LeetCode #3224 — MEDIUM

Minimum Array Changes to Make Differences Equal

Move from brute-force thinking to an efficient approach using array strategy.

The Problem

Problem Statement

You are given an integer array nums of size n where n is even, and an integer k.

You can perform some changes on the array, where in one change you can replace any element in the array with any integer in the range from 0 to k.

You need to perform some changes (possibly none) such that the final array satisfies the following condition:

There exists an integer X such that abs(a[i] - a[n - i - 1]) = X for all (0 <= i < n).

Return the minimum number of changes required to satisfy the above condition.

Example 1:

Input: nums = [1,0,1,2,4,3], k = 4

Output: 2

Explanation:
We can perform the following changes:

Replace nums[1] by 2. The resulting array is nums = [1,2,1,2,4,3].
Replace nums[3] by 3. The resulting array is nums = [1,2,1,3,4,3].

The integer X will be 2.

Example 2:

Input: nums = [0,1,2,3,3,6,5,4], k = 6

Output: 2

Explanation:
We can perform the following operations:

Replace nums[3] by 0. The resulting array is nums = [0,1,2,0,3,6,5,4].
Replace nums[4] by 4. The resulting array is nums = [0,1,2,0,4,6,5,4].

The integer X will be 4.

Constraints:

2 <= n == nums.length <= 10⁵
n is even.
0 <= nums[i] <= k <= 10⁵

Step 01

Brute Force Baseline

Problem summary: You are given an integer array nums of size n where n is even, and an integer k. You can perform some changes on the array, where in one change you can replace any element in the array with any integer in the range from 0 to k. You need to perform some changes (possibly none) such that the final array satisfies the following condition: There exists an integer X such that abs(a[i] - a[n - i - 1]) = X for all (0 <= i < n). Return the minimum number of changes required to satisfy the above condition.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Hash Map

Example 1

[1,0,1,2,4,3]
4

Example 2

[0,1,2,3,3,6,5,4]
6

Step 02

Core Insight

What unlocks the optimal approach

There are at most <code>k + 1</code> possible values of the integer <code>X</code>.
How do we calculate the minimum number of changes efficiently if we fix the value of <code>X</code> before applying any changes?

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #3224: Minimum Array Changes to Make Differences Equal
class Solution {
    public int minChanges(int[] nums, int k) {
        int[] d = new int[k + 2];
        int n = nums.length;
        for (int i = 0; i < n / 2; ++i) {
            int x = Math.min(nums[i], nums[n - i - 1]);
            int y = Math.max(nums[i], nums[n - i - 1]);
            d[0] += 1;
            d[y - x] -= 1;
            d[y - x + 1] += 1;
            d[Math.max(y, k - x) + 1] -= 1;
            d[Math.max(y, k - x) + 1] += 2;
        }
        int ans = n, s = 0;
        for (int x : d) {
            s += x;
            ans = Math.min(ans, s);
        }
        return ans;
    }
}

// Accepted solution for LeetCode #3224: Minimum Array Changes to Make Differences Equal
func minChanges(nums []int, k int) int {
	d := make([]int, k+2)
	n := len(nums)
	for i := 0; i < n/2; i++ {
		x, y := nums[i], nums[n-1-i]
		if x > y {
			x, y = y, x
		}
		d[0] += 1
		d[y-x] -= 1
		d[y-x+1] += 1
		d[max(y, k-x)+1] -= 1
		d[max(y, k-x)+1] += 2
	}
	ans, s := n, 0
	for _, x := range d {
		s += x
		ans = min(ans, s)
	}
	return ans
}

# Accepted solution for LeetCode #3224: Minimum Array Changes to Make Differences Equal
class Solution:
    def minChanges(self, nums: List[int], k: int) -> int:
        d = [0] * (k + 2)
        n = len(nums)
        for i in range(n // 2):
            x, y = nums[i], nums[-i - 1]
            if x > y:
                x, y = y, x
            d[0] += 1
            d[y - x] -= 1
            d[y - x + 1] += 1
            d[max(y, k - x) + 1] -= 1
            d[max(y, k - x) + 1] += 2
        return min(accumulate(d))

// Accepted solution for LeetCode #3224: Minimum Array Changes to Make Differences Equal
impl Solution {
    pub fn min_changes(nums: Vec<i32>, k: i32) -> i32 {
        let n = nums.len();
        let mut d = vec![0; (k + 2) as usize];
        for i in 0..n / 2 {
            let x = nums[i].min(nums[n - i - 1]);
            let y = nums[i].max(nums[n - i - 1]);
            d[0] += 1;
            d[(y - x) as usize] -= 1;
            d[(y - x + 1) as usize] += 1;
            let idx = (y.max(k - x) + 1) as usize;
            d[idx] -= 1;
            d[idx] += 2;
        }
        let mut ans = n as i32;
        let mut s = 0;
        for x in d {
            s += x;
            ans = ans.min(s);
        }
        ans
    }
}

// Accepted solution for LeetCode #3224: Minimum Array Changes to Make Differences Equal
function minChanges(nums: number[], k: number): number {
    const d: number[] = Array(k + 2).fill(0);
    const n = nums.length;
    for (let i = 0; i < n >> 1; ++i) {
        const x = Math.min(nums[i], nums[n - 1 - i]);
        const y = Math.max(nums[i], nums[n - 1 - i]);
        d[0] += 1;
        d[y - x] -= 1;
        d[y - x + 1] += 1;
        d[Math.max(y, k - x) + 1] -= 1;
        d[Math.max(y, k - x) + 1] += 2;
    }
    let [ans, s] = [n, 0];
    for (const x of d) {
        s += x;
        ans = Math.min(ans, s);
    }
    return ans;
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n)

Space

O(n)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(1) space

Two nested loops check every pair or subarray. The outer loop fixes a starting point, the inner loop extends or searches. For n elements this gives up to n²/2 operations. No extra space, but the quadratic time is prohibitive for large inputs.

OPTIMIZED

O(n) time

O(1) space

Most array problems have an O(n²) brute force (nested loops) and an O(n) optimal (single pass with clever state tracking). The key is identifying what information to maintain as you scan: a running max, a prefix sum, a hash map of seen values, or two pointers.

Shortcut: If you are using nested loops on an array, there is almost always an O(n) solution. Look for the right auxiliary state.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Mutating counts without cleanup

Wrong move: Zero-count keys stay in map and break distinct/count constraints.

Usually fails on: Window/map size checks are consistently off by one.

Fix: Delete keys when count reaches zero.