Problem Description

In this problem, you are given two inputs: an integer n, specifying the length of a 0-indexed array nums, and a 2D-array ranges. Each entry in ranges represents a sub-range of the array nums with a start and end index (inclusive). These ranges can overlap each other. The task is to identify and return all the sub-ranges of nums that are maximally uncovered by any of the ranges provided in ranges. An uncovered range is considered maximal if:

1. Each cell within the uncovered range belongs to just one uncovered sub-range. This implies that uncovered sub-ranges cannot overlap.
2. There are no such consecutive uncovered ranges that the end of one is immediately followed by the start of another. Essentially, no two uncovered ranges should be adjacent.

The expected output is a 2D-array of the uncovered ranges, sorted by their start index in ascending order.

Intuition

The intuition behind the solution approach can be broken down into the following steps:

1. Sorting the Ranges: First, we sort the given ranges by their start indices. Sorting helps us to process the ranges sequentially, thus making it simpler to find any gaps between successive ranges.

2. Finding Uncovered Ranges: We initialize a variable last that keeps track of the end index of the last covered sub-range (initialized to -1 since the array is 0-indexed). We then iterate over the sorted ranges, and for each range (l, r):

   a. We check if there is an uncovered range that starts after the end of the last covered range (last + 1) and ends before the start of the current range (l - 1). If such an uncovered range exists, we add it to the answer.

   b. We update last to be the maximum of the current end index r and the previously stored last, as this keeps last pointing to the end of the current furthest covered range.

3. Checking for Rightmost Uncovered Range: After processing all the given ranges, there might still be an uncovered range right at the end of the nums array. We check if the index one more than the last covered index (last + 1) is less than n. If it is, an uncovered sub-range exists from last + 1 to n - 1, and we append this range to our answer.

4. Returning the Result: Finally, the 2D-array ans contains all the maximally uncovered ranges and is returned as the solution.

Learn more about Sorting patterns.

Solution Approach

The implementation of the solution uses a straightforward approach to solve the problem. Here's a step-by-step explanation:

1. Sort Ranges: The first step is to sort the ranges array. This is done with the sort() method, which sorts the list in place. Sorting is based on the first element of each sub-list, which represents the start of the range. Sorting is critical as it allows us to easily compare the current range with the last uncovered range found.

   ranges.sort()

2. Initialize Variables: We initialize a list ans to collect the answers and a variable last to keep track of the end of the last covered range. The last variable starts at -1 since the array is 0-indexed and we want to find if there's an uncovered range starting from the first index (0).

   last = -1
   ans = []

3. Iterate and Find Uncovered Ranges: We loop through each range (l, r) in the sorted ranges list. For each range, two main checks occur:

   a. Uncovered Range Check: If the start of the current range (l) is greater than last + 1, there is an uncovered range between last + 1 and l - 1. This range is added to the ans list.

    ```python
    if last + 1 < l:
        ans.append([last + 1, l - 1])
    ```

   b. Update Last Covered: We then update last to be the maximum of last and the current range's end r. This step effectively skips over any overlap and ensures that we extend our covered area to include the current range.

    ```python
    last = max(last, r)
    ```

4. Check for Tail Uncovered Range: After the loop, we check if there's an uncovered range from the end of the last covered range to the end of the array nums. If such an uncovered range exists, it is added to ans.

   if last + 1 < n:
       ans.append([last + 1, n - 1])

5. Return Answer: The ans list now contains all the maximal uncovered ranges, sorted by starting point, and can be returned as the final answer.

   return ans

The simple but effective approach works due to the properties of sorting and the greedy method of extending the covered range to the furthest end reached by any range. The implementation is efficient and straightforward, not requiring any complex algorithms or data structures.

Example Walkthrough

Let's consider a small example to illustrate the solution approach.

Suppose we are given an integer n = 10, representing an array nums of length 10, and a 2D-array ranges = [[1, 3], [6, 9], [2, 5]]. We want to find all the maximally uncovered sub-ranges in nums.

Following the solution approach:

1. Sort Ranges: Sort the ranges array to process in order. After sorting based on start index, we get [[1, 3], [2, 5], [6, 9]].

2. Initialize Variables: We initialize last = -1 and ans = [].

3. Iterate and Find Uncovered Ranges:

   • For the first range [1, 3], since last + 1 (0) is less than 1, there's an uncovered range [0, 0]. We add it to ans, resulting in ans = [[0, 0]]. Update last to 3.
   • For the second range [2, 5], last + 1 (4) is not less than 2, so no new uncovered range is added. Update last to 5 (the current end index is not larger than last).
   • For the third range [6, 9], last + 1 (6) is equal to the start index, so again, no uncovered range is added. Update last to 9.

4. Check for Tail Uncovered Range: After processing all ranges, we check for any uncovered range from the end of the last covered range (last = 9) to the length of nums (n = 10). Since last + 1 (10) is less than n, there's an uncovered range [10, 9], which is invalid because the start is greater than the end. Thus, no range is added to ans.

5. Return Answer: The final answer is ans = [[0, 0]], which is the list of all maximally uncovered ranges in nums.

This example helps us understand how the solution approach works step by step to identify uncovered ranges in the array nums.

Solution Implementation

Time and Space Complexity

The time complexity of the provided code primarily depends on the sorting of the ranges and the subsequent iteration through the sorted ranges list.

1. Sorting the ranges: The sort operation for the ranges at the beginning of the function has a time complexity of O(m log m), where m is the number of intervals in the ranges list.

2. Iterating through the sorted ranges: The iteration through ranges involves comparing and possibly appending to the ans list. This iteration occurs once for each element in the ranges list, giving it a time complexity of O(m).

3. Combining the above steps: Since the sort operation dominates the overall time complexity, we can state that the overall time complexity of the function is O(m log m).

The space complexity of the code comes from the additional memory used to store the ans list.

1. Space for ans list: In the worst case, the ans list could potentially store every single uncovered range between adjacent ranges in the sorted list as well as the uncovered range before the first range and after the last range if applicable. This worst-case scenario happens when none of the ranges overlap at all, which theoretically could create up to m+1 entries in the ans list in the case where each interval has a gap with adjacent intervals. Thus, the space complexity is O(m).

In summary:

• The Time Complexity of the code is O(m log m).
• The Space Complexity of the code is O(m).

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