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1674. Minimum Moves to Make Array Complementary

Description

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.

Solutions

Solution 1: Difference Array

Assume that in the final array, the sum of the pair $\textit{nums}[i]$ and $\textit{nums}[n-i-1]$ is $s$.

Let's denote $x$ as the smaller value between $\textit{nums}[i]$ and $\textit{nums}[n-i-1]$, and $y$ as the larger value.

For each pair of numbers, we have the following scenarios:

  • If no replacement is needed, then $x + y = s$.
  • If one replacement is made, then $x + 1 \le s \le y + \textit{limit}$.
  • If two replacements are made, then $2 \le s \le x$ or $y + \textit{limit} + 1 \le s \le 2 \times \textit{limit}$.

That is:

  • In the range $[2,..x]$, $2$ replacements are needed.
  • In the range $[x+1,..x+y-1]$, $1$ replacement is needed.
  • At $[x+y]$, no replacement is needed.
  • In the range $[x+y+1,..y + \textit{limit}]$, $1$ replacement is needed.
  • In the range $[y + \textit{limit} + 1,..2 \times \textit{limit}]$, $2$ replacements are needed.

We enumerate each pair of numbers and use a difference array to update the number of replacements needed in different ranges for each pair.

Finally, we find the minimum value among the prefix sums from index $2$ to $2 \times \textit{limit}$, which is the minimum number of replacements needed.

The time complexity is $O(n)$, and the space complexity is $O(n)$. Here, $n$ is the length of the array $\textit{nums}$.

Similar problems:

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class Solution:
    def minMoves(self, nums: List[int], limit: int) -> int:
        d = [0] * (2 * limit + 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[2] += 2
            d[x + 1] -= 2
            d[x + 1] += 1
            d[x + y] -= 1
            d[x + y + 1] += 1
            d[y + limit + 1] -= 1
            d[y + limit + 1] += 2
        return min(accumulate(d[2:]))

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class Solution {
    public int minMoves(int[] nums, int limit) {
        int[] d = new int[2 * limit + 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[2] += 2;
            d[x + 1] -= 2;
            d[x + 1] += 1;
            d[x + y] -= 1;
            d[x + y + 1] += 1;
            d[y + limit + 1] -= 1;
            d[y + limit + 1] += 2;
        }
        int ans = n;
        for (int i = 2, s = 0; i < d.length; ++i) {
            s += d[i];
            ans = Math.min(ans, s);
        }
        return ans;
    }
}

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class Solution {
public:
    int minMoves(vector<int>& nums, int limit) {
        int n = nums.size();
        int d[limit * 2 + 2];
        memset(d, 0, sizeof(d));
        for (int i = 0; i < n / 2; ++i) {
            int x = nums[i], y = nums[n - i - 1];
            if (x > y) {
                swap(x, y);
            }
            d[2] += 2;
            d[x + 1] -= 2;
            d[x + 1] += 1;
            d[x + y] -= 1;
            d[x + y + 1] += 1;
            d[y + limit + 1] -= 1;
            d[y + limit + 1] += 2;
        }
        int ans = n;
        for (int i = 2, s = 0; i <= limit * 2; ++i) {
            s += d[i];
            ans = min(ans, s);
        }
        return ans;
    }
};

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func minMoves(nums []int, limit int) int {
    n := len(nums)
    d := make([]int, 2*limit+2)
    for i := 0; i < n/2; i++ {
        x, y := nums[i], nums[n-1-i]
        if x > y {
            x, y = y, x
        }
        d[2] += 2
        d[x+1] -= 2
        d[x+1] += 1
        d[x+y] -= 1
        d[x+y+1] += 1
        d[y+limit+1] -= 1
        d[y+limit+1] += 2
    }
    ans, s := n, 0
    for _, x := range d[2:] {
        s += x
        ans = min(ans, s)
    }
    return ans
}

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function minMoves(nums: number[], limit: number): number {
    const n = nums.length;
    const d: number[] = Array(limit * 2 + 2).fill(0);
    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[2] += 2;
        d[x + 1] -= 2;
        d[x + 1] += 1;
        d[x + y] -= 1;
        d[x + y + 1] += 1;
        d[y + limit + 1] -= 1;
        d[y + limit + 1] += 2;
    }
    let ans = n;
    let s = 0;
    for (let i = 2; i < d.length; ++i) {
        s += d[i];
        ans = Math.min(ans, s);
    }
    return ans;
}

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