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  1. Could you confirm that the input number ( N ) is always a positive integer?
  2. Is there any constraint on the maximum value of ( N )?

This information will help me determine any edge cases or constraints necessary for the solution.

Strategy

To solve the problem of finding ( N ) unique integers that sum up to zero, we can exploit the properties of arithmetic sequences and symmetry. Here’s the approach:

  1. Observation:
    • For the sum of ( N ) integers to be zero, an easy way is to use negative and positive pairs of numbers, along with zero if ( N ) is odd.
    • For example, for ( N = 5 ), we can use ([-2, -1, 0, 1, 2]).
  2. Approach:
    • If ( N ) is even, simply generate pairs ([-N/2, -N/2 + 1, …, -1, 1, …, N/2 - 1, N/2]).
    • If ( N ) is odd, do the same, and add zero into the list.

Code

Here’s the code to implement this strategy:

def sumZero(N):
    result = []
    
    # If N is even or odd, the strategy works with the loop below
    for i in range(1, N // 2 + 1):
        result.append(i)
        result.append(-i)
    
    # If N is odd, add a zero
    if N % 2 != 0:
        result.append(0)
    
    return result

Time Complexity

The time complexity of this solution is ( O(N) ) because we are creating the list of ( N ) numbers and each insertion operation is ( O(1) ).

Let’s run a few examples to see how the function performs:

These examples should validate the correctness of the function.

Let me know if you need anything else!

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