The problem “Find N Unique Integers Sum up to Zero” from Leetcode offers an excellent opportunity to explore Python language features, algorithmic thinking, and mathematical reasoning. The problem serves as a great exercise for beginners and offers a variety of solutions, each with its own advantages and disadvantages. This comprehensive guide will dissect the problem and present multiple approaches for solving it.

## Table of Contents

- Problem Description
- Initial Thoughts and Observations
- Basic Approach
- Mathematical Approach
- Optimized One-liner
- Testing Scenarios
- Complexity Analysis
- Conclusion

## 1. Problem Description

You are tasked with generating an array of `N`

unique integers that sum up to zero. There are multiple valid answers, and you may return any of them.

### Example

- Input:
`N = 5`

- Output:
`[-2, -1, 0, 1, 2]`

or`[-3, -1, 1, 2, 3]`

etc.

### Constraints

`1 <= N <= 1000`

## 2. Initial Thoughts and Observations

- The problem is asking for unique integers.
- The sum of all the integers in the array should be zero.
- When
`N`

is odd, zero should be part of the array. - When
`N`

is even, the numbers can be evenly distributed across zero.

## 3. Basic Approach

One of the simplest ways to tackle this problem is to start appending integers from 1 to `N-1`

and then add their negative sum as the last element.

### Algorithm

- Initialize an empty list
`result`

. - Loop from 1 to
`N-1`

and append each number to`result`

. - Append the negative sum of
`result`

to`result`

. - Return
`result`

.

### Python Code

```
def sumZero(N):
result = []
for i in range(1, N):
result.append(i)
result.append(-sum(result))
return result
```

## 4. Mathematical Approach

Since we know that the sum of integers from 1 to `N-1`

is `(N-1) * N / 2`

, we can directly append the negative of this sum to our list without having to calculate the sum dynamically.

### Algorithm

- Initialize an empty list
`result`

. - Append numbers from 1 to
`N-1`

to`result`

. - Append
`-((N - 1) * N // 2)`

to`result`

. - Return
`result`

.

### Python Code

```
def sumZero(N):
result = list(range(1, N))
result.append(-((N - 1) * N // 2))
return result
```

## 5. Optimized One-liner

Using Python’s list comprehensions and the built-in `sum`

function, we can solve this problem in a single line.

### Python Code

```
def sumZero(N):
return list(range(1, N)) + [-sum(range(1, N))]
```

## 6. Testing Scenarios

- Test with
`N = 1`

. Expected output:`[0]`

. - Test with
`N = 2`

. Expected output:`[1, -1]`

. - Test with
`N = 3`

. Expected output:`[0, 1, -1]`

or similar. - Test with a large
`N`

to see if the algorithm holds up.

## 7. Complexity Analysis

**Basic Approach**:- Time Complexity: O(N)
- Space Complexity: O(N)

**Mathematical Approach**:- Time Complexity: O(N)
- Space Complexity: O(N)

**Optimized One-liner**:- Time Complexity: O(N)
- Space Complexity: O(N)

All three algorithms exhibit linear time and space complexity, which makes them scalable for large inputs as allowed by the problem’s constraints.

## 8. Conclusion

The “Find N Unique Integers Sum up to Zero” problem on Leetcode is a good introductory problem for practicing algorithmic thinking and Python programming. It allows us to explore different algorithms with varying degrees of complexity and optimization, thus offering a comprehensive learning experience. While the problem might appear simple, the multiple approaches to solving it help in understanding the subtleties involved in algorithm design and Python syntax.