Queries on count of points lie inside a circle

Last Updated : 11 Sep, 2023

Given n coordinate (x, y) of points on 2D plane and Q queries. Each query contains an integer r, the task is to count the number of points lying inside or on the circumference of the circle having radius r and centered at the origin.
Examples :

Input : n = 5  Coordinates:   1 1  2 2  3 3  -1 -1  4 4    Query 1: 3  Query 2: 32    Output :  3  5  For first query radius = 3, number of points lie  inside or on the circumference are (1, 1), (-1, -1),  (2, 2). There are only 3 points lie inside or on   the circumference of the circle.  For second query radius = 32, all five points are  inside the circle.

The equation for the circle centered at origin (0, 0) with radius r, x² + y² = r². And condition for a point at (x₁, y₁) to lie inside or on the circumference, x₁² + y₁² <= r².
A Naive approach can be for each query, traverse through all points and check the condition. This take O(n*Q) time complexity.
An Efficient approach is to precompute x² + y² for each point coordinate and store them in an array p[]. Now, sort the array p[]. Then apply binary search on the array to find last index with condition p[i] <= r² for each query.
Below is the implementation of this approach:

C++

   // C++ program to find number of points lie inside or 
// on the circumference of circle for Q queries. 
#include <bits/stdc++.h> 
using namespace std; 
  
// Computing the x^2 + y^2 for each given points 
// and sorting them. 
void preprocess(int p[], int x[], int y[], int n) 
{ 
    for (int i = 0; i < n; i++) 
        p[i] = x[i] * x[i] + y[i] * y[i]; 
  
    sort(p, p + n); 
} 
  
// Return count of points lie inside or on circumference 
// of circle using binary search on p[0..n-1] 
int query(int p[], int n, int rad) 
{ 
    int start = 0, end = n - 1; 
    while ((end - start) > 1) { 
        int mid = (start + end) / 2; 
        double tp = sqrt(p[mid]); 
  
        if (tp > (rad * 1.0)) 
            end = mid - 1; 
        else
            start = mid; 
    } 
  
    double tp1 = sqrt(p[start]), tp2 = sqrt(p[end]); 
  
    if (tp1 > (rad * 1.0)) 
        return 0; 
    else if (tp2 <= (rad * 1.0)) 
        return end + 1; 
    else
        return start + 1; 
} 
  
// Driven Program 
int main() 
{ 
    int x[] = { 1, 2, 3, -1, 4 }; 
    int y[] = { 1, 2, 3, -1, 4 }; 
    int n = sizeof(x) / sizeof(x[0]); 
  
    // Compute distances of all points and keep 
    // the distances sorted so that query can 
    // work in O(logn) using Binary Search. 
    int p[n]; 
    preprocess(p, x, y, n); 
  
    // Print number of points in a circle of radius 3. 
    cout << query(p, n, 3) << endl; 
  
    // Print number of points in a circle of radius 32. 
    cout << query(p, n, 32) << endl; 
    return 0; 
} 
 
  

Java

   // JAVA Code for Queries on count of 
// points lie inside a circle 
import java.util.*; 
  
class GFG { 
  
    // Computing the x^2 + y^2 for each given points 
    // and sorting them. 
    public static void preprocess(int p[], int x[], 
                                  int y[], int n) 
    { 
        for (int i = 0; i < n; i++) 
            p[i] = x[i] * x[i] + y[i] * y[i]; 
  
        Arrays.sort(p); 
    } 
  
    // Return count of points lie inside or on 
    // circumference of circle using binary 
    // search on p[0..n-1] 
    public static int query(int p[], int n, int rad) 
    { 
        int start = 0, end = n - 1; 
        while ((end - start) > 1) { 
            int mid = (start + end) / 2; 
            double tp = Math.sqrt(p[mid]); 
  
            if (tp > (rad * 1.0)) 
                end = mid - 1; 
            else
                start = mid; 
        } 
  
        double tp1 = Math.sqrt(p[start]); 
        double tp2 = Math.sqrt(p[end]); 
  
        if (tp1 > (rad * 1.0)) 
            return 0; 
        else if (tp2 <= (rad * 1.0)) 
            return end + 1; 
        else
            return start + 1; 
    } 
  
    /* Driver program to test above function */
    public static void main(String[] args) 
    { 
        int x[] = { 1, 2, 3, -1, 4 }; 
        int y[] = { 1, 2, 3, -1, 4 }; 
        int n = x.length; 
  
        // Compute distances of all points and keep 
        // the distances sorted so that query can 
        // work in O(logn) using Binary Search. 
        int p[] = new int[n]; 
        preprocess(p, x, y, n); 
  
        // Print number of points in a circle of 
        // radius 3. 
        System.out.println(query(p, n, 3)); 
  
        // Print number of points in a circle of 
        // radius 32. 
        System.out.println(query(p, n, 32)); 
    } 
} 
// This code is contributed by Arnav Kr. Mandal. 
 
  

Python 3

   # Python 3 program to find number of  
# points lie inside or on the circumference  
# of circle for Q queries. 
import math 
  
# Computing the x^2 + y^2 for each  
# given points and sorting them. 
def preprocess(p, x, y, n): 
    for i in range(n): 
        p[i] = x[i] * x[i] + y[i] * y[i] 
  
    p.sort() 
  
# Return count of points lie inside  
# or on circumference of circle using 
# binary search on p[0..n-1] 
def query(p, n, rad): 
  
    start = 0
    end = n - 1
    while ((end - start) > 1): 
        mid = (start + end) // 2
        tp = math.sqrt(p[mid]) 
  
        if (tp > (rad * 1.0)): 
            end = mid - 1
        else: 
            start = mid 
  
    tp1 = math.sqrt(p[start]) 
    tp2 = math.sqrt(p[end]) 
  
    if (tp1 > (rad * 1.0)): 
        return 0
    else if (tp2 <= (rad * 1.0)): 
        return end + 1
    else: 
        return start + 1
  
# Driver Code 
if __name__ == "__main__": 
      
    x = [ 1, 2, 3, -1, 4 ] 
    y = [ 1, 2, 3, -1, 4 ] 
    n = len(x) 
  
    # Compute distances of all points and keep 
    # the distances sorted so that query can 
    # work in O(logn) using Binary Search. 
    p = [0] * n 
    preprocess(p, x, y, n) 
  
    # Print number of points in a  
    # circle of radius 3. 
    print(query(p, n, 3)) 
  
    # Print number of points in a  
    # circle of radius 32. 
    print(query(p, n, 32)) 
  
# This code is contributed by ita_c 
 
  

C#

   // C# Code for Queries on count of 
// points lie inside a circle 
using System; 
  
class GFG { 
  
    // Computing the x^2 + y^2 for each  
    // given points and sorting them. 
    public static void preprocess(int[] p, int[] x, 
                                    int[] y, int n) 
    { 
        for (int i = 0; i < n; i++) 
            p[i] = x[i] * x[i] + y[i] * y[i]; 
  
        Array.Sort(p); 
    } 
  
    // Return count of points lie inside or on 
    // circumference of circle using binary 
    // search on p[0..n-1] 
    public static int query(int[] p, int n, int rad) 
    { 
        int start = 0, end = n - 1; 
        while ((end - start) > 1) { 
            int mid = (start + end) / 2; 
            double tp = Math.Sqrt(p[mid]); 
  
            if (tp > (rad * 1.0)) 
                end = mid - 1; 
            else
                start = mid; 
        } 
  
        double tp1 = Math.Sqrt(p[start]); 
        double tp2 = Math.Sqrt(p[end]); 
  
        if (tp1 > (rad * 1.0)) 
            return 0; 
        else if (tp2 <= (rad * 1.0)) 
            return end + 1; 
        else
            return start + 1; 
    } 
  
    /* Driver program to test above function */
    public static void Main() 
    { 
        int[] x = { 1, 2, 3, -1, 4 }; 
        int[] y = { 1, 2, 3, -1, 4 }; 
        int n = x.Length; 
  
        // Compute distances of all points and keep 
        // the distances sorted so that query can 
        // work in O(logn) using Binary Search. 
        int[] p = new int[n]; 
        preprocess(p, x, y, n); 
  
        // Print number of points in a circle of 
        // radius 3. 
        Console.WriteLine(query(p, n, 3)); 
  
        // Print number of points in a circle of 
        // radius 32. 
        Console.WriteLine(query(p, n, 32)); 
    } 
} 
  
// This code is contributed by vt_m. 
 
  

Javascript

   <script> 
  
// Javascript Code for Queries on count of 
// points lie inside a circle 
      
    // Computing the x^2 + y^2 for each given points 
    // and sorting them. 
    function preprocess(p,x,y,n) 
    { 
        for (let i = 0; i < n; i++) 
            p[i] = x[i] * x[i] + y[i] * y[i]; 
          
        p.sort(function(a,b){return a-b;}); 
          
    } 
      
    // Return count of points lie inside or on 
    // circumference of circle using binary 
    // search on p[0..n-1] 
    function query(p,n,rad) 
    { 
        let start = 0, end = n - 1; 
        while ((end - start) > 1) { 
            let mid = Math.floor((start + end) / 2); 
            let tp = Math.sqrt(p[mid]); 
    
            if (tp > (rad * 1.0)) 
                end = mid - 1; 
            else
                start = mid; 
        } 
    
        let tp1 = Math.sqrt(p[start]); 
        let tp2 = Math.sqrt(p[end]); 
    
        if (tp1 > (rad * 1.0)) 
            return 0; 
        else if (tp2 <= (rad * 1.0)) 
            return end + 1; 
        else
            return start + 1; 
    } 
      
    /* Driver program to test above function */
    let x=[1, 2, 3, -1, 4 ]; 
    let y=[1, 2, 3, -1, 4]; 
    let n = x.length; 
      
    // Compute distances of all points and keep 
    // the distances sorted so that query can 
    // work in O(logn) using Binary Search. 
    let p=new Array(n); 
    for(let i=0;i<n;i++) 
    { 
        p[i]=0; 
    } 
    preprocess(p, x, y, n); 
  
    // Print number of points in a circle of 
    // radius 3. 
    document.write(query(p, n, 3)+"<br>"); 
  
    // Print number of points in a circle of 
    // radius 32. 
    document.write(query(p, n, 32)+"<br>"); 
      
    // This code is contributed by rag2127 
      
</script>
 
  

Output:

3  5

Time Complexity: O(n log n) for preprocessing and O(Q Log n) for Q queries.
Auxiliary Space: O(n) it is using extra space for array p

Count Integral points inside a Triangle

Aarti_Rathi and Anuj Chauhan

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Practice Tags :

Queries on count of points lie inside a circle

C++

Java

Python 3

C#

Javascript

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