Maximum Team Size with Overlapping Intervals - Solution & Explanation

Problem Overview: You are given multiple time intervals representing when each member is available. The task is to determine the largest number of intervals that overlap at any moment. That value represents the maximum possible team size that can work together simultaneously.

Approach 1: Brute Force Interval Comparison (O(n²) time, O(1) space)

The straightforward method checks every interval against all others and counts how many overlap with it. For each interval i, iterate through the remaining intervals and test overlap conditions such as start_j ≤ end_i and end_j ≥ start_i. Track the maximum overlap count encountered. This approach is easy to reason about but becomes slow when the number of intervals grows, since every pair is compared.

Approach 2: Sweep Line with Sorted Events (O(n log n) time, O(n) space)

Transform each interval into two events: a start event (time, +1) and an end event (time, -1). Sort all events by time. Then iterate through the events while maintaining a running counter of active intervals. Each start increases the counter and each end decreases it. The maximum value reached during the scan is the largest overlapping group. This pattern is known as the sweep line technique and is commonly used in interval problems.

Approach 3: Two Sorted Arrays (Starts and Ends) (O(n log n) time, O(n) space)

Extract all start times into one array and all end times into another. Sort both arrays. Use two pointers: one iterating through starts and one through ends. When the next start time is less than or equal to the current end time, a new interval begins before the previous one ends, so increment the active team count and move the start pointer. Otherwise move the end pointer to close an interval. Track the maximum active count during the scan. This approach is conceptually similar to the sweep line but avoids building explicit event objects and relies only on sorting.

Recommended for interviews: The sweep line or two‑pointer sorted approach is what interviewers usually expect. Brute force shows you understand how overlaps work, but the optimal solution demonstrates knowledge of event ordering and efficient interval processing. Both optimal approaches run in O(n log n) due to sorting and handle large datasets comfortably.