Mastering the Merge Intervals Pattern: A Comprehensive Guide

The merge intervals pattern is a powerful algorithmic technique that’s frequently used in coding interviews and real-world programming scenarios. In this blog post, we’ll dive deep into this pattern, exploring its concepts, implementation, and common variations.

What is the Merge Intervals Pattern?

The merge intervals pattern is used to deal with overlapping intervals. It’s particularly useful when you need to either combine overlapping intervals or find the overlapping parts between intervals.

Some common problems that can be solved using this pattern include:

1. Merging overlapping intervals
2. Finding the intersection of intervals
3. Determining if a point is covered by any interval
4. Calculating the total time covered by a set of intervals

The Basic Algorithm

The core idea behind the merge intervals pattern is simple:

1. Sort the intervals based on their start times.
2. Iterate through the sorted intervals and merge overlapping ones.

Let’s look at a Python implementation for merging overlapping intervals:

def merge_intervals(intervals):
    # Sort intervals based on start time
    intervals.sort(key=lambda x: x[0])
    
    merged = []
    for interval in intervals:
        # If merged is empty or if there's no overlap,
        # simply append the current interval
        if not merged or merged[-1][1] < interval[0]:
            merged.append(interval)
        else:
            # There is an overlap, so we merge the current and previous intervals
            merged[-1][1] = max(merged[-1][1], interval[1])
    
    return merged

# Example usage
intervals = [[1,3],[2,6],[8,10],[15,18]]
print(merge_intervals(intervals))
# Output: [[1,6],[8,10],[15,18]]

Time and Space Complexity

• Time Complexity: O(n log n), where n is the number of intervals. This is due to the sorting step.
• Space Complexity: O(n) to store the merged intervals.

Variations and Related Problems

1. Insert Interval: Given a set of non-overlapping intervals and a new interval, insert the new interval and merge if necessary.

2. Interval List Intersections: Given two lists of closed intervals, find the intersection of these two lists.

3. Employee Free Time: Given the working hours of multiple employees, find the common free time for all employees.

4. Meeting Rooms: Determine if a person can attend all meetings given a list of meeting time intervals.

Real-world Applications

The merge intervals pattern has numerous practical applications:

1. Calendar applications for finding free time slots
2. Resource allocation in operating systems
3. Network bandwidth allocation
4. Flight scheduling systems

Conclusion

Mastering the merge intervals pattern is crucial for tackling a wide range of problems involving time, scheduling, and resource allocation. By understanding the core concept and practicing various problem variations, you’ll be well-equipped to handle these types of challenges in both coding interviews and real-world scenarios.

Remember, the key to solving merge interval problems efficiently is to sort the intervals first and then process them in order. Happy coding!