Check if a subsequence of length K with odd sum exists

Given an array arr[] of integers, the task is to check if it is possible to obtain a subsequence of K elements from the array such that their sum is odd. If it is possible, print Yes. Otherwise, print No.

Examples:

Input: arr[] = { 2, 5, 6, 7, 4 }, K = 3
Output: Yes
Explanation:
Subsequences {2, 5, 6}, {2, 6, 7} and {2, 7, 4} have odd sum

Input: arr[] = { 1, 5, 7, 11 }, K = 4
Output: No
Explanation:
Only subsequence of length 4 is {1, 5, 7, 11} which has even sum (24). Hence no such subsequence exists.

Naive Approach:
The simplest method to solve the problem is to generate all subsequences of length K and check if any of these subsequences have an odd sum. The time complexity for such an approach will be exponential and thus inefficient.



Efficient Approach:
The efficient method to approach the above problem will be to count the number of odd elements in the array and then, simply checking for all the edge cases when it will not be possible to find a subsequence with odd sum.

The edge cases to be considered when such a subsequence cannot be generated are as follows:

  • If there are no odd elements in the array, any subsequence will contain only even elements and an even sum will be obtained. So, it is not possible to generate a subsequence with odd sum.
  • If K is even and there are no even elements present in the array, a subsequence with odd sum is not possible.

For all other cases, it will be possible to generate a subsequence with an odd sum.

Below is the implementation of above approach:

C++

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// C++ program to check if a
// subsequence of length K
// with odd sum exists in the
// given array
#include <bits/stdc++.h>
using namespace std;
  
// Function to check if any required
// subsequence exists or not
bool isSubseqPossible(int arr[], int N, int K)
{
    int i;
    // Store count of odd and
    // even elements in the array
    int odd = 0, even = 0;
  
    // Calculate the count of
    // odd and even elements
    for (i = 0; i < N; i++) {
        if (arr[i] % 2 == 1)
            odd++;
        else
            even++;
    }
  
    // If no odd elements exists
    // or no even elements exists
    // when K is even
    if (odd == 0 
|| (even == 0 && K % 2 == 0))
  
        // Subsequence is not possible
        return false;
  
    // Possible otherwise
    return true;
}
  
// Driver Code
int main()
{
    int arr[] = { 2, 3, 5, 7, 4 };
    int N = sizeof(arr) / sizeof(arr[0]);
    int K = 3;
    cout << (isSubseqPossible(arr, N, K)
                 ? "Yes"
                 : "No");
    return 0;
}

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Java

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// Java program to check if a
// subsequence of length K
// with odd sum exists in the
// given array
class GFG{
  
// Function to check if any required
// subsequence exists or not
static boolean isSubseqPossible(int []arr, 
                             int N, int K)
{
    int i;
      
    // Store count of odd and
    // even elements in the array
    int odd = 0, even = 0;
  
    // Calculate the count of
    // odd and even elements
    for (i = 0; i < N; i++) 
    {
        if (arr[i] % 2 == 1)
            odd++;
        else
            even++;
    }
  
    // If no odd elements exists
    // or no even elements exists
    // when K is even
    if (odd == 0 || (even == 0 && K % 2 == 0))
  
        // Subsequence is not possible
        return false;
  
    // Possible otherwise
    return true;
}
  
// Driver Code
public static void main(String args[])
{
    int []arr = { 2, 3, 5, 7, 4 };
    int N = arr.length;
    int K = 3;
    System.out.print(isSubseqPossible(arr, N, K) ? 
                                           "Yes" : "No");
}
}
  
// This code is contributed by Code_Mech

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Python3

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# Python3 program to check if a subsequence
# of length K with odd sum exists in the
# given array
  
# Function to check if any required
# subsequence exists or not
def isSubseqPossible(arr, N, K):
  
    i = 0
      
    # Store count of odd and
    # even elements in the array
    odd = 0
    even = 0
  
    # Calculate the count of
    # odd and even elements
    for i in range(N):
        if (arr[i] % 2 == 1):
            odd += 1
        else:
            even += 1
  
    # If no odd element exists or no
    # even element exists when K even
    if (odd == 0 or (even == 0 and K % 2 == 0)):
  
        # Subsequence is not possible
        return False
  
    # Otherwise possible
    return True
  
# Driver code
if __name__ == '__main__':
  
    arr = [ 2, 3, 5, 7, 4 ]
    N = len(arr)
    K = 3
  
    print("Yes" if isSubseqPossible(arr, N, K) else "No")
  
# This code is contributed by himanshu77

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C#

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// C# program to check if a
// subsequence of length K
// with odd sum exists in the
// given array
using System;
class GFG{
  
// Function to check if any required
// subsequence exists or not
static bool isSubseqPossible(int []arr, 
                             int N, int K)
{
    int i;
      
    // Store count of odd and
    // even elements in the array
    int odd = 0, even = 0;
  
    // Calculate the count of
    // odd and even elements
    for (i = 0; i < N; i++) 
    {
        if (arr[i] % 2 == 1)
            odd++;
        else
            even++;
    }
  
    // If no odd elements exists
    // or no even elements exists
    // when K is even
    if (odd == 0 || (even == 0 && K % 2 == 0))
  
        // Subsequence is not possible
        return false;
  
    // Possible otherwise
    return true;
}
  
// Driver Code
public static void Main()
{
    int []arr = { 2, 3, 5, 7, 4 };
    int N = arr.Length;
    int K = 3;
    Console.Write(isSubseqPossible(arr, N, K) ? 
                                        "Yes" : "No");
}
}
  
// This code is contributed by Code_Mech

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Output:

Yes

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

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