Count Distinct Non-Negative Integer Pairs (x, y) that Satisfy the Inequality x*x + y*y < n - GeeksforGeeks

03 Nov, 2021
Given a positive number n, count all distinct Non-Negative Integer pairs (x, y) that satisfy the inequality x*x + y*y < n. 
Examples:
Input:  n = 5
Output: 6
The pairs are (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (0, 2)

Input: n = 6
Output: 8
The pairs are (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (0, 2),
              (1, 2), (2, 1)
A Simple Solution is to run two loops. The outer loop goes for all possible values of x (from 0 to √n). The inner loops pick all possible values of y for the current value of x (picked by outer loop). 
Following is the implementation of a simple solution.  
C++
Java
Python3
C#
PHP
Javascript
#include <iostream>using namespace std; // This function counts number of pairs (x, y) that satisfy// the inequality x*x + y*y < n.int countSolutions(int n){   int res = 0;   for (int x = 0; x*x < n; x++)      for (int y = 0; x*x + y*y < n; y++)         res++;   return res;} // Driver program to test above functionint main(){    cout << "Total Number of distinct Non-Negative pairs is "         << countSolutions(6) << endl;    return 0;}
// Java code to Count Distinct// Non-Negative Integer Pairs// (x, y) that Satisfy the// inequality x*x + y*y < nimport java.io.*; class GFG{    // This function counts number    // of pairs (x, y) that satisfy    // the inequality x*x + y*y < n.    static int countSolutions(int n)    {        int res = 0;        for (int x = 0; x * x < n; x++)            for (int y = 0; x * x + y * y < n; y++)                res++;                         return res;    }     // Driver program    public static void main(String args[])    {        System.out.println ( "Total Number of distinct Non-Negative pairs is "                                                        +countSolutions(6));             }} // This article is contributed by vt_m.
# Python3 implementation of above approach # This function counts number of pairs# (x, y) that satisfy# the inequality x*x + y*y < n.def countSolutions(n):     res = 0    x = 0    while(x * x < n):        y = 0        while(x * x + y * y < n):            res = res + 1            y = y + 1        x = x + 1     return res # Driver program to test above functionif __name__=='__main__':    print("Total Number of distinct Non-Negative pairs is ",         countSolutions(6)) # This code is contributed by# Sanjit_Prasad
// C# code to Count Distinct// Non-Negative Integer Pairs// (x, y) that Satisfy the// inequality x*x + y*y < nusing System; class GFG {         // This function counts number    // of pairs (x, y) that satisfy    // the inequality x*x + y*y < n.    static int countSolutions(int n)    {        int res = 0;                 for (int x = 0; x*x < n; x++)            for (int y = 0; x*x + y*y < n; y++)                res++;                         return res;    }     // Driver program    public static void Main()    {        Console.WriteLine( "Total Number of "          + "distinct Non-Negative pairs is "                        + countSolutions(6));    }} // This code is contributed by Sam007.
<?php// PHP program Count Distinct// Non-Negative Integer Pairs// (x, y) that Satisfy the// Inequality x*x + y*y < n // function counts number of// pairs (x, y) that satisfy// the inequality x*x + y*y < n.function countSolutions($n){    $res = 0;    for($x = 0; $x * $x < $n; $x++)        for($y = 0; $x * $x + $y * $y < $n; $y++)            $res++;    return $res;} // Driver Code{    echo "Total Number of distinct Non-Negative pairs is ";    echo countSolutions(6) ;    return 0;} // This code is contributed by nitin mittal.?>
<script> // This function counts number// of pairs (x, y) that satisfy// the inequality x*x + y*y < n. function countSolutions( n){   let res = 0;   for (let x = 0; x*x < n; x++){      for (let y = 0; x*x + y*y < n; y++){         res++;      }   }   return res;} // Driver program to test above functiondocument.write("Total Number of distinct Non-Negative pairs is "+countSolutions(6)); </script>
Output: 
Total Number of distinct Non-Negative pairs is 8
An upper bound for the time complexity of the above solution is O(n). The outer loop runs √n times. The inner loop runs at most √n times. 
Auxiliary Space: O(1) 
Using an Efficient Solution, we can find the count in O(√n) time. The idea is to first find the count of all y values corresponding to the 0 value of x. Let count of distinct y values be yCount. We can find yCount by running a loop and comparing yCount*yCount with n. After we have the initial yCount, we can one by one increase the value of x and find the next value of yCount by reducing yCount.  
C++
Java
Python3
C#
PHP
Javascript
// An efficient C program to find different (x, y) pairs that// satisfy x*x + y*y < n.#include <iostream>using namespace std; // This function counts number of pairs (x, y) that satisfy// the inequality x*x + y*y < n.int countSolutions(int n){   int x = 0, yCount, res = 0;    // Find the count of different y values for x = 0.   for (yCount = 0; yCount*yCount < n; yCount++) ;    // One by one increase value of x, and find yCount for   // current x.  If yCount becomes 0, then we have reached   // maximum possible value of x.   while (yCount != 0)   {       // Add yCount (count of different possible values of y       // for current x) to result       res  +=  yCount;        // Increment x       x++;        // Update yCount for current x. Keep reducing yCount while       // the inequality is not satisfied.       while (yCount != 0 && (x*x + (yCount-1)*(yCount-1) >= n))         yCount--;   }    return res;} // Driver program to test above functionint main(){    cout << "Total Number of distinct Non-Negative pairs is "         << countSolutions(6) << endl;    return 0;}
// An efficient Java program to// find different (x, y) pairs// that satisfy x*x + y*y < n.import java.io.*; class GFG{    // This function counts number    //of pairs (x, y) that satisfy    // the inequality x*x + y*y < n.    static int countSolutions(int n)    {        int x = 0, yCount, res = 0;                 // Find the count of different        // y values for x = 0.        for (yCount = 0; yCount * yCount < n; yCount++) ;                 // One by one increase value of x,        // and find yCount forcurrent x. If        // yCount becomes 0, then we have reached        // maximum possible value of x.        while (yCount != 0)        {            // Add yCount (count of different possible            // values of y for current x) to result            res += yCount;                         // Increment x            x++;                         // Update yCount for current x. Keep reducing            // yCount while the inequality is not satisfied.            while (yCount != 0 && (x * x + (yCount - 1)                                  * (yCount - 1) >= n))            yCount--;                     }        return res;             }         // Driver program    public static void main(String args[])    {        System.out.println ( "Total Number of distinct Non-Negative pairs is "                             +countSolutions(6)) ;             }} // This article is contributed by vt_m.
# An efficient python program to# find different (x, y) pairs# that satisfy x*x + y*y < n. # This function counts number of# pairs (x, y) that satisfy the# inequality x*x + y*y < n.def countSolutions(n):         x = 0    res = 0    yCount = 0     # Find the count of different    # y values for x = 0.    while(yCount * yCount < n):        yCount = yCount + 1             # One by one increase value of    # x, and find yCount for current    # x. If yCount becomes 0, then    # we have reached maximum    # possible value of x.    while (yCount != 0):        # Add yCount (count of        # different possible values        # of y for current x) to        # result        res = res + yCount         # Increment x        x = x + 1         # Update yCount for current        # x. Keep reducing yCount        # while the inequality is        # not satisfied.        while (yCount != 0 and (x * x                     + (yCount - 1) *                     (yCount - 1) >= n)):            yCount = yCount - 1             return res # Driver program to test# above functionprint ("Total Number of distinct ",           "Non-Negative pairs is "               , countSolutions(6)) # This code is contributed by Sam007.
// An efficient C# program to// find different (x, y) pairs// that satisfy x*x + y*y < n.using System; class GFG {         // This function counts number    //of pairs (x, y) that satisfy    // the inequality x*x + y*y < n.    static int countSolutions(int n)    {        int x = 0, yCount, res = 0;                 // Find the count of different        // y values for x = 0.        for (yCount = 0; yCount * yCount < n;                                   yCount++) ;                 // One by one increase value of x,        // and find yCount forcurrent x. If        // yCount becomes 0, then we have        // reached maximum possible value        // of x.        while (yCount != 0)        {                         // Add yCount (count of different            // possible values of y for            // current x) to result            res += yCount;                         // Increment x            x++;                         // Update yCount for current x.            // Keep reducing yCount while the            // inequality is not satisfied.            while (yCount != 0 && (x * x +                            (yCount - 1) *                         (yCount - 1) >= n))            yCount--;        }                 return res;    }         // Driver program    public static void Main()    {        Console.WriteLine( "Total Number of "          + "distinct Non-Negative pairs is "                       + countSolutions(6)) ;    }} // This code is contributed by Sam007.
<?php// An efficient C program to find different// (x, y) pairs that satisfy x*x + y*y < n. // This function counts number of pairs// (x, y) that satisfy the inequality// x*x + y*y < n.function countSolutions( $n){    $x = 0; $yCount; $res = 0;         // Find the count of different y values    // for x = 0.    for ($yCount = 0; $yCount*$yCount < $n;                               $yCount++) ;         // One by one increase value of x, and    // find yCount for current x. If yCount    // becomes 0, then we have reached    // maximum possible value of x.    while ($yCount != 0)    {        // Add yCount (count of different        // possible values of y        // for current x) to result        $res += $yCount;             // Increment x        $x++;             // Update yCount for current x. Keep        // reducing yCount while the        // inequality is not satisfied.        while ($yCount != 0 and ($x * $x +           ($yCount-1) * ($yCount-1) >= $n))            $yCount--;    }         return $res;} // Driver program to test above functionecho "Total Number of distinct Non-Negative",        "pairs is ", countSolutions(6) ,"\n"; // This code is contributed by anuj_67.?>
<script>    // An efficient Javascript program to    // find different (x, y) pairs    // that satisfy x*x + y*y < n.         // This function counts number    //of pairs (x, y) that satisfy    // the inequality x*x + y*y < n.    function countSolutions(n)    {        let x = 0, yCount, res = 0;                  // Find the count of different        // y values for x = 0.        for (yCount = 0; yCount * yCount < n;                                   yCount++) ;                  // One by one increase value of x,        // and find yCount forcurrent x. If        // yCount becomes 0, then we have        // reached maximum possible value        // of x.        while (yCount != 0)        {                          // Add yCount (count of different            // possible values of y for            // current x) to result            res += yCount;                          // Increment x            x++;                          // Update yCount for current x.            // Keep reducing yCount while the            // inequality is not satisfied.            while (yCount != 0 && (x * x +                            (yCount - 1) *                         (yCount - 1) >= n))            yCount--;        }                  return res;    }         document.write( "Total Number of "          + "distinct Non-Negative pairs is "                       + countSolutions(6)) ; </script>
Output: 
Total Number of distinct Non-Negative pairs is 8
Time Complexity of the above solution seems more but if we take a closer look, we can see that it is O(√n). In every step inside the inner loop, the value of yCount is decremented by 1. The value yCount can decrement at most O(√n) times as yCount is counted y values for x = 0. In the outer loop, the value of x is incremented. The value of x can also increment at most O(√n) times as the last x is for yCount equals 1. 
Auxiliary Space: O(1)This article is contributed by Sachin Gupta. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.  
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