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: 

C++

// C++ program to find minimum radius
// such that atleast k point lie inside
// the circle
#include<bits/stdc++.h>
using namespace std;

// Return minimum distance required so that
// atleast k point lie inside the circle.
int minRadius(int k, int x[], int y[], int n)
{
   int dis[n];

   // Finding distance between of each
   // point from origin
   for (int i = 0; i < n; i++)
       dis[i] = x[i] * x[i] + y[i] * y[i];

    // Sorting the distance
    sort(dis, dis + n);

    return dis[k - 1];
}

// Driven Program
int main()
{
  int k = 3;
  int x[] = { 1, -1, 1 };
  int y[] = { 1, -1, -1 };
  int n = sizeof(x)/sizeof(x[0]);

  cout << minRadius(k, x, y, n) << endl;

  return 0;
}

Java

// Java program to find minimum radius
// such that atleast k point lie inside
// the circle
import java.util.Arrays;

class GFG
{

    // Return minimum distance required so that
    // atleast k point lie inside the circle.
    static int minRadius(int k, int[] x, int[] y,
                                          int n)
    {
        int[] dis=new int[n];

        // Finding distance between of each
        // point from origin
        for (int i = 0; i < n; i++)
            dis[i] = x[i] * x[i] + y[i] * y[i];

        // Sorting the distance
        Arrays.sort(dis);

        return dis[k - 1];
    }

    // Driven Program
    public static void main (String[] args) {

    int k = 3;
    int[] x = { 1, -1, 1 };
    int[] y = { 1, -1, -1 };
    int n = x.length;

    System.out.println(minRadius(k, x, y, n));

    }
}

/* This code is contributed by Mr. Somesh Awasthi */

Python3

# Python3 program to find minimum radius
# such that atleast k point lie inside
# the circle


# Return minimum distance required so
# that atleast k point lie inside the
# circle.
def minRadius(k, x, y, n):
    dis = [0] * n

    # Finding distance between of each
    # point from origin

    for i in range(0, n):
        dis[i] = x[i] * x[i] + y[i] * y[i]

    # Sorting the distance
    dis.sort()

    return dis[k - 1]

# Driver Program
k = 3
x = [1, -1, 1]
y = [1, -1, -1]
n = len(x)

print(minRadius(k, x, y, n))

# This code is contributed by
# Prasad Kshirsagar

C#

// C# program to find minimum radius
// such that atleast k point lie inside
// the circle
using System;

class GFG {

    // Return minimum distance required
    // so that atleast k point lie inside
    // the circle.
    static int minRadius(int k, int []x,
                          int[] y, int n)
    {
        int[] dis = new int[n];

        // Finding distance between of
        // each point from origin
        for (int i = 0; i < n; i++)
            dis[i] = x[i] * x[i] +
                       y[i] * y[i];

        // Sorting the distance
        Array.Sort(dis);

        return dis[k - 1];
    }

    // Driven Program
    public static void Main ()
    {
        int k = 3;
        int[] x = { 1, -1, 1 };
        int[] y = { 1, -1, -1 };
        int n = x.Length;

        Console.WriteLine(
              minRadius(k, x, y, n));
    }
}

// This code is contributed by vt_m.

PHP

<?php
// PHP program to find minimum radius
// such that atleast k point lie inside
// the circle

// Return minimum distance required
// so that atleast k point lie
// inside the circle.
function minRadius($k, $x, $y, $n)
{
    $dis =array();

    // Finding distance between
    // of each point from origin
    for ($i = 0; $i < $n; $i++)
        $dis[$i] = $x[$i] * $x[$i] +
                   $y[$i] * $y[$i];

        // Sorting the distance
        sort($dis);

        return $dis[$k - 1];
}

// Driver Code
$k = 3;
$x = array(1, -1, 1);
$y = array(1, -1, -1);
$n = count($x);

echo minRadius($k, $x, $y, $n) ;

// This code is contributed by anuj_67.
?>

Javascript

<script>
// Javascript program to find minimum radius
// such that atleast k point lie inside
// the circle

    // Return minimum distance required so that
    // atleast k point lie inside the circle.
    function minRadius(k, x, y, n) {

        let dis = Array.from({length: n}, (_, i) => 0);

        // Finding distance between of each
        // point from origin
        for (let i = 0; i < n; i++)
            dis[i] = x[i] * x[i] + y[i] * y[i];

        // Sorting the distance
        dis.sort();

        return dis[k - 1];
    }

// driver function

    let k = 3;
    let x = [ 1, -1, 1 ];
    let y = [ 1, -1, -1 ];
    let n = x.length;

    document.write(minRadius(k, x, y, n));

 // This code is contributed by code_hunt.
</script>

Output

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.

C++

#include <cmath>
#include <iostream>
#include <vector>

using namespace std;

double dist(pair<int, int> p1, pair<int, int> p2)
{
    // Calculate Euclidean distance between two points
    return sqrt(pow(p1.first - p2.first, 2)
                + pow(p1.second - p2.second, 2));
}

int count_points_in_circle(vector<pair<int, int> > points,
                           pair<int, int> center,
                           double radius)
{
    // Count the number of points inside or on the
    // circumference of the circle
    int count = 0;
    for (auto point : points) {
        if (dist(point, center) <= radius) {
            count++;
        }
    }
    return count;
}

int MinimumRadius(vector<pair<int, int> > points, int k)
{
    double left = 0.0;
    double right = 0.0;
    // Find the maximum distance between any two points
    for (int i = 0; i < points.size(); i++) {
        for (int j = i + 1; j < points.size(); j++) {
            double d = dist(points[i], points[j]);
            if (d > right) {
                right = d;
            }
        }
    }
    while (left <= right) {
        double mid = (left + right) / 2.0;
        bool found = false;
        // Check if there exist k points inside or on the
        // circumference of a circle with radius mid
        for (int i = 0; i < points.size(); i++) {
            if (count_points_in_circle(points, points[i],
                                       mid)
                >= k) {
                found = true;
                break;
            }
        }
        if (found) {
            right = mid - 1.0;
        }
        else {
            left = mid + 1.0;
        }
    }
    return static_cast<int>(left);
}

// Example usage
int main()
{
    vector<pair<int, int> > points{ { 1, 1 },
                                    { -1, -1 },
                                    { 1, -1 } };
    int k = 3;
    cout << MinimumRadius(points, k) << endl;
}

Python3

import math

def dist(p1, p2):
    # Calculate Euclidean distance between two points
    return math.sqrt((p1[0] - p2[0])**2 + (p1[1] - p2[1])**2)

def count_points_in_circle(points, center, radius):
    # Count the number of points inside or on the circumference of the circle
    count = 0
    for point in points:
        if dist(point, center) <= radius:
            count += 1
    return count

def minimum_radius(points, k):
    left, right = 0, 0
    # Find the maximum distance between any two points
    for i in range(len(points)):
        for j in range(i+1, len(points)):
            d = dist(points[i], points[j])
            if d > right:
                right = d
    while left <= right:
        mid = (left + right) / 2
        found = False
        # Check if there exist k points inside or on the circumference of a circle with radius mid
        for i in range(len(points)):
            if count_points_in_circle(points, points[i], mid) >= k:
                found = True
                break
        if found:
            right = mid - 1
        else:
            left = mid + 1
    return int(left)

# Example usage
points = [(1, 1), (-1, -1), (1, -1)]
k = 3
print(minimum_radius(points, k))

Javascript

// JavaScript implementation of Minimum Radius problem
function dist(p1, p2) {
    // Calculate Euclidean distance between two points
    return Math.sqrt(Math.pow(p1[0] - p2[0], 2) + Math.pow(p1[1] - p2[1], 2));
}

function count_points_in_circle(points, center, radius) {
    // Count the number of points inside or on the
    // circumference of the circle
    let count = 0;
    for (let point of points) {
        if (dist(point, center) <= radius) {
            count++;
        }
    }
    return count;
}

function MinimumRadius(points, k) {
    let left = 0.0;
    let right = 0.0;
    // Find the maximum distance between any two points
    for (let i = 0; i < points.length; i++) {
        for (let j = i + 1; j < points.length; j++) {
            let d = dist(points[i], points[j]);
            if (d > right) {
                right = d;
            }
        }
    }
    while (left <= right) {
        let mid = (left + right) / 2.0;
        let found = false;
        // Check if there exist k points inside or on the
        // circumference of a circle with radius mid
        for (let i = 0; i < points.length; i++) {
            if (count_points_in_circle(points, points[i], mid) >= k) {
                found = true;
                break;
            }
        }
        if (found) {
            right = mid - 1.0;
        } else {
            left = mid + 1.0;
        }
    }
    return Math.floor(left);
}

// Example usage
const points = [
    [1, 1],
    [-1, -1],
    [1, -1]
];
const k = 3;
console.log(MinimumRadius(points, k));

Output

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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