This problem requires finding the smallest interval that contains a query. By sorting the intervals and utilizing binary search, we can efficiently solve the problem. The main challenge is to balance between interval sorting and query lookup to ensure minimal computational time.
Problem Statement
You are given a 2D integer array intervals, where each element represents an interval described by two integers: a starting point and an endpoint. The size of an interval is determined by the number of integers it contains, or more formally, by the difference between the endpoint and the start point plus one.
You are also given an array of integer queries. For each query, find the smallest interval that contains the query. If no interval contains the query, return -1. The task is to return an array containing the answers for all the queries.
Examples
Example 1
Input: intervals = [[1,4],[2,4],[3,6],[4,4]], queries = [2,3,4,5]
Output: [3,3,1,4]
The queries are processed as follows:
- Query = 2: The interval [2,4] is the smallest interval containing 2. The answer is 4 - 2 + 1 = 3.
- Query = 3: The interval [2,4] is the smallest interval containing 3. The answer is 4 - 2 + 1 = 3.
- Query = 4: The interval [4,4] is the smallest interval containing 4. The answer is 4 - 4 + 1 = 1.
- Query = 5: The interval [3,6] is the smallest interval containing 5. The answer is 6 - 3 + 1 = 4.
Example 2
Input: intervals = [[2,3],[2,5],[1,8],[20,25]], queries = [2,19,5,22]
Output: [2,-1,4,6]
The queries are processed as follows:
- Query = 2: The interval [2,3] is the smallest interval containing 2. The answer is 3 - 2 + 1 = 2.
- Query = 19: None of the intervals contain 19. The answer is -1.
- Query = 5: The interval [2,5] is the smallest interval containing 5. The answer is 5 - 2 + 1 = 4.
- Query = 22: The interval [20,25] is the smallest interval containing 22. The answer is 25 - 20 + 1 = 6.
Constraints
- 1 <= intervals.length <= 105
- 1 <= queries.length <= 105
- intervals[i].length == 2
- 1 <= lefti <= righti <= 107
- 1 <= queries[j] <= 107
Solution Approach
Sort the Intervals
First, sort the intervals by their starting points. Sorting ensures that we can process the intervals in an ordered manner, allowing us to make efficient queries.
Binary Search for Query
For each query, apply binary search to find the smallest interval that contains it. The binary search minimizes the time it takes to search through the intervals.
Optimize with a Min-Heap
Use a min-heap to efficiently manage the intervals that overlap the current query. The heap helps in finding the smallest interval with the least computation overhead.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity depends on the sorting of intervals and the binary search process. Sorting the intervals takes O(n log n), and for each query, binary search can be applied in O(log n). The total complexity is O(n log n + m log n), where n is the number of intervals and m is the number of queries.
What Interviewers Usually Probe
- Candidate understands how to use sorting to optimize interval processing.
- Candidate demonstrates knowledge of binary search in finding the correct interval.
- Candidate correctly applies a heap to minimize computational time during interval queries.
Common Pitfalls or Variants
Common pitfalls
- Not sorting the intervals before applying binary search leads to inefficient solutions.
- Failing to use the heap structure efficiently could result in higher-than-expected complexity.
- Not handling cases where no interval contains the query could lead to incorrect results.
Follow-up variants
- What if the intervals are already sorted?
- How would you handle queries that are out of bounds of all the intervals?
- What if you need to handle a larger range of values, say up to 10^9?
How GhostInterview Helps
- GhostInterview helps by guiding you through the steps to optimize your approach for sorting, searching, and using data structures.
- It encourages focusing on the binary search pattern and teaches you how to integrate additional optimization techniques like heap usage.
- GhostInterview provides hints and suggests better time complexities, helping you streamline your approach to solve this problem efficiently.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the core pattern in solving 'Minimum Interval to Include Each Query'?
The core pattern involves binary search to find the smallest interval containing each query, with optimizations like sorting intervals and using a heap.
How does binary search help in solving this problem?
Binary search helps by efficiently narrowing down the search space for each query, minimizing the number of comparisons needed to find the valid interval.
What is the time complexity of this problem?
The time complexity is O(n log n + m log n), where n is the number of intervals and m is the number of queries.
What are common pitfalls when solving this problem?
Common pitfalls include not sorting intervals, not using a heap to manage active intervals, and missing edge cases where no interval contains a query.
Can GhostInterview help with advanced variations of this problem?
Yes, GhostInterview can provide hints and optimizations for handling larger ranges, pre-sorted intervals, and other advanced variations of this problem.
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