<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>

from typing import List
from fractions import Fraction

class Solution:
    # a line can be described by y = mx + b, e.g. two numbers: m and b.
    # for each point i, compare with all other points j > i, calculate slop and intercept
    # bin the results, keep track of the max bin size.
    # very simple, O(n^2), impossible to do better.
    def maxPoints(self, points: List[List[int]]) -> int:
        n = len(points)
        if n == 1:
            return 1
        
        dict = {}
        max_points = 0
        for i in range(n):
            for j in range(i+1, n):
                xi, yi = points[i] if points[i][0] < points[j][0] else points[j]
                xj, yj = points[j] if points[i][0] < points[j][0] else points[i]

                if xj != xi:
                    #m = float((yj - yi) / (xj - xi))
                    m = Fraction((yj - yi), (xj - xi))
                else:
                    m = float('inf')

                if m == float('inf'):
                    b = float(xi)
                elif m == 0:
                    b = "inf, " + str(yi)
                else:
                    #b = float(yi - m*xi)
                    b = Fraction(yi - m*xi)

                b = str(b)
                m = str(m)

                if (m, b) not in dict:
                    dict[(m, b)] = set()
                dict[(m, b)].add(i)
                dict[(m, b)].add(j)
                
                max_points = max(max_points, len(dict[(m, b)]))

        print(dict)

        return max_points
40436