I went with the classic event-based sweep: split each interval into a +1 at start and -1 at end, sort the events, then scan.
Use a sweep-line algorithm: create events for each trip's start (+1) and end (-1), sort them by time, and sweep through to track the current number of active trips. Handle half-open intervals by processing all starts before ends at the same timestamp. Track the maximum count and the timestamp when it first occurs.
Pro tip: Clarify that the peak timestamp can be any time during the peak interval, and mention that if multiple peaks exist, returning the earliest is a common convention. Also, note that O(1) extra space is achievable by sorting the events in-place or using the input arrays with two pointers after sorting.
Confirm the definition of half-open intervals, what to return if multiple peaks occur, and whether the timestamp should be the start of the peak or any time during it. Discuss edge cases like no trips, all trips overlapping, or trips with zero duration.
Explain creating events: for each trip, a start event at start time with +1 and an end event at end time with -1. Sort events by time, and for ties, process starts before ends to respect half-open intervals.
Initialize current count and max count to 0, and peak time to None. Iterate through sorted events, updating current count, and when current count exceeds max count, update max count and record the event time as peak time.
State that sorting takes O(n log n) time, and sweeping takes O(n) time, so overall O(n log n). For O(1) extra space, avoid creating a separate events array by sorting the input intervals and using two pointers to simulate the sweep.
Walk through a small example to verify correctness, especially the half-open interval handling. Discuss trade-offs: the two-pointer approach may be more complex but meets O(1) space, while the events array is simpler but uses O(n) space.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the original problem: likely given a list of intervals, find the maximum number of overlapping intervals (max concurrency). Then, to find the smallest contiguous time range [L, R) where this maximum holds without dropping, use a sweep line algorithm with events (start and end times). Track the current concurrency and the start of the current maximal segment; when concurrency reaches the maximum, record the segment's start; when it drops below the maximum, compute the segment's length and update the best range if it's the smallest so far.
Pro tip: Mention that you would handle edge cases like multiple disjoint maximal segments and ties by length, and that you would confirm whether the range should be inclusive/exclusive and whether zero-length ranges are allowed. Also, note that if the maximum concurrency is zero (no intervals), the range is undefined or empty.
Confirm the input format (list of intervals), definition of concurrency (number of overlapping intervals at a time), and what 'smallest contiguous time range' means (minimum length R-L). Ask about edge cases: empty input, multiple segments, ties.
Use a sweep line algorithm: create events for interval starts (+1) and ends (-1), sort by time, and track the running sum to find the maximum concurrency value.
During the sweep, whenever the concurrency reaches the maximum, mark the start of a segment; when it drops below the maximum, mark the end. Collect all such [start, end) segments.
Among all maximal segments, compute their lengths (end - start) and select the one with the smallest length. If there are ties, decide which to return (e.g., the earliest).
If no intervals or maximum concurrency is 0, return an appropriate value (e.g., null or empty range). Otherwise, return the smallest [L, R).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Acknowledge the shift from batch to streaming and propose a streaming algorithm that processes data in one pass with limited memory. Focus on maintaining approximate statistics or sketches, and discuss trade-offs between accuracy, memory, and latency. Highlight the need for incremental updates and potential use of windowing or decay.
Pro tip: Mention that you would first clarify the business goal (e.g., real-time monitoring vs. offline analytics) because it determines whether approximate answers suffice or if exact results are required, which influences algorithm choice.
Ask about the specific use case, required accuracy, latency constraints, and available memory. Determine if exact answers are needed or if approximations are acceptable.
Select appropriate algorithms like reservoir sampling, count-min sketch, HyperLogLog, or t-digest for quantiles, depending on the metric (e.g., average, distinct count, percentiles).
Ensure the algorithm can update its state with each new data point in O(1) or O(log n) time and memory, and handle out-of-order or late data if necessary.
Discuss the trade-offs between accuracy, memory usage, and computational complexity. Explain how to tune parameters (e.g., sketch size) to balance these.
If the stream is unbounded, propose sliding windows or exponential decay to focus on recent data and bound memory usage.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.