# Leetcode – Find N Unique Integers Sum up to Zero Solution

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.

1. Problem Description
2. Initial Thoughts and Observations
3. Basic Approach
4. Mathematical Approach
5. Optimized One-liner
6. Testing Scenarios
7. Complexity Analysis
8. 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

1. Initialize an empty list result.
2. Loop from 1 to N-1 and append each number to result.
3. Append the negative sum of result to result.
4. 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

1. Initialize an empty list result.
2. Append numbers from 1 to N-1 to result.
3. Append -((N - 1) * N // 2) to result.
4. 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

1. Test with N = 1. Expected output: [0].
2. Test with N = 2. Expected output: [1, -1].
3. Test with N = 3. Expected output: [0, 1, -1] or similar.
4. Test with a large N to see if the algorithm holds up.

## 7. Complexity Analysis

1. Basic Approach:
• Time Complexity: O(N)
• Space Complexity: O(N)
2. Mathematical Approach:
• Time Complexity: O(N)
• Space Complexity: O(N)
3. 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.