Minimum Number of Removals to Make Mountain Array

1671. Minimum Number of Removals to Make Mountain Array

Difficulty: Hard

Topics: Array, Binary Search, Dynamic Programming, Greedy

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.

  Hint:

  1. Think the opposite direction instead of minimum elements to remove the maximum mountain subsequence
  2. Think of LIS it's kind of close

  Solution:

  We can use a dynamic programming approach with the idea of finding the maximum mountain subsequence rather than directly counting elements to remove. This approach is based on finding two Longest Increasing Subsequences (LIS) for each position in the array: one going left-to-right and the other going right-to-left. Once we have the longest possible mountain subsequence, the difference between the original array length and this subsequence length will give us the minimum elements to remove.

  Solution Outline

  1. Identify increasing subsequence lengths:

     • Compute the leftLIS array, where leftLIS[i] represents the length of the longest increasing subsequence ending at i (going left to right).

  2. Identify decreasing subsequence lengths:

     • Compute the rightLIS array, where rightLIS[i] represents the length of the longest decreasing subsequence starting at i (going right to left).

  3. Calculate maximum mountain length:

     • For each index i where 0 < i < n - 1, check if there exists a valid peak (i.e., leftLIS[i] > 1 and rightLIS[i] > 1). Calculate the mountain length at i as leftLIS[i] rightLIS[i] - 1.

  4. Get the minimum removals:

     • The minimum elements to remove will be the original array length minus the longest mountain length found.

  Let's implement this solution in PHP: 1671. Minimum Number of Removals to Make Mountain Array
  

  <?php
  /**
   * @param Integer[] $nums
   * @return Integer
   */
  function minimumMountainRemovals($nums) {
      ...
      ...
      ...
      /**
       * go to ./solution.php
       */
  }
  
  // Example usage
  $nums1 = [1, 3, 1];
  echo minimumMountainRemovals($nums1); // Output: 0
  
  $nums2 = [2, 1, 1, 5, 6, 2, 3, 1];
  echo minimumMountainRemovals($nums2); // Output: 3
  ?>
  

  Explanation:

  1. Left LIS Calculation:

     • leftLIS[i] stores the maximum length of an increasing subsequence ending at i. We iterate through each i, and for each i, iterate from 0 to i-1 to find the longest subsequence ending at i.

  2. Right LIS Calculation:

     • rightLIS[i] stores the maximum length of a decreasing subsequence starting at i. We iterate backward from n-2 to 0, and for each i, iterate from n-1 down to i 1 to find the longest subsequence starting at i.

  3. Mountain Calculation:

     • A valid peak at index i must have leftLIS[i] > 1 and rightLIS[i] > 1. The length of a mountain with peak at i is leftLIS[i] rightLIS[i] - 1.

  4. Final Calculation:

     • The minimum removals needed is the difference between the original array length and the maximum mountain length found.

  Complexity Analysis

  • Time Complexity: O(n²), due to the double loop in the LIS calculations.
  • Space Complexity: O(n), for storing the leftLIS and rightLIS arrays.

  This solution ensures that we find the maximum mountain subsequence and compute the minimum removals required to achieve a mountain array.

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