()) {
                    auto p = m.equal_range(length);
                    for (auto k = p.first; k != p.second; k++) {
                        ans = std::min(ans, std::abs(x1 * k->second.at(1) - y1 * k->second.at(0)));
                    }
                }
                m.insert(std::make_pair(length, vector<int>{int(x1), int(y1)}));
            }
        }
        return ans == LONG_MAX ? 0 : ans / 2;
    }

};
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