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Divide array into two parts with equal sum according to the given constraints
  • Difficulty Level : Medium
  • Last Updated : 06 May, 2019

Given an array arr[] of N integers, the task is to select an integer x (which may or may not be present in the array) and remove all of its occurrences from the array and divide the remaining array into two non-empty sub-sets such that:

  1. The elements of the first set are strictly smaller than x.
  2. The elements of the second set are strictly greater than x.
  3. The sum of the elements of both the sets is equal.
      If such an integer exists then print Yes otherwise print No.

      Examples:

      Input: arr[] = {1, 2, 2, 5}
      Output: Yes
      Choose x = 3, after removing all of its occurrences the array becomes arr[] = {1, 2, 2, 5}
      {1, 2, 2} and {5} are the required sub-sets.

      Input: arr[] = {2, 1}
      Output: No



      Approach: The idea is to first sort the array and for all the numbers lying between 1 to maximum number present in the array, apply binary search and check if on removing all its occurrences from the array, sum of elements present on its left side (which are smaller than it) and sum of elements present on the right side (which are greater than it) is equal.

      Below is the implementation of the above approach:

      C++




      // C++ implementation of the approach
      #include <bits/stdc++.h>
      using namespace std;
        
      // Function that checks if the given
      // conditions are satisfied
      void IfExists(int arr[], int n)
      {
          // To store the prefix sum
          // of the array elements
          int sum[n];
        
          // Sort the array
          sort(arr, arr + n);
        
          sum[0] = arr[0];
        
          // Compute the prefix sum array
          for (int i = 1; i < n; i++)
              sum[i] = sum[i - 1] + arr[i];
        
          // Maximum element in the array
          int max = arr[n - 1];
        
          // Variable to check if there exists any number
          bool flag = false;
        
          for (int i = 1; i <= max; i++) {
        
              // Stores the index of the largest
              // number present in the array
              // smaller than i
              int findex = 0;
        
              // Stores the index of the smallest
              // number present in the array
              // greater than i
              int lindex = 0;
        
              int l = 0;
              int r = n - 1;
        
              // Find index of smallest number
              // greater than i
              while (l <= r) {
                  int m = (l + r) / 2;
        
                  if (arr[m] < i) {
                      findex = m;
                      l = m + 1;
                  }
                  else
                      r = m - 1;
              }
        
              l = 1;
              r = n;
              flag = false;
        
              // Find index of smallest number
              // greater than i
              while (l <= r) {
                  int m = (r + l) / 2;
        
                  if (arr[m] > i) {
                      lindex = m;
                      r = m - 1;
                  }
                  else
                      l = m + 1;
              }
        
              // If there exists a number
              if (sum[findex] == sum[n - 1] - sum[lindex - 1]) {
                  flag = true;
                  break;
              }
          }
        
          // If no such number exists
          // print no
          if (flag)
              cout << "Yes";
          else
              cout << "No";
      }
        
      // Driver code
      int main()
      {
          int arr[] = { 1, 2, 2, 5 };
          int n = sizeof(arr) / sizeof(int);
          IfExists(arr, n);
        
          return 0;
      }

      Java




      // Java implementation of the approach 
      import java.util.*;
        
      class GFG
      {
        
      // Function that checks if the given 
      // conditions are satisfied 
      static void IfExists(int arr[], int n) 
          // To store the prefix sum 
          // of the array elements 
          int sum[] = new int[n]; 
        
          // Sort the array 
          Arrays.sort(arr); 
        
          sum[0] = arr[0]; 
        
          // Compute the prefix sum array 
          for (int i = 1; i < n; i++) 
              sum[i] = sum[i - 1] + arr[i]; 
        
          // Maximum element in the array 
          int max = arr[n - 1]; 
        
          // Variable to check if there exists any number 
          boolean flag = false
        
          for (int i = 1; i <= max; i++) 
          
        
              // Stores the index of the largest 
              // number present in the array 
              // smaller than i 
              int findex = 0
        
              // Stores the index of the smallest 
              // number present in the array 
              // greater than i 
              int lindex = 0
        
              int l = 0
              int r = n - 1
        
              // Find index of smallest number 
              // greater than i 
              while (l <= r) 
              
                  int m = (l + r) / 2
        
                  if (arr[m] < i) 
                  
                      findex = m; 
                      l = m + 1
                  
                  else
                      r = m - 1
              
        
              l = 1
              r = n; 
              flag = false
        
              // Find index of smallest number 
              // greater than i 
              while (l <= r) 
              
                  int m = (r + l) / 2
        
                  if (arr[m] > i) 
                  
                      lindex = m; 
                      r = m - 1
                  
                  else
                      l = m + 1
              
        
              // If there exists a number 
              if (sum[findex] == sum[n - 1] - sum[lindex - 1]) 
              
                  flag = true
                  break
              
          
        
          // If no such number exists 
          // print no 
          if (flag) 
              System.out.println("Yes"); 
          else
              System.out.println("No"); 
        
      // Driver code 
      public static void main(String args[])
          int arr[] = { 1, 2, 2, 5 }; 
          int n = arr.length; 
          IfExists(arr, n); 
      }
        
      // This code is contributed by Arnab Kundu

      Python3




      # Python3 implementation of the approach 
        
      # Function that checks if the given 
      # conditions are satisfied 
      def IfExists(arr, n) :
        
          # To store the prefix sum 
          # of the array elements 
          sum = [0] * n; 
        
          # Sort the array 
          arr.sort(); 
        
          sum[0] = arr[0]; 
        
          # Compute the prefix sum array 
          for i in range(1, n) : 
              sum[i] = sum[i - 1] + arr[i]; 
        
          # Maximum element in the array 
          max = arr[n - 1]; 
        
          # Variable to check if there 
          # exists any number 
          flag = False
        
          for i in range(1, max + 1) :
        
              # Stores the index of the largest 
              # number present in the array 
              # smaller than i 
              findex = 0
        
              # Stores the index of the smallest 
              # number present in the array 
              # greater than i 
              lindex = 0
        
              l = 0
              r = n - 1
        
              # Find index of smallest number 
              # greater than i 
              while (l <= r) :
                  m = (l + r) // 2
        
                  if (arr[m] < i) :
                      findex = m; 
                      l = m + 1
                    
                  else :
                      r = m - 1
                
              l = 1
              r = n; 
              flag = False
        
              # Find index of smallest number 
              # greater than i 
              while (l <= r) : 
                  m = (r + l) // 2
        
                  if (arr[m] > i) : 
                      lindex = m; 
                      r = m - 1
                    
                  else :
                      l = m + 1
                
              # If there exists a number 
              if (sum[findex] == sum[n - 1] - 
                                 sum[lindex - 1]) : 
                  flag = True
                  break
                
          # If no such number exists 
          # print no 
          if (flag) : 
              print("Yes"); 
          else :
              print("No"); 
        
      # Driver code 
      if __name__ == "__main__"
        
          arr = [ 1, 2, 2, 5 ]; 
            
          n = len(arr) ;
          IfExists(arr, n); 
            
      # This code is contributed by Ryuga

      C#




      // C# implementation of the approach 
      using System;
        
      class GFG
      {
            
      // Function that checks if the given 
      // conditions are satisfied 
      static void IfExists(int[] arr, int n) 
          // To store the prefix sum 
          // of the array elements 
          int[] sum = new int[n]; 
        
          // Sort the array 
          Array.Sort(arr); 
        
          sum[0] = arr[0]; 
        
          // Compute the prefix sum array 
          for (int i = 1; i < n; i++) 
              sum[i] = sum[i - 1] + arr[i]; 
        
          // Maximum element in the array 
          int max = arr[n - 1]; 
        
          // Variable to check if there exists any number 
          bool flag = false
        
          for (int i = 1; i <= max; i++) 
          
        
              // Stores the index of the largest 
              // number present in the array 
              // smaller than i 
              int findex = 0; 
        
              // Stores the index of the smallest 
              // number present in the array 
              // greater than i 
              int lindex = 0; 
        
              int l = 0; 
              int r = n - 1; 
        
              // Find index of smallest number 
              // greater than i 
              while (l <= r) 
              
                  int m = (l + r) / 2; 
        
                  if (arr[m] < i) 
                  
                      findex = m; 
                      l = m + 1; 
                  
                  else
                      r = m - 1; 
              
        
              l = 1; 
              r = n; 
              flag = false
        
              // Find index of smallest number 
              // greater than i 
              while (l <= r) 
              
                  int m = (r + l) / 2; 
        
                  if (arr[m] > i) 
                  
                      lindex = m; 
                      r = m - 1; 
                  
                  else
                      l = m + 1; 
              
        
              // If there exists a number 
              if (sum[findex] == sum[n - 1] - sum[lindex - 1]) 
              
                  flag = true
                  break
              
          
        
          // If no such number exists 
          // print no 
          if (flag) 
              Console.WriteLine("Yes"); 
          else
              Console.WriteLine("No"); 
        
      // Driver code 
      public static void Main()
          int[] arr = { 1, 2, 2, 5 }; 
          int n = arr.Length; 
          IfExists(arr, n); 
      }
        
      // This code is contributed by Code_Mech.

      PHP




      <?php
      // PHP implementation of the approach
        
      // Function that checks if the given
      // conditions are satisfied
      function IfExists($arr, $n)
      {
          // To store the prefix $sum
          // of the array elements
          $sum = array_fill(0, $n, 0);
        
          // Sort the array
          sort($arr);
        
          $sum[0] = $arr[0];
        
          // Compute the prefix sum array
          for ($i = 1; $i < $n; $i++)
              $sum[$i] = $sum[$i - 1] + $arr[$i];
        
          // Maximum element in the array
          $max = $arr[$n - 1];
        
          // Variable to check if there exists any number
          $flag = false;
        
          for ($i = 1; $i <= $max; $i++) 
          {
        
              // Stores the index of the largest
              // number present in the array
              // smaller than i
              $findex = 0;
        
              // Stores the index of the smallest
              // number present in the array
              // greater than i
              $lindex = 0;
        
              $l = 0;
              $r = $n - 1;
        
              // Find index of smallest number
              // greater than i
              while ($l <= $r
              {
                  $m = ($l + $r) / 2; 
                  if ($arr[$m] < $i
                  {
                      $findex = $m;
                      $l = $m + 1;
                  }
                  else
                      $r = $m - 1;
              }
        
              $l = 1;
              $r = $n;
              $flag = false;
        
              // Find index of smallest number
              // greater than i
              while ($l <= $r)
              {
                  $m = ($r + $l) / 2;
        
                  if ($arr[$m] > $i
                  {
                      $lindex = $m;
                      $r = $m - 1;
                  }
                  else
                      $l = $m + 1;
              }
        
              // If there exists a number
              if ($sum[$findex] == $sum[$n - 1] -
                                   $sum[$lindex - 1])
              {
                  $flag = true;
                  break;
              }
          }
        
          // If no such number exists
          // print no
          if ($flag == true)
              echo "Yes";
          else
              echo "No";
      }
        
      // Driver code
      $arr = array(1, 2, 2, 5 );
      $n = sizeof($arr);
      IfExists($arr, $n);
        
      // This code is contributed by ihritik
      ?>
      Output:
      Yes
      

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