> For the complete documentation index, see [llms.txt](https://hao-fu-1.gitbook.io/oj/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://hao-fu-1.gitbook.io/oj/array_and_numbers/sao-miao-xian/1674.-minimum-moves-to-make-array-complementary.md).

# 1674. Minimum Moves to Make Array Complementary

You are given an integer array `nums` of **even** length `n` and an integer `limit`. In one move, you can replace any integer from `nums` with another integer between `1` and `limit`, inclusive.

The array `nums` is **complementary** if for all indices `i` (**0-indexed**), `nums[i] + nums[n - 1 - i]` equals the same number. For example, the array `[1,2,3,4]` is complementary because for all indices `i`, `nums[i] + nums[n - 1 - i] = 5`.

Return the ***minimum** number of moves required to make* `nums` ***complementary***.

**Example 1:**

```
Input: nums = [1,2,4,3], limit = 4
Output: 1
Explanation: In 1 move, you can change nums to [1,2,2,3] (underlined elements are changed).
nums[0] + nums[3] = 1 + 3 = 4.
nums[1] + nums[2] = 2 + 2 = 4.
nums[2] + nums[1] = 2 + 2 = 4.
nums[3] + nums[0] = 3 + 1 = 4.
Therefore, nums[i] + nums[n-1-i] = 4 for every i, so nums is complementary.
```

**Example 2:**

```
Input: nums = [1,2,2,1], limit = 2
Output: 2
Explanation: In 2 moves, you can change nums to [2,2,2,2]. You cannot change any number to 3 since 3 > limit.
```

**Example 3:**

```
Input: nums = [1,2,1,2], limit = 2
Output: 0
Explanation: nums is already complementary.
```

**Constraints:**

* `n == nums.length`
* `2 <= n <= 105`
* `1 <= nums[i] <= limit <= 105`
* `n` is even.

数组nums如果对所有元素都满足nums\[i] + nums\[n - 1 - i]则称作互补，现能让每个数都变成\[1, limit]之中任意数，问让nums互补的最少变动次数。让T为互补时的可能值，a和b为nums\[i]和nums\[n - 1 - i]，分5种情况：

1. 2 <= T < min(a, b) + 1, 需要a和b都变小，两次变动；
2. min(a, b) + 1 <= T < a + b 需要max(a, b)变小，一次变动
3. a + b = T 无需变动
4. a + b < T <= max(a, b) + limit 需要min(a, b)变大，一次变动
5. max(a, b) + limit < T <= 2 \* limit 需要a和b都变大，两次变大

根据以上观察遍历所有ab对，并更新delta，delta为T从i-1到i时变动数目的变化量。再用扫描线遍历T的所有可能\[2, 2 \* limit]，cur\_i += delta\[i]为T = i时所需变动数目。min(cur\[i])为所求。

参考了[这](https://link.zhihu.com/?target=https%3A//leetcode.com/problems/minimum-moves-to-make-array-complementary/discuss/952773/PythonJava-simple-O%28max%28n-k%29)。

```python
class Solution:
    def minMoves(self, nums: List[int], limit: int) -> int:
        N, delta = len(nums), collections.Counter()
        for i in range(N // 2):
            a, b = nums[i], nums[N - 1 - i]
            # 2 <= T < min(A, B) + 1, we need 2 operations to make both A, B smaller
            delta[2] += 2
            # min(A, B) + 1 <= T < A + B, we need 1 operation to make the larger one out of A and B smaller
            delta[min(a, b) + 1] -= 1
            # T = A + B, we need 0 operation
            delta[a + b] -= 1
            # A + B < T < max(A, B) + limit, we need 1 operation to make the smaller one out of A and B larger
            delta[a + b + 1] += 1
            # max(A, B) + limit < T <= 2 * limit, we need 2 operation to make both A, B larger
            delta[max(a, b) + limit + 1] += 1
            cur, res = 0, math.inf
        for i in range(2, 2 * limit + 1):
            cur += delta[i]
            res = min(res, cur)
        return res
        
```


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://hao-fu-1.gitbook.io/oj/array_and_numbers/sao-miao-xian/1674.-minimum-moves-to-make-array-complementary.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.