Find minimum radius such that atleast k point lie inside the circle

Given a positive integer K, a circle center at (0, 0) and coordinates of some points. The task is to find minimum radius of the circle so that at-least k points lie inside the circle. Output the square of the minimum radius. 

Examples: 

Input : (1, 1), (-1, -1), (1, -1), 
 k = 3
Output : 2
We need a circle of radius at least 2
to include 3 points.
Input : (1, 1), (0, 1), (1, -1), 
 k = 2
Output : 1
We need a circle of radius at least 1
to include 2 points. The circle around
(0, 0) of radius 1 would include (1, 1)
and (0, 1).

The idea is to find square of Euclidean Distance of each point from origin (0, 0). Now, sort these distance in increasing order. Now the k^th element of distance is the required minimum radius.
Below is the implementation of this approach: 

2

Time complexity: O(n + nlogn)
Auxiliary Space: O(n)ve.

Approach#2: Using binary search

This code uses binary search to find the minimum radius such that at least k points lie inside or on the circumference of the circle. It first finds the maximum distance between any two points, then performs binary search on the range [0, max_distance] to find the minimum radius.

Algorithm

1. Initialize left = 0 and right = maximum distance between any two points in the given set of points.
2. While left <= right, find mid = (left + right) / 2
3. Check if there exist k points inside or on the circumference of a circle with radius mid using a simple linear search. 4. If k or more points are inside or on the circumference of the circle, set right = mid - 1.
5. If less than k points are inside or on the circumference of the circle, set left = mid + 1.
6. After the binary search, the value of left will be the minimum radius required to include k points.

2

Time Complexity: O(n^2 * log(r)) where n is the number of points and r is the maximum distance between any two points.
Space complexity: O(1) as it uses only a constant amount of extra space irrespective of the size of the input.

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