Find minimum area of rectangle with given set of coordinates

Given an array arr of set of points in the X-Y plane. The task is to find the minimum area of a rectangle that can be formed from these points. The sides of the rectangle should be parallel to the X and Y axes. If a rectangle cannot be formed with the given points then print 0.

Examples:

Input: arr[][] = [[1, 1], [1, 3], [3, 1], [3, 3], [2, 2]]
Output: 4
The only rectangle possible will be formed with the points (1, 1), (1, 3), (3, 1) and (3, 3)

Input: arr[][] = [[1, 1], [1, 3], [3, 1], [3, 3], [4, 1], [4, 3]]
Output: 2

Approach: Group the points by X coordinates, so that points on straight vertical lines are grouped together. Then, for every pair of points in a group, for eg. coordinates (X, Y1) and (X, Y2), we check for the smallest rectangle with this pair of points as the rightmost edge of the rectangle to be formed. We can do this by keeping track of all other pairs of points we’ve visited before. Finally return the minimum possible area of the rectangle obtained.

Below is the implementation of the above approach:

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# Python Implementation of above approach
import collections
  
# function to find minimum area of Rectangle
def minAreaRect(A):
  
    # creating empty columns
    columns = collections.defaultdict(list)
  
    # fill columns with coordinates
    for x, y in A:
        columns[x].append(y)
  
    lastx = {}
    ans = float('inf')
  
    for x in sorted(columns):
        column = columns[x]
        column.sort()
        for j, y2 in enumerate(column):
            for i in range(j):
                y1 = column[i]
  
                # check if rectangle can be formed
                if (y1, y2) in lastx:
                    ans = min(ans, (x - lastx[y1, y2]) * (y2 - y1))
                lastx[y1, y2] = x
  
    if ans < float('inf'):
        return ans
    else:
        return 0
  
# Driver code
A = [[1, 1], [1, 3], [3, 1], [3, 3], [2, 2]]
print(minAreaRect(A))

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Output:

4


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