Problem Description

In this LeetCode problem, we are given an array of integers (nums). The task is to identify the smallest segment (continuous subarray) of this array that, if we sort it, the entire array becomes sorted in non-decreasing order (i.e., ascending order with duplicates allowed). The goal is to find the length of this smallest segment.

We are looking for the subarray where once sorted, the values before and after this selected part fit in order and don't disrupt the non-decreasing sequence. To solve this problem, we have to distinguish between the parts of the array which are already in the correct position and those that are not.

Intuition

There are different strategies that we could use to solve this problem, some being more optimal than others. The initial, more straightforward approach is to sort a copy of the array and to compare it to the original. The first and last positions where they differ will give us the boundaries of the subarray that needs to be sorted.

However, the actual implemented solution follows a more optimal and intuitive approach that avoids using extra space and reduces the time complexity. The solution involves iterating over the array while keeping track of the maximum and minimum values seen from the left and the right, respectively.

While scanning from the left, we maintain the maximum value encountered so far - let's call it mx. If at any index i, the value x is less than mx, this suggests x is out of its required sorted position because everything before it is already part of a non-decreasing sequence. Hence, x should be included in the subarray that needs sorting. The rightmost index r where this occurs marks the endpoint of the unsorted subarray.

Simultaneously, we scan from the right and maintain the minimum value encountered so far - referenced as mi. Upon finding an element greater than mi at index n - i - 1 from the end, it indicates this element is also out of order. The leftmost index l where this situation arises becomes the beginning of the unsorted subarray.

Initially, l and r are set to -1, and if they remain unchanged after the iterations, it means the entire array is already sorted, and we return length 0. If they have been updated, the length of the minimum subarray required to be sorted is calculated by the difference r - l + 1.

The intuition behind updating r and l only when we find values out of their required position is that only these values define the boundaries of the shortest unsorted subarray—and only by sorting them can we achieve a fully sorted array.

Learn more about Stack, Greedy, Two Pointers, Sorting and Monotonic Stack patterns.

Solution Approach

The solution involves a single pass approach that cleverly makes use of two properties, namely tracking the maximum and minimum values from the left and right, respectively. By doing this, we can identify the bounds of the unsorted subarray without having to sort the array or use extra space.

Let's understand this algorithm step by step:

1. Initialize Variables: Two pointers l and r are initialized to -1 which will represent the left and right bounds of the unsorted subarray. We also initialize two variables, mi to +infinity, representing the minimum value seen from the right; and mx to -infinity, representing the maximum value seen from the left.

2. Iterate Over the Array: We loop over all elements in the array using a variable i that goes from 0 to n - 1, where n is the length of the array. While iterating, we perform the following:

   • Update Right Boundary (r): For the i-th element x, we compare it with the maximum value seen so far mx. If x is smaller than mx, it means x is out of place and should be part of the subarray to be sorted, so we update r to be the current index i.
   • Update Maximum (mx): If x is greater than or equal to mx, we update mx to be x because it's the new maximum value seen so far.
   • At each iteration of i, we also consider the mirror position from the right end of the array: index n - i - 1.
   • Update Left Boundary (l): We compare nums[n - i - 1] with the minimum value seen from the right mi. If nums[n - i - 1] is bigger than mi, it's out of place and should be included in the subarray to be sorted, so we update l to be n - i - 1.
   • Update Minimum (mi): If nums[n - i - 1] is less than or equal to mi, we update mi to this value because it's now the minimum seen from the right.

3. Calculate the Subarray Length: After iterating through the array, we check if r was updated. If r is not updated (r == -1), it means the array was already sorted, and we return 0. Otherwise, we return the length of the subarray that needs to be sorted, which is r - l + 1.

By iterating from both ends towards the center and updating l and r on-the-go, we are leveraging the information of where the sequence breaks from being non-decreasing. mx and mi act as sentinels to detect any number that doesn't fit into the non-decreasing order up to that point, thus efficiently finding the shortest unsorted window in one single pass with a time complexity of O(n) and space complexity of O(1).

Example Walkthrough

Let's walk through a small example using the described solution approach. Consider the array nums = [2, 6, 4, 8, 10, 9, 15].

Step 1: Initialize Variables

• l = -1
• r = -1
• mi = +infinity
• mx = -infinity

Step 2: Iterate Over the Array

• We start iterating from the first element to the last.
• For each element, we perform the checks and updates as specified.

Let's see this in action:

i. When i = 0, element nums[i] = 2.

• mx = max(-infinity, 2) = 2.
• The mirror index from the right is n - 0 - 1 = 6. Element nums[6] = 15.
• mi = min(+infinity, 15) = 15.

ii. Moving to i = 1, nums[i] = 6.

• Element 6 is not less than mx (2). Thus, mx becomes 6.
• The mirror index from the right is n - 1 - 1 = 5. Element nums[5] = 9.
• mi becomes min(15, 9) = 9.

iii. Next, i = 2, nums[i] = 4.

• 4 is less than mx (6), hence we update r = 2.
• The mirror index from the right is n - 2 - 1 = 4. Element nums[4] = 10.
• mi is now the min(9, 10) = 9.

iv. For i = 3, nums[i] = 8.

• 8 is not less than mx (6). So, mx becomes 8.
• The mirror index is n - 3 - 1 = 3. Element nums[3] = 8.
• mi stays at 9 since 8 is less than mi.

v. When i = 4, nums[i] = 10.

• 10 is not less than mx. Update mx to 10.
• The mirror index is n - 4 - 1 = 2. Element nums[2] = 4.
• 4 is less than mi (9), so we update l = 2.

As we continue, mx and mi will not update any further because the remaining elements (nums[5] = 9 and nums[6] = 15) are greater than all seen max values and less than all seen min values, respectively.

Step 3: Calculate the Subarray Length

• After completing the iteration, l is found to be 2 and r is also 2.
• They both referred to the same element because we performed the l update after the r update.
• We need to find the full range of the unsorted segment.
• We already know r should be at least 5, because nums[5] = 9 was out of place.
• Correcting for the left boundary due to the two-pointer check in the algorithm, l should be 1, not 2. The element nums[1] = 6 initiated the descending sequence.
• Thus, the length of the subarray is r - l + 1 = 5 - 1 + 1 = 5.

The subarray [6, 4, 8, 10, 9] from indices 1 to 5 is the smallest segment that, once sorted, the entire array nums becomes sorted. The length of this segment is 5.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(n). This is because the code contains only one for loop that iterates over all elements of the array, nums, exactly once. Inside the loop, it performs a constant number of operations for each element, which does not depend on the size of the array.

The space complexity of the code is O(1) which means constant space. No additional space is utilized that is dependent on the input size n, besides a fixed number of individual variables (mi, mx, l, r, and n) that occupy space which does not scale with the size of the input array.

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