Find Maximal Uncovered Ranges

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ArraysGreedy Algorithms

You are given a set of integer ranges. Your task is to find the minimum number of ranges that cover the entire range from the minimum start point to the maximum end point of all given ranges. A range is considered covered if every point within it is included in at least one of the given ranges.

For example, given the ranges [[1, 5], [3, 6], [7, 9]], the overall range to be covered is from 1 to 9. These ranges can cover it:

Range 1: [1, 5]
Range 2: [3, 6]
Range 3: [7, 9]

You need to determine a minimal set of these ranges to cover [1, 9].

However, the actual task is slightly different. Given an array of integer ranges ranges, your task is to return the maximal length of the uncovered regions.

Example 1:

Input: ranges = [[1,4],[6,8]]
Output: 2
Explanation: The range [1, 8] is the range to be covered.
It has sub-ranges [1,4] and [6,8]. Therefore, the uncovered range is [5,5] with length 1.  Hence the result is 1 (8-1+1) - (4-1+1) - (8-6+1) = 8 - 4 - 3 = 1.

Example 2:

Input: ranges = [[1,3],[6,9],[4,5]]
Output: 2
Explanation: The range [1, 9] is the range to be covered.
It has sub-ranges [1,3], [4,5], and [6,9]. The uncovered ranges are empty so the result is 9-1+1 - (3-1+1) - (5-4+1) - (9-6+1) = 9 - 3 - 2 - 4 = 0

Example 3:

Input: ranges = [[0,2],[3,4],[5,5],[7,9]]
Output: 2
Explanation: The range [0, 9] is the range to be covered.
It has sub-ranges [0,2], [3,4], [5,5], and [7,9]. The uncovered ranges are [6, 6] so the result is 0.

Constraints:

1 <= ranges.length <= 1000
ranges[i].length == 2
0 <= start_i <= end_i <= 10⁵

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

Can you provide more details about the range objects? Specifically, what data types are used for the start and end points of each range, and what are the possible ranges for these values?
What should I return if there are no uncovered ranges? Should I return an empty list, null, or some other indicator?
If there are overlapping ranges in the input, should I merge them first to simplify the identification of uncovered ranges?
How should the uncovered ranges be represented in the output? Should it be a list of range objects, or some other format?
Are the input ranges guaranteed to be sorted in any particular order (e.g., by start point), or should I assume they are unsorted?

Brute Force Solution

Approach

We're given a number line and a set of ranges. The brute force approach is about trying every possible combination of ranges and seeing which combination covers the most of the number line without any overlap. It's like trying all possible outfits to see which one looks the best.

Here's how the algorithm would work step-by-step:

Start by considering no ranges at all and see how much of the number line is covered.
Then, try using only the first range and see how much is covered.
Next, try using only the second range, then only the third range, and so on, checking the coverage for each individual range.
After that, try using the first and second range together, then the first and third, then the second and third, and all other possible pairs of ranges.
While trying different ranges together, check if they overlap. If they do, skip that combination because the problem description states that the ranges must not overlap.
Continue this process, trying all possible combinations of three ranges, four ranges, and so on, up to using all of the ranges.
For each combination, calculate how much of the number line is covered without any overlapping regions. Keep track of the combination that covers the most area.
Finally, compare the total coverage of all the valid combinations and select the one with the maximum coverage. This gives us the maximal uncovered ranges.

Code Implementation

def find_maximal_uncovered_ranges_brute_force(ranges):
    max_coverage = 0
    best_combination = []
    number_of_ranges = len(ranges)

    for i in range(1 << number_of_ranges):
        current_combination = []

        for j in range(number_of_ranges):
            if (i >> j) & 1:
                current_combination.append(ranges[j])

        is_valid_combination = True
        for first_index in range(len(current_combination)):
            for second_index in range(first_index + 1, len(current_combination)):
                range_one_start, range_one_end = current_combination[first_index]
                range_two_start, range_two_end = current_combination[second_index]

                # Check for overlap; if overlap exists, skip this combination
                if not (range_one_end <= range_two_start or range_two_end <= range_one_start):
                    is_valid_combination = False
                    break

            if not is_valid_combination:
                break

        if is_valid_combination:
            coverage = 0
            combined_ranges = []

            for current_range in current_combination:
                combined_ranges.append(current_range)

            combined_ranges.sort()
            covered_points = set()

            # Calculate the coverage of the current combination
            for current_range_start, current_range_end in combined_ranges:
                for point in range(current_range_start, current_range_end):
                    covered_points.add(point)

            coverage = len(covered_points)

            # Update best combination
            if coverage > max_coverage:
                max_coverage = coverage
                best_combination = current_combination

    return best_combination

Big(O) Analysis

Time Complexity

O(2^n) – The described algorithm iterates through all possible subsets of the input ranges. Given 'n' ranges, there are 2^n possible subsets (each range can either be included or excluded). For each subset, the algorithm checks for overlaps, which takes O(n) time in the worst case to compare each range within the subset. Calculating the coverage for each valid (non-overlapping) subset also takes O(n) time since we might have to iterate through all the ranges in the subset. Thus, we iterate through 2^n subsets and perform O(n) operations for each, resulting in a time complexity of O(n * 2^n). However, the dominant term is 2^n, so the overall time complexity simplifies to O(2^n).

Space Complexity

O(2^N) – The brute force approach explores all possible combinations of ranges, which leads to a power set. To generate and store these combinations, temporary lists are created, potentially holding up to N ranges each, where N is the total number of input ranges. Since there are 2^N possible subsets (combinations), the space required to store all these combinations can grow exponentially with the input size. Therefore, the auxiliary space used is O(2^N).

Optimal Solution

Approach

The key idea is to sort all the ranges first and then merge overlapping ranges. After merging we need to identify the gaps left by the merged ranges, those gaps are the uncovered maximal ranges. We will process these sorted and merged ranges sequentially to find and report those maximal uncovered ranges.

Here's how the algorithm would work step-by-step:

First, organize all the given ranges by their starting points, putting them in ascending order.
Go through the organized ranges one by one, and if any ranges overlap, merge them into a single, larger range. This ensures we have the minimal set of non-overlapping covered sections.
Now, look at the spaces between the merged ranges. These spaces represent the uncovered sections.
Identify the maximal gaps between the merged ranges to define the ranges that are uncovered and have the largest possible span.
Report the starting and ending points of these maximal uncovered ranges, which answer our question.

Code Implementation

def find_maximal_uncovered_ranges(ranges):
    # Sort ranges based on start points to easily merge overlaps.
    sorted_ranges = sorted(ranges, key=lambda x: x[0])

    merged_ranges = []
    if not sorted_ranges:
        return []

    current_range = sorted_ranges[0]

    for start, end in sorted_ranges[1:]:
        if start <= current_range[1]:
            # Merge overlapping ranges by extending the current range.
            current_range = (current_range[0], max(current_range[1], end))
        else:
            merged_ranges.append(current_range)
            current_range = (start, end)

    merged_ranges.append(current_range)

    uncovered_ranges = []

    # Identify uncovered ranges by checking gaps between merged ranges.
    for i in range(len(merged_ranges) - 1):
        uncovered_start = merged_ranges[i][1]
        uncovered_end = merged_ranges[i+1][0]

        if uncovered_end > uncovered_start:
            uncovered_ranges.append((uncovered_start, uncovered_end))

    return uncovered_ranges

Big(O) Analysis

Time Complexity

O(n log n) – The most significant operation impacting the time complexity is sorting the input ranges, which takes O(n log n) time. The subsequent merging of overlapping ranges and identifying uncovered ranges involves iterating through the sorted ranges, taking O(n) time. Since O(n log n) dominates O(n), the overall time complexity of the algorithm is O(n log n).

Space Complexity

O(N) – The algorithm sorts the input ranges, which may require O(N) auxiliary space depending on the sorting algorithm used. Also, merging the overlapping ranges may create a new list of merged ranges. In the worst-case scenario where no ranges overlap, the merged list will contain all N ranges, thus requiring O(N) space. Therefore, the overall auxiliary space complexity is O(N).

Edge Cases

Case	How to Handle
Empty list of ranges	Return an empty list since there are no ranges to uncover.
List of ranges contains only one range	Return the single range as the maximal uncovered range as no other ranges exist to compare against.
Ranges are adjacent or overlapping.	Merge or consider as partially covered to accurately identify maximal uncovered portions.
Ranges completely cover the entire possible input space (e.g., all integers).	Return an empty list because there is no uncovered portion.
Ranges contain negative numbers and/or zero.	Ensure the solution handles negative indices or values appropriately, possibly by using an offset or a different data structure (e.g., a map).
Ranges have identical start and end points (single points).	Treat these as ranges with a length of zero and process them like any other range.
Very large ranges (potentially causing integer overflow during calculations).	Use appropriate data types (e.g., long) or consider using a different method of comparing lengths/distances to prevent overflow.
The uncovered ranges are disconnected.	The solution must identify and return all disjoint uncovered intervals.

Empty list of ranges

How to Handle:

Return an empty list since there are no ranges to uncover.

List of ranges contains only one range

How to Handle:

Return the single range as the maximal uncovered range as no other ranges exist to compare against.

Ranges are adjacent or overlapping.

How to Handle:

Merge or consider as partially covered to accurately identify maximal uncovered portions.

Ranges completely cover the entire possible input space (e.g., all integers).

How to Handle:

Return an empty list because there is no uncovered portion.

Ranges contain negative numbers and/or zero.

How to Handle:

Ensure the solution handles negative indices or values appropriately, possibly by using an offset or a different data structure (e.g., a map).

Ranges have identical start and end points (single points).

How to Handle:

Treat these as ranges with a length of zero and process them like any other range.

Very large ranges (potentially causing integer overflow during calculations).

How to Handle:

Use appropriate data types (e.g., long) or consider using a different method of comparing lengths/distances to prevent overflow.

The uncovered ranges are disconnected.

How to Handle:

The solution must identify and return all disjoint uncovered intervals.