From 97d2f2ba412f73904abe1c87ac8634d79fd51ce0 Mon Sep 17 00:00:00 2001 From: Lusina <12752833+BrianLusina@users.noreply.github.com> Date: Sat, 20 Dec 2025 14:28:48 +0300 Subject: [PATCH 1/2] feat(algorithms, intervals): remove covered intervals --- .../intervals/remove_intervals/README.md | 72 +++++++++++++ .../intervals/remove_intervals/__init__.py | 43 ++++++++ .../remove_covered_intervals_example_1.png | Bin 0 -> 93657 bytes .../remove_covered_intervals_example_2.png | Bin 0 -> 101005 bytes .../remove_covered_intervals_example_3.png | Bin 0 -> 68592 bytes .../remove_covered_intervals_example_4.png | Bin 0 -> 79382 bytes .../remove_covered_intervals_example_5.png | Bin 0 -> 77855 bytes .../remove_covered_intervals_solution_1.png | Bin 0 -> 48157 bytes .../remove_covered_intervals_solution_2.png | Bin 0 -> 78258 bytes .../remove_covered_intervals_solution_3.png | Bin 0 -> 78428 bytes .../remove_covered_intervals_solution_4.png | Bin 0 -> 101079 bytes .../remove_covered_intervals_solution_5.png | Bin 0 -> 96064 bytes .../remove_covered_intervals_solution_6.png | Bin 0 -> 74919 bytes .../remove_covered_intervals_solution_7.png | Bin 0 -> 64922 bytes .../test_remove_covered_intervals.py | 98 ++++++++++++++++++ 15 files changed, 213 insertions(+) create mode 100644 algorithms/intervals/remove_intervals/README.md create mode 100644 algorithms/intervals/remove_intervals/__init__.py create mode 100644 algorithms/intervals/remove_intervals/images/examples/remove_covered_intervals_example_1.png create mode 100644 algorithms/intervals/remove_intervals/images/examples/remove_covered_intervals_example_2.png create mode 100644 algorithms/intervals/remove_intervals/images/examples/remove_covered_intervals_example_3.png create mode 100644 algorithms/intervals/remove_intervals/images/examples/remove_covered_intervals_example_4.png create mode 100644 algorithms/intervals/remove_intervals/images/examples/remove_covered_intervals_example_5.png create mode 100644 algorithms/intervals/remove_intervals/images/solutions/remove_covered_intervals_solution_1.png create mode 100644 algorithms/intervals/remove_intervals/images/solutions/remove_covered_intervals_solution_2.png create mode 100644 algorithms/intervals/remove_intervals/images/solutions/remove_covered_intervals_solution_3.png create mode 100644 algorithms/intervals/remove_intervals/images/solutions/remove_covered_intervals_solution_4.png create mode 100644 algorithms/intervals/remove_intervals/images/solutions/remove_covered_intervals_solution_5.png create mode 100644 algorithms/intervals/remove_intervals/images/solutions/remove_covered_intervals_solution_6.png create mode 100644 algorithms/intervals/remove_intervals/images/solutions/remove_covered_intervals_solution_7.png create mode 100644 algorithms/intervals/remove_intervals/test_remove_covered_intervals.py diff --git a/algorithms/intervals/remove_intervals/README.md b/algorithms/intervals/remove_intervals/README.md new file mode 100644 index 00000000..db18d4a1 --- /dev/null +++ b/algorithms/intervals/remove_intervals/README.md @@ -0,0 +1,72 @@ +# Remove Covered Intervals + +Given an array of intervals, where each interval is represented as intervals[i]=[li,ri) (indicating the range from +li to ri, inclusive of li and exclusive of ri), remove all intervals that are completely covered by another interval in +the list. Return the count of intervals that remain after removing the covered ones. + +> Note An interval [a, b) is considered covered by another interval [c,d) if and only if c ⇐ a and b ⇐ d. + +## Constraints + +- 1 <= intervals.length <= 10^4 +- intervals[i].length == 2 +- 0 <= li < ri <= 10^5 +- All the give intervals are unique + +## Examples + +![Example 1](./images/examples/remove_covered_intervals_example_1.png) +![Example 2](./images/examples/remove_covered_intervals_example_2.png) +![Example 3](./images/examples/remove_covered_intervals_example_3.png) +![Example 4](./images/examples/remove_covered_intervals_example_4.png) +![Example 5](./images/examples/remove_covered_intervals_example_5.png) + + +## Solution + +The first step is to simplify the process by sorting the intervals. Sorting by the start point in ascending order is +straightforward and simplifies the iteration process. However, an important edge case arises when two intervals share +the same start point. In such scenarios, sorting solely by the start point would fail to correctly identify covered +intervals. To handle this, we sort intervals with the same start point by their endpoint in descending order, ensuring +that longer intervals come first. This sorting strategy guarantees that if one interval covers another, it will be +positioned earlier in the sorted list. Once the intervals are sorted, we iterate through them while keeping track of the +maximum endpoint seen so far. If the current interval’s end point exceeds this maximum, it is not covered, so we increment +the count and update the maximum end. The interval is covered and skipped if the endpoint is less than or equal to the +maximum. After completing the iteration, the final count reflects the remaining non-covered intervals. + +Now, let’s look at the solution steps below: + +1. If the start points are the same, sort the intervals by the start point in ascending order, otherwise, sort by the + endpoint in descending order to prioritize longer intervals. +2. Initialize the count with zero to track the remaining (non-covered) intervals. +3. Initialize prev_end with zero to track the maximum end value we’ve seen.. +4. Start iterating through intervals for each interval [start, end] in the sorted list: + - If end > prev_end, any previous interval does not cover the interval. + - Increment count by 1 + - Update prev_end to end. + - Else: + - A previous interval covers the interval, so we skip it. + +5. After iterating through all the intervals, the return count is the final value, representing the remaining intervals. + +Let’s look at the following illustration to get a better understanding of the solution: + +![Solution 1](./images/solutions/remove_covered_intervals_solution_1.png) +![Solution 2](./images/solutions/remove_covered_intervals_solution_2.png) +![Solution 3](./images/solutions/remove_covered_intervals_solution_3.png) +![Solution 4](./images/solutions/remove_covered_intervals_solution_4.png) +![Solution 5](./images/solutions/remove_covered_intervals_solution_5.png) +![Solution 6](./images/solutions/remove_covered_intervals_solution_6.png) +![Solution 7](./images/solutions/remove_covered_intervals_solution_7.png) + +### Time Complexity + +The time complexity of the solution is O(n logn), where n is the number of intervals. This is because sorting the +intervals takes O(n logn) time, and the subsequent iteration through the intervals takes O(n) time. Therefore, the +overall time complexity is dominated by the sorting step, resulting in O(n logn). + +### Space Complexity + +The sorting operation has a space complexity of O(n) in the worst case due to additional memory required for temporary +arrays during the sorting process. Apart from the space used by the built-in sorting algorithm, the algorithm’s space +complexity is constant, O(1). Therefore, the overall space complexity of the solution is O(n). diff --git a/algorithms/intervals/remove_intervals/__init__.py b/algorithms/intervals/remove_intervals/__init__.py new file mode 100644 index 00000000..22f71975 --- /dev/null +++ b/algorithms/intervals/remove_intervals/__init__.py @@ -0,0 +1,43 @@ +from typing import List + + +def remove_covered_intervals(intervals: List[List[int]]) -> int: + """ + Finds the number of intervals that are not covered by any other interval in the list. + + This uses a greedy approach to find the number of intervals that are not covered by any other interval in the list. + Note An interval [a, b) is considered covered by another interval [c,d) if and only if c ⇐ a and b ⇐ d. + + So, the timeline will look something like this: + c---a---b---d + + If b <= d, then [a, b) is covered by [c, d) + + Args: + intervals (List[List[int]]): A list of intervals, where each interval is represented as [start, end] + Returns: + int: The number of intervals that are not covered by any other interval in the list + """ + # early return if there are no intervals to begin with + if not intervals: + return 0 + + # Sort intervals by start time in ascending order and then by end time in descending order. We sort by the end time + # to remove a tie-breaker where two intervals have the same start time. + # This will incur a time complexity of O(nlogn). We sort in place, so, we do not + # incur any additional space complexity by copying over to a new list. The assumption made here is that it is okay + # to mutate the input list. + intervals.sort(key=lambda x: (x[0], -x[1])) + + # keep track of the last max end seen so far, we use a large negative infinity to cover all possible numbers + max_end_seen = float('-inf') + count = 0 + + # We then iterate through the given intervals + for _, current_end in intervals: + if current_end > max_end_seen: + count += 1 + max_end_seen = current_end + + # return the count of non-overlapping intervals + return count diff --git a/algorithms/intervals/remove_intervals/images/examples/remove_covered_intervals_example_1.png b/algorithms/intervals/remove_intervals/images/examples/remove_covered_intervals_example_1.png new file mode 100644 index 0000000000000000000000000000000000000000..1ac2260a116b73a15c7c271b115b17187f8dc3ba GIT binary patch literal 93657 zcmeEucRbbq7r!ziB$W_uR4S{?Y?9SL$aba3PPXeBAtSP9_P(~r=F%{(y~$qJ9ygck zX8&G3>s#MX|Nee|{2t%?cwG11_xrWp=XG9ZJkL4aswl}*k~5GK5D-u*D%?{gARxjL 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