Overlapping Intervals

The Overlapping Intervals pattern is a common technique used to solve problems related to merging, inserting, and manipulating intervals.

This pattern involves sorting intervals and then processing them to find overlaps, merge them, or insert new intervals.

Overlapping intervals are pairs of numbers that represent a range, where some ranges may overlap with others.

For example, in the intervals [(1,3), (2,6), (8,10), (15,18)], we can see that (1,3) and (2,6) overlap because they share common values.

Key Techniques for Handling Overlapping Intervals

1. Sorting: Most interval problems begin by sorting the intervals based on their start times. This allows for efficient processing of the intervals in a single pass.

2. Merging: When intervals overlap, we often need to merge them into a single, larger interval.

3. Comparison: We frequently need to compare the end of one interval with the start of the next to determine if they overlap.

4. Stack or List: We often use a stack or list to keep track of merged intervals or to store results.

Here's a general approach to solving overlapping interval problems:

1. Sort the intervals based on start time.

2. Initialize a result list (or stack).

3. Iterate through the sorted intervals:

   • If the result is empty or the current interval doesn't overlap with the previous interval, add it to the result.

   • If there is an overlap, merge the current interval with the previous one.

4. Return the result.

LeetCode Examples

1. Merge Intervals (LeetCode 56)

Problem Statement: Given an array of intervals where intervals[i] = [starti, endi], merge all overlapping intervals, and return an array of the non-overlapping intervals that cover all the intervals in the input.

Intuition:

To merge overlapping intervals, we need to first sort the intervals by their start times. Then, we can iterate through the sorted intervals and merge any overlapping intervals by comparing the end time of the current interval with the start time of the next interval.

Solution:

1. Sort the intervals by their start times.

2. Initialize a result list with the first interval.

3. Iterate through the sorted intervals and compare the end time of the last interval in the result list with the start time of the current interval.

4. If they overlap, merge them by updating the end time of the last interval in the result list.

5. If they do not overlap, add the current interval to the result list.

Code:

Time Complexity: O(n log n) due to sorting

Space Complexity: O(n) to store the result

2. Insert Interval (LeetCode 57)

Problem Statement: Given a set of non-overlapping intervals and a new interval, insert the new interval into the list (merge if necessary). You may assume that the intervals were initially sorted according to their start times.

Intuition:

To insert a new interval, we need to find the correct position to insert it and then merge any overlapping intervals.

By iterating through the list, we can insert the new interval and handle merging at the same time.

Approach:

1. Initialize a result list and a variable to track the position to insert the new interval.

2. Iterate through the intervals and add all intervals that end before the new interval starts to the result list.

3. Merge the new interval with any overlapping intervals.

4. Add the merged interval to the result list.

5. Add the remaining intervals to the result list.

Code:

Time Complexity: O(n)

Space Complexity: O(n) to store the result

2. Non-overlapping Intervals (LeetCode 435)

Problem Statement: Given an array of intervals, find the minimum number of intervals you need to remove to make the rest of the intervals non-overlapping.

Intuition:

To minimize the number of intervals to remove, we should always keep intervals with the earliest end time, as this gives the maximum room for subsequent intervals to fit without overlapping.

Approach:

1. Sort the intervals by their end times.

2. Initialize a variable to track the end time of the last added interval and a counter for the number of removed intervals.

3. Iterate through the sorted intervals and compare the start time of the current interval with the end time of the last added interval.

4. If they overlap, increment the counter for removed intervals.

5. If they do not overlap, update the end time of the last added interval.

Advanced Techniques

1. Sweep Line Algorithm: For more complex interval problems, especially those involving multiple types of events (like start and end of intervals), the sweep line algorithm can be very effective.

2. Interval Trees: For problems involving large numbers of intervals and frequent queries, an interval tree data structure can provide efficient solutions.

3. Segment Trees: When dealing with problems that involve range queries or updates, segment trees can be a powerful tool.

Conclusion

The Overlapping Intervals pattern is a powerful technique for solving a wide range of problems involving ranges or intervals.

Key points to remember:

1. Sorting intervals is often the first step in solving these problems.

2. Merging overlapping intervals is a common operation.

3. Using appropriate data structures (like heaps or trees) can significantly optimize solutions for more complex problems.

4. This pattern is particularly useful for scheduling, resource allocation, and time management problems.

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