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Split first N natural numbers into two sets with minimum absolute difference of their sums

  • Difficulty Level : Medium
  • Last Updated : 26 Apr, 2021

Given an integer N, split the first N natural numbers into two sets such that the absolute difference between their sum is minimum. The task is to print the minimum absolute difference that can be obtained.

Examples:

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Input: N = 5
Output: 1
Explanation:
Split the first N (= 5) natural numbers into sets {1, 2, 5} (sum = 8) and {3, 4} (sum = 7).
Therefore, the required output is 1.

Input: N = 6
Output: 1



Naive Approach: This problem can be solved using the Greedy technique. Follow the steps below to solve the problem:

  • Initialize two variables, say sumSet1 and sumSet2 to store the sum of the elements from the two sets.
  • Traverse first N natural numbers from N to 1. For every number, check if the current sum of elements in set1 is less than or equal to the sum of elements in set2. If found to be true, add the currently traversed number into set1 and update sumSet1.
  • Otherwise, add the value of the current natural number to set2 and update sumSet2.
  • Finally, print abs(sumSet1 – sumSet2) as the required answer.

Below is the implementation of the above approach:

C++14




// C++ program to implement
// the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to split the first N
// natural numbers into two sets
// having minimum absolute
// difference of their sums
int minAbsDiff(int N)
{
    // Stores the sum of
    // elements of set1
    int sumSet1 = 0;
 
    // Stores the sum of
    // elements of set2
    int sumSet2 = 0;
 
    // Traverse first N
    // natural numbers
    for (int i = N; i > 0; i--) {
 
        // Check if sum of elements of
        // set1 is less than or equal
        // to sum of elements of set2
        if (sumSet1 <= sumSet2) {
            sumSet1 += i;
        }
        else {
            sumSet2 += i;
        }
    }
    return abs(sumSet1 - sumSet2);
}
 
// Driver Code
int main()
{
    int N = 6;
    cout << minAbsDiff(N);
}

Java




// Java program to implement
// the above approach
import java.io.*;
  
class GFG{
     
// Function to split the first N
// natural numbers into two sets
// having minimum absolute
// difference of their sums
static int minAbsDiff(int N)
{
     
    // Stores the sum of
    // elements of set1
    int sumSet1 = 0;
  
    // Stores the sum of
    // elements of set2
    int sumSet2 = 0;
  
    // Traverse first N
    // natural numbers
    for(int i = N; i > 0; i--)
    {
         
        // Check if sum of elements of
        // set1 is less than or equal
        // to sum of elements of set2
        if (sumSet1 <= sumSet2)
        {
            sumSet1 += i;
        }
        else
        {
            sumSet2 += i;
        }
    }
    return Math.abs(sumSet1 - sumSet2);
}
 
// Driver code
public static void main (String[] args)
{
    int N = 6;
     
    System.out.println(minAbsDiff(N));
}
}
 
// This code is contributed by offbeat

Python3




# Python3 program to implement
# the above approach
 
# Function to split the first N
# natural numbers into two sets
# having minimum absolute
# difference of their sums
def minAbsDiff(N):
     
    # Stores the sum of
    # elements of set1
    sumSet1 = 0
 
    # Stores the sum of
    # elements of set2
    sumSet2 = 0
 
    # Traverse first N
    # natural numbers
    for i in reversed(range(N + 1)):
         
        # Check if sum of elements of
        # set1 is less than or equal
        # to sum of elements of set2
        if sumSet1 <= sumSet2:
           sumSet1 = sumSet1 + i
        else:
           sumSet2 = sumSet2 + i
       
    return abs(sumSet1 - sumSet2)
 
# Driver Code
N = 6
 
print(minAbsDiff(N))
 
# This code is contributed by sallagondaavinashreddy7

C#




// C# program to implement
// the above approach
using System;
 
class GFG{
 
// Function to split the first N
// natural numbers into two sets
// having minimum absolute
// difference of their sums
static int minAbsDiff(int N)
{
     
    // Stores the sum of
    // elements of set1
    int sumSet1 = 0;
  
    // Stores the sum of
    // elements of set2
    int sumSet2 = 0;
  
    // Traverse first N
    // natural numbers
    for(int i = N; i > 0; i--)
    {
         
        // Check if sum of elements of
        // set1 is less than or equal
        // to sum of elements of set2
        if (sumSet1 <= sumSet2)
        {
            sumSet1 += i;
        }
        else
        {
            sumSet2 += i;
        }
    }
    return Math.Abs(sumSet1 - sumSet2);
}
 
// Driver code
static void Main()
{
    int N = 6;
     
    Console.Write(minAbsDiff(N));
}
}
 
// This code is contributed by divyeshrabadiya07

Javascript




<script>
 
// Javascript program to implement
// the above approach
 
// Function to split the first N
// natural numbers into two sets
// having minimum absolute
// difference of their sums
function minAbsDiff(N)
{
     
    // Stores the sum of
    // elements of set1
    var sumSet1 = 0;
 
    // Stores the sum of
    // elements of set2
    var sumSet2 = 0;
 
    // Traverse first N
    // natural numbers
    for(i = N; i > 0; i--)
    {
         
        // Check if sum of elements of
        // set1 is less than or equal
        // to sum of elements of set2
        if (sumSet1 <= sumSet2)
        {
            sumSet1 += i;
        }
        else
        {
            sumSet2 += i;
        }
    }
    return Math.abs(sumSet1 - sumSet2);
}
 
// Driver code
var N = 6;
 
document.write(minAbsDiff(N));
 
// This code is contributed by umadevi9616
 
</script>
Output
1

Time Complexity: O(N)
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach the idea is based on the following observations:

Splitting any 4 consecutive integers into 2 sets gives the minimum absolute difference of their sum equal to 0.

Mathematical proof:
Considering 4 consecutive integers {a1, a2, a3, a4}
a4 = a3 + 1
a1=a2  – 1
=> a4 + a1 = a3 + 1 + a2  – 1
=> a4 + a1 = a2 + a3

Follow the steps below to solve the problem:

  • If N % 4 == 0 or N % 4 == 3, then print 0.
  • Otherwise, print 1.

Below is the implementation of the above approach:

C++




// C++ program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to split the first N
// natural numbers into two sets
// having minimum absolute
// difference of their sums
int minAbsDiff(int N)
{
    if (N % 4 == 0 || N % 4 == 3) {
        return 0;
    }
    return 1;
}
 
// Driver Code
int main()
{
    int N = 6;
    cout << minAbsDiff(N);
}

Java




// Java program to implement
// the above approach
import java.io.*;
import java.util.*;
 
class GFG{
     
// Function to split the first N
// natural numbers into two sets
// having minimum absolute
// difference of their sums
static int minAbsDiff(int N)
{
    if (N % 4 == 0 || N % 4 == 3)
    {
        return 0;
    }
    return 1;
}
 
// Driver Code
public static void main (String[] args)
{
    int N = 6;
     
    System.out.println(minAbsDiff(N));
}
}
 
// This code is contributed by sallagondaavinashreddy7

Python3




# Python3 program to implement
# the above approach
 
# Function to split the first N
# natural numbers into two sets
# having minimum absolute
# difference of their sums
def minAbsDiff(N):
     
    if (N % 4 == 0 or N % 4 == 3):
        return 0
         
    return 1
 
# Driver Code
N = 6
 
print(minAbsDiff(N))
 
# This code is contributed by sallagondaavinashreddy7

C#




// C# program to implement
// the above approach
using System;
class GFG{
     
// Function to split the first N
// natural numbers into two sets
// having minimum absolute
// difference of their sums
static int minAbsDiff(int N)
{
  if (N % 4 == 0 ||
      N % 4 == 3)
  {
    return 0;
  }
  return 1;
}
 
// Driver Code
public static void Main(String[] args)
{
  int N = 6;
  Console.WriteLine(minAbsDiff(N));
}
}
 
// This code is contributed by 29AjayKumar

Javascript




<script>
// javascript program to implement
// the above approach   
// Function to split the first N
    // natural numbers into two sets
    // having minimum absolute
    // difference of their sums
    function minAbsDiff(N) {
        if (N % 4 == 0 || N % 4 == 3) {
            return 0;
        }
        return 1;
    }
 
    // Driver Code
     
        var N = 6;
 
        document.write(minAbsDiff(N));
 
// This code contributed by gauravrajput1
</script>
Output
1

Time Complexity: O(1)
Auxiliary Space: O(1)




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