Check if every Subarray of even length has sum 0

Given an array A[ ] of size N, the task is to check if the sum of every even-sized subarray is 0 or not.

Examples:

Input: N = 4, A[] = {8, -8, 7, 9}
Output: NO
Explanation: Sum of subarray {7, 9} is not 0.

Input: N = 2, A[] = {0, 0}
Output: YES
Explanation: The only possible even length subarray is {0, 0} and its sum is 0.

Naive Approach: The basic way to solve the problem is as follows:

Generate all possible subarrays,then choose all even length subarray from that.Then, check if everytime its sum  is 0 or not. If anytime it's sum is not zero then return "NO" else return "YES".

Steps to implement-

• Run two loops to find all subarrays
• Simultaneously calculate the length of all subarray
• If any subarray's length is even then find its sum
• If anyone subarray of even length has non-zero sum then print "NO" else print "YES"

Code-

NO

Time Complexity: O(N³), because of two nested loops to find all subarray and a third loop to find the sum of all elements of the subarray
Auxiliary Space: O(1), because no extra space has been used

Efficient Approach: To solve the problem follow the below idea:

The idea is to check the complete array once for all possible subarrays of length 2 because all other even sized subarrays of length greater than 2 can be made by combining subarrays of length 2. So if all subarrays of length 2 have sum 0, all other even sized subarrays will also have sum 0.

Follow the steps mentioned below to implement the idea:

• Start iterating from i = 1 to N-1:
  • Check if the sum of A[i] and A[i-1] is 0 or not.
  • If it is not 0, return the answer as "NO" and no need to calculate further.
• If the iteration is over and the condition is satisfied for all the subarrays, return "YES" as the required answer.

Below is the implementation of the above approach.

NO

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