leetcode

Find N Unique Integers Sum up to Zero - Link

Question Description

Given an integer n, return any array containing n unique integers such that they add up to 0.

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Constraints

• 1 <= n <= 1000

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Approaches

Approach 1: Flag-Based Alternating Approach

Use a flag to alternate between negative and positive numbers while building the array.

Algorithm:

1. Handle edge case: if n == 1, return empty array (no unique integers sum to 0)
2. Create array of size n
3. If n is odd, start from index 1 (reserve index 0 for 0)
4. Use a flag to alternate between negative and positive numbers
5. Start with num = 1 and increment after placing positive numbers
6. Place 0 at the beginning if n is odd

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Dry Run (Approach 1)

Example Input: n = 5

Step-by-step execution:

• n = 5 (odd), so index = 1, arr = [0, 0, 0, 0, 0]
• flag = true, num = 1
• index = 1: flag=true, place -1, index=2, flag=false
• index = 2: flag=false, place 1, index=3, flag=true, num=2
• index = 3: flag=true, place -2, index=4, flag=false
• index = 4: flag=false, place 2, index=5, flag=true, num=3
• index = 5: reached end

Result: [0, -1, 1, -2, 2] (sum = 0)

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Solution (Approach 1)

class Solution {
    public int[] sumZero(int n) {
        if (n == 1) return new int[n];

        int[] arr = new int[n];
        int index = 0;
        if (n % 2 == 1) {
            index = 1; // Leave space for 0 if n is odd
        }

        boolean flag = true;
        int num = 1;

        while (index < n) {
            if (flag) {
                arr[index++] = -num;
                flag = false;
            } else {
                arr[index++] = num;
                flag = true;
                num++;
            }
        }

        return arr;
    }
}

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Time and Space Complexity (Approach 1)

• Time Complexity: O(n) - single pass to fill the array
• Space Complexity: O(n) - for the output array

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Approach 2: Pair-Based Approach (Cleaner)

Create pairs of numbers that sum to zero: (-1, 1), (-2, 2), etc.

Algorithm:

1. Calculate number of pairs: pairs = n / 2
2. Create array of size n
3. Use a loop to place pairs: (-i, i) for i from 1 to pairs
4. If n is odd, place 0 at the end
5. This ensures sum is always 0

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Dry Run (Approach 2)

Example Input: n = 5

Step-by-step execution:

• pairs = 5 / 2 = 2
• index = 0
• i=1: arr[0] = -1, arr[1] = 1, index = 2
• i=2: arr[2] = -2, arr[3] = 2, index = 4
• n is odd, so arr[4] = 0

Result: [-1, 1, -2, 2, 0] (sum = 0)

Example Input: n = 4

Step-by-step execution:

• pairs = 4 / 2 = 2
• index = 0
• i=1: arr[0] = -1, arr[1] = 1, index = 2
• i=2: arr[2] = -2, arr[3] = 2, index = 4
• n is even, no 0 added

Result: [-1, 1, -2, 2] (sum = 0)

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Solution (Approach 2)

class Solution {
    public int[] sumZero(int n) {
        int[] arr = new int[n];
        int pairs = n / 2;
        int index = 0;

        for (int i = 1; i <= pairs; i++) {
            arr[index++] = -i;
            arr[index++] = i;
        }

        if (n % 2 == 1) {
            arr[index] = 0;
        }

        return arr;
    }
}

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Time and Space Complexity (Approach 2)

• Time Complexity: O(n) - single loop for n/2 iterations
• Space Complexity: O(n) - for the output array

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Comparison of Approaches

Approach	Time Complexity	Space Complexity	Readability	Edge Case Handling
Approach 1	O(n)	O(n)	Medium	Good
Approach 2	O(n)	O(n)	High	Excellent

Key Insight: Both approaches generate the same mathematical pattern: pairs of numbers that cancel each other out, with 0 added for odd lengths.

Why This Works:

• For even n: n/2 pairs of (i, -i) sum to 0
• For odd n: n/2 pairs of (i, -i) + 0 = 0
• All numbers are unique by construction

Alternative Mathematical Approaches:

1. Arithmetic Sequence: [0, 1, 2, ..., n-1] transformed to sum to 0
2. Random Selection: Any n-1 numbers and their negative sum

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