Problem Description

The goal is to find the minimum length of a subarray that, when removed from an array arr, leaves the remaining elements in a non-decreasing order (i.e., each element is less than or equal to the next). In other words, after removing the subarray, the resulting array should be sorted in non-decreasing order. This problem also includes the possibility of not removing any subarray at all if arr is already non-decreasing. The term "subarray" refers to a sequence of elements that are contiguous within the arr.

Intuition

The key to solving this problem lies in identifying parts of the array that are already sorted in non-decreasing order. Once we've identified such parts, we can find the minimum subarray to be removed. The solution approach can be broken down into the following steps:

1. Find the longest non-decreasing subarray from the start (left sorted subarray).
2. Find the longest non-decreasing subarray from the end (right sorted subarray).
3. Evaluate if the entire array is already non-decreasing by checking if the left and right overlap or touch each other. If they do, the shortest subarray to remove would be of length 0, which means we don't have to remove anything.
4. If a removal is needed, we can consider two potential solutions:
   • Remove the elements from the end of the left sorted subarray to the beginning of the array, leaving only the right sorted subarray.
   • Remove the elements from the start of the right sorted subarray to the end of the array, leaving only the left sorted subarray.
5. It's possible that by combining some portion of the left subarray with some portion of the right subarray, we could actually remove a shorter subarray in between and still maintain the non-decreasing order. Therefore, we iterate through the left sorted subarray and try to match its end with the beginning of the right sorted subarray, minimizing the length of the subarray to be removed.

Following these steps, we can determine the shortest subarray to remove, ensuring the array remains sorted in non-decreasing order after the removal.

Learn more about Stack, Two Pointers, Binary Search and Monotonic Stack patterns.

Solution Approach

The solution approach consists of several key steps that use loops and variables to track the progress through the array arr. Here's how the implementation works:

1. Initialize two pointers, i at the beginning of the array and j at the end. These pointers are used to find the left and right sorted subarrays, respectively.
2. Progress i forward through the array until we find the first element that is not in non-decreasing order. Until that point, the elements rest in a sorted subarray from the start.
3. Similarly, move j backwards through the array to find the first element from the end that breaks the non-decreasing order. Until that point, the elements are in a sorted subarray from the end.
4. If i has passed j, return 0, as the entire array is already non-decreasing or it has only one element that is out of order, which can be removed by itself.
5. Compute the initial potential answers:
   • The length of a subarray from i to the end of the array: n - i - 1
   • The length of a subarray from the start of the array to j: j
   We are interested in the minimal length of the subarray to be removed, so we take the minimum of these two potential answers.
6. Then comes the crucial step: trying to find the shortest subarray for removal that possibly lies between the sorted subarrays identified in steps 2 and 3. Initialize a new pointer r (short for right) to j.
7. Now, iterate through the array arr using the left pointer from 0 to i (inclusive). For each position of the left pointer, progress the right pointer until arr[r] is not less than arr[l], ensuring that elements to the left and right are in non-decreasing order.
8. Update answer ans each time to reflect the minimal value: the current ans and the number of elements between the left and right pointers, denoted by r - l - 1.

By following these steps, the function concludes by returning ans, which represents the length of the shortest subarray to remove to achieve a non-decreasing array after its removal.

This implementation is efficient and makes clever use of two-pointer technique along with a simple for loop and while loop constructs to keep track of the non-decreasing subarrays from both the start and end of the input array and to calculate the minimum length of the subarray that needs to be removed.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach. Consider the array arr = [1, 3, 2, 3, 5].

1. Starting from the left, we see that 1 <= 3, but 3 > 2, so the longest non-decreasing subarray from the start is [1, 3] with i = 1.
2. Starting from the right, we see that 5 >= 3, 3 >= 2, but 2 < 3, so the longest non-decreasing subarray from the end is [2, 3, 5] with j = 2.
3. Since i < j, they do not overlap, and we must remove a subarray to make the entire array non-decreasing.
4. If we remove elements starting from the end of the left sorted subarray to the beginning of the array, we would remove [1, 3], leaving [2, 3, 5], which is in non-decreasing order. However, this results in removing 2 elements.
5. Conversely, if we remove elements from the start of the right sorted subarray to the end of the array, we would remove [3, 5], leaving [1, 3, 2], which is not in non-decreasing order, so this is not a valid option.
6. Now, we check to see if it's possible to maintain a part of the left subarray [1, 3] and combine it with the right subarray [2, 3, 5] to minimize the length of the subarray to be removed. To find the shortest subarray for removal, initialize pointer r (short for right) to j, which is 2 at the moment.
7. We iterate through the array from the left pointer l = 0 to i = 1. When l = 0, arr[l] = 1 is less than arr[r] = 2 (since r is at j), so we don't need to move r. Next, when l = 1, arr[l] = 3 is greater than arr[r] = 2, so we increment r to ensure that arr[r] is not less than arr[l]. Since arr[r] = 3 is now greater than arr[l] = 3, we can stop.
8. The minimal length of the subarray to be removed lies in between pointer l and pointer r, which in this case is the subarray [2] (since r = 3 and l = 1, we have r - l - 1 = 3 - 1 - 1 = 1 element to be removed).

Thus, by following the solution steps, the smallest subarray we need to remove to make arr sorted in non-decreasing order is [2] of length 1. Hence, the function returns 1 as the answer.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided code can be broken down as follows:

1. Two while loops (before the if statement) are executed sequentially, each advancing at most n steps. The worst-case complexity for this part is O(n).
2. The if statement is a constant time check O(1).
3. The minimum of n - i - 1 and j is also a constant time operation O(1).
4. A for loop runs from 0 to i + 1, and inside it, there is a while loop that could iterate from j to n in the worst case. In the worst-case scenario, this nested loop could run O(n^2) times because for each iteration of the for loop (at most n times), the while loop could also iterate n times.

Thus, the overall time complexity is dominated by the nested loop, giving us a worst-case time complexity of O(n^2).

Space Complexity

The space complexity is determined by the extra space used by the algorithm besides the input. In this case:

1. Variables i, j, n, ans, and r use constant space O(1).
2. There are no additional data structures used that grow with the size of the input.

Therefore, the space complexity is O(1), which corresponds to constant space usage.

Learn more about how to find time and space complexity quickly using problem constraints.