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The key move is reducing all the intervals down to a single range: take the max of all the left endpoints and the min of all the right endpoints.
Clarify that 'every interval simultaneously' means the intersection of all intervals, then compute that intersection. Sort the list of integers and use binary search to count how many fall within the intersection bounds, handling duplicates naturally.
Pro tip: Mention that if the intersection is empty, the answer is zero, and that sorting the integers once allows O(log n) counting per query if the problem is extended to multiple interval sets.
Confirm that 'every interval simultaneously' means the intersection of all intervals. Ask about edge cases: empty intervals, empty list, or intervals with no overlap.
Find the maximum of all lower bounds and the minimum of all upper bounds. If max_lower > min_upper, the intersection is empty and the answer is 0.
Sort the list of integers in ascending order. This enables efficient counting of elements within a range using binary search.
Use binary search to find the first index where the value is >= max_lower and the last index where the value is <= min_upper. The count is the difference between these indices plus one.
State the time complexity: O(n log n) for sorting plus O(log n) for binary search, and O(1) extra space. Discuss handling duplicates and empty inputs.
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