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What is the time complexity of the binary search + preprocessing approach for the Minimum Interval to Include Each Query problem with n intervals and m queries?

medium🪤 Complexity Trap Q5 of Q15

Intervals - Minimum Interval to Include Each Query

What is the time complexity of the binary search + preprocessing approach for the Minimum Interval to Include Each Query problem with n intervals and m queries?

AO(n log n + m n log n)

BO((n + m) log n)

CO(n m)

DO(n log n + m log n)

Step-by-Step Solution

Solution:

Step 1: Analyze preprocessing
Sorting intervals by length takes O(n log n).
Step 2: Analyze query processing
For each query, binary search over lengths is O(log n), but coverage check scans up to O(n) intervals, resulting in O(n log n) per query.
Step 3: Combine complexities
Total time is O(n log n + m * n log n).
Final Answer:
Option A -> Option A
Quick Check:
Coverage check inside binary search causes O(n) factor per query [OK]

Quick Trick: Coverage check inside binary search causes O(n) per query [OK]

Common Mistakes:

MISTAKES

Assuming coverage check is O(log n) leading to O((n+m) log n)

Trap Explanation:

PITFALL

Candidates often forget coverage check scans intervals linearly, inflating complexity.

Interviewer Note:

CONTEXT

Tests understanding of nested binary search with linear coverage checks

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