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1671. Minimum Number of Removals to Make Mountain Array

Description

You may recall that an array arr is a mountain array if and only if:

  • arr.length >= 3
  • There exists some index i (0-indexed) with 0 < i < arr.length - 1 such that:
    • arr[0] < arr[1] < ... < arr[i - 1] < arr[i]
    • arr[i] > arr[i + 1] > ... > arr[arr.length - 1]

Given an integer array nums​​​, return the minimum number of elements to remove to make nums​​​ a mountain array.

 

Example 1:

Input: nums = [1,3,1]
Output: 0
Explanation: The array itself is a mountain array so we do not need to remove any elements.

Example 2:

Input: nums = [2,1,1,5,6,2,3,1]
Output: 3
Explanation: One solution is to remove the elements at indices 0, 1, and 5, making the array nums = [1,5,6,3,1].

 

Constraints:

  • 3 <= nums.length <= 1000
  • 1 <= nums[i] <= 109
  • It is guaranteed that you can make a mountain array out of nums.

Solutions

Solution 1: Dynamic Programming

This problem can be transformed into finding the longest increasing subsequence and the longest decreasing subsequence.

We define $left[i]$ as the length of the longest increasing subsequence ending with $nums[i]$, and define $right[i]$ as the length of the longest decreasing subsequence starting with $nums[i]$.

Then the final answer is $n - \max(left[i] + right[i] - 1)$, where $1 \leq i \leq n$, and $left[i] \gt 1$ and $right[i] \gt 1$.

The time complexity is $O(n^2)$, and the space complexity is $O(n)$. Here, $n$ is the length of the array $nums$.

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class Solution:
    def minimumMountainRemovals(self, nums: List[int]) -> int:
        n = len(nums)
        left = [1] * n
        right = [1] * n
        for i in range(1, n):
            for j in range(i):
                if nums[i] > nums[j]:
                    left[i] = max(left[i], left[j] + 1)
        for i in range(n - 2, -1, -1):
            for j in range(i + 1, n):
                if nums[i] > nums[j]:
                    right[i] = max(right[i], right[j] + 1)
        return n - max(a + b - 1 for a, b in zip(left, right) if a > 1 and b > 1)

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class Solution {
    public int minimumMountainRemovals(int[] nums) {
        int n = nums.length;
        int[] left = new int[n];
        int[] right = new int[n];
        Arrays.fill(left, 1);
        Arrays.fill(right, 1);
        for (int i = 1; i < n; ++i) {
            for (int j = 0; j < i; ++j) {
                if (nums[i] > nums[j]) {
                    left[i] = Math.max(left[i], left[j] + 1);
                }
            }
        }
        for (int i = n - 2; i >= 0; --i) {
            for (int j = i + 1; j < n; ++j) {
                if (nums[i] > nums[j]) {
                    right[i] = Math.max(right[i], right[j] + 1);
                }
            }
        }
        int ans = 0;
        for (int i = 0; i < n; ++i) {
            if (left[i] > 1 && right[i] > 1) {
                ans = Math.max(ans, left[i] + right[i] - 1);
            }
        }
        return n - ans;
    }
}

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class Solution {
public:
    int minimumMountainRemovals(vector<int>& nums) {
        int n = nums.size();
        vector<int> left(n, 1), right(n, 1);
        for (int i = 1; i < n; ++