How would you solve the 'Merge Intervals' problem: merge all overlapping intervals in an array of given numerical intervals?

Overview

This interview question is a staple for evaluating a software engineer's foundational algorithmic skills and their ability to design efficient solutions for array manipulation problems. It directly assesses a candidate's approach to problem decomposition, algorithmic choice, and the implementation of robust, performant code, which are critical for building scalable systems.

Interview Question:

How would you solve the 'Merge Intervals' problem: merge all overlapping intervals in an array of given numerical intervals?

Why Interviewers Ask This:

Interviewers pose the Merge Intervals problem to assess several key competencies vital for a Software Engineer role, particularly in FAANG-style interviews. First, it evaluates a candidate's algorithmic thinking and ability to identify the most efficient approach, often involving sorting as a prerequisite for a linear scan. Second, it tests their understanding of data structures and how to manipulate them effectively. Crucially, it measures their proficiency in Big O analysis, expecting them to articulate the time and space complexity of their proposed solution and justify its optimality. Finally, it gauges their capacity to write clean, production-grade code that handles edge cases gracefully, demonstrating attention to detail and a commitment to software quality and maintainability.

Expert Answer:

To efficiently solve the Merge Intervals problem, the most critical first step is to sort the given array of intervals. We should sort them based on their start times. If two intervals have the same start time, their end times can be used as a secondary sorting criterion, though it's often not strictly necessary for correctness if the primary sort is by start time. This sorting step is crucial because it ensures that we can process intervals in a sequential manner, guaranteeing that if an interval overlaps with a previous one, it will appear immediately after it in the sorted list.

Next, we initialize an empty list, let's call it merged_intervals, to store our results. We then iterate through the sorted intervals. For each interval, we compare it with the last interval added to merged_intervals. If merged_intervals is empty, or if the current interval does not overlap with the last interval in merged_intervals, we simply add the current interval to merged_intervals. An overlap occurs if the current interval's start time is less than or equal to the end time of the last interval in merged_intervals.

If an overlap is detected, we need to merge the current interval with the last one in merged_intervals. This is done by updating the end time of the last interval in merged_intervals to be the maximum of its current end time and the current interval's end time. We take the maximum to ensure that the merged interval encompasses the full range of both overlapping intervals. This process effectively extends the last merged interval to cover the current one.

This approach guarantees an optimal time complexity. The sorting step dominates the complexity, taking O(N log N) time, where N is the number of intervals. The subsequent single pass through the sorted intervals takes O(N) time. Therefore, the overall time complexity is O(N log N). The space complexity is O(N) for storing the merged_intervals list, which in the worst case (no overlaps) could contain all original intervals, plus the space required by the sorting algorithm (which can be O(log N) or O(N) depending on the implementation). This solution is robust, handles various edge cases like empty input, single intervals, or completely non-overlapping intervals, and is highly scalable for large datasets.

Finally, when implementing this, attention to code clarity and edge case handling is paramount. Ensure the comparison logic for merging is precise and that the initial state of merged_intervals is handled correctly. Thorough unit testing with diverse interval sets (including edge cases) would be essential to validate the correctness and robustness of the solution before it reaches production.

Speaking Blueprint:

[The Hook] This is a classic problem that tests how we handle ordered data and identify overlaps. My approach would leverage sorting to simplify the merging process, ensuring an optimal solution.

[The Core Execution] First, I would sort the input array of intervals based on their start times. This is a critical preprocessing step that allows us to process intervals sequentially. Once sorted, I'd initialize an empty list to store the merged intervals. I'd then iterate through the sorted intervals. For each interval, I'd check if it overlaps with the last interval in my merged list. If there's no overlap, or if the merged list is empty, I simply add the current interval. If there is an overlap – meaning the current interval's start is less than or equal to the last merged interval's end – I'd update the end of the last merged interval to be the maximum of its current end and the current interval's end. This ensures the merged interval covers the full range. This strategy yields an optimal time complexity of O(N log N) due to sorting, with O(N) space complexity for the result.

[The Punchline] This solution is efficient, robust, and handles all edge cases gracefully, from empty inputs to fully overlapping or non-overlapping sets. It's a production-ready approach that balances performance with code clarity and maintainability, making it suitable for scalable systems.

Common Mistakes:

• Not sorting the intervals: Attempting to merge intervals without first sorting them by their start times leads to a much more complex and often incorrect algorithm, typically requiring nested loops and resulting in a suboptimal O(N^2) time complexity.
• Incorrect overlap condition: A common error is using an imprecise condition for overlap, such as current.start < last_merged.end instead of current.start <= last_merged.end. This can fail to merge intervals that touch at their boundaries, like [1,2] and [2,3].
• Failing to update the maximum end: When merging, candidates sometimes forget to take max(last_merged.end, current.end), instead simply using current.end. This can result in a merged interval that doesn't fully encompass the original range if the current interval is contained within the last merged one but has a smaller end time.
• Poor handling of edge cases: Not considering an empty input array, an array with a single interval, or an array where no intervals overlap can lead to runtime errors or incorrect results. The solution should gracefully handle these scenarios.
• Modifying the original array in-place incorrectly: While an in-place modification might seem appealing for space optimization, it often complicates the logic and can lead to subtle bugs if not handled with extreme care, especially when elements are removed or shifted. For clarity and safety, returning a new list is generally preferred unless strict memory constraints dictate otherwise.
• Lack of clarity and readability: Even with a correct algorithm, poorly structured code, unclear variable names, or missing comments can make the solution difficult to understand, debug, and maintain, which is a significant red flag in a technical interview.