<p>Given an array of <code>points</code> where <code>points[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> represents a point on the <strong>X-Y</strong> plane, return <em>the maximum number of points that lie on the same straight line</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/02/25/plane1.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> points = [[1,1],[2,2],[3,3]]
<strong>Output:</strong> 3
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/02/25/plane2.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> points = [[1,1],[3,2],[5,3],[4,1],[2,3],[1,4]]
<strong>Output:</strong> 4
</pre>

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

<ul>
	<li><code>1 &lt;= points.length &lt;= 300</code></li>
	<li><code>points[i].length == 2</code></li>
	<li><code>-10<sup>4</sup> &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 10<sup>4</sup></code></li>
	<li>All the <code>points</code> are <strong>unique</strong>.</li>
</ul>

class Solution:
    def maxPoints(self, points: List[List[int]]) -> int:
        #y = mx + c
        #c = y - mx
        #c_x = -c/m
        def grad(p1, p2):
            y1,x1 = p1
            y2,x2 = p2
            if (x2 - x1) == 0:
                m = "inf"
                c = None
                c_x = x1
            else:
                m = (y2 - y1) / (x2 - x1)
                if m == 0:
                    c = y1
                    c_x = None
                else:
                    c = y1 - m * x1
                    c_x = -c / m
            return m,c,c_x

        lines = {} # (m,c) -> freq
        visited = {}
        points.sort(key=lambda x: x[0])
        n = len(points)
        for i in range(n):
            for j in range(i+1, n):
                m,c,c_x = grad(points[i], points[j])
                if (m,c,c_x) in visited:
                    continue
                else:
                    lines[(m,c,c_x)] = lines.get((m,c,c_x), 1) + 1
            visited.update(lines)
        if n < 2:
            return n
        return max(lines.values())


        
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