Mastering the "Merge Intervals" Problem: Approach and Complexity Analysis...

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.

Example 1:

Input: intervals = [[1,3],[2,6],[8,10],[15,18]]
Output: [[1,6],[8,10],[15,18]]
Explanation: Since intervals [1,3] and [2,6] overlap, merge them into [1,6].
        

Example 2:

Input: intervals = [[1,4],[4,5]]
Output: [[1,5]]
Explanation: Intervals [1,4] and [4,5] are considered overlapping.        

The "Merge Intervals" problem is a common challenge in coding interviews and on platforms such as LeetCode. It involves merging all overlapping intervals in a collection to produce a list where no intervals overlap. This problem is key in understanding array manipulation and sorting algorithms. Let's dissect the solution steps for this problem and evaluate its time and space complexity.

Solution Steps

1. Sort the Intervals: Begin by sorting the intervals based on their starting points. This is crucial as it aligns intervals in a sequence where overlapping intervals are next to each other.
2. Initialize a Result Array: Create an array to hold the merged intervals.
3. Iterate Through the Sorted Intervals: Go through each interval in the sorted array and merge them as needed.Start with the first interval and compare it with the next one.If the current interval overlaps with the next (the end of the current interval is greater than or equal to the start of the next), merge them by updating the end of the current interval to the maximum of the ends of both intervals.If there’s no overlap, add the current interval to the result array and move to the next interval.
4. Finalize the Result: Once all intervals have been processed, add the last interval to the result array.
5. Return the Merged Intervals: The result array now contains all merged intervals without any overlaps.

Time Complexity

1. Sorting the Intervals: The initial sorting of the intervals typically uses an algorithm like QuickSort or MergeSort, with a time complexity of O(n log n), where n is the number of intervals.
2. Single Iteration for Merging: After sorting, the algorithm goes through the intervals only once. This iteration is O(n), as each interval is checked at most once.

Combining these, the overall time complexity of the "Merge Intervals" solution is O(n log n).

Space Complexity

1. Space for Sorting: Depending on the sorting algorithm used, the space complexity can vary. For instance, if an in-place sort is used, the space complexity can be O(1). However, if a sort like MergeSort is used, it could be O(n).
2. Space for the Result Array: The space needed for the result array is O(n) in the worst case, where no intervals are merged.

Hence, the worst-case space complexity can be O(n), considering the space required for sorting and the result array.

Conclusion

The "Merge Intervals" problem's solution is an elegant demonstration of the effectiveness of sorting in simplifying complex problems. With a time complexity of O(n log n) and a space complexity that can be as low as O(1) depending on the sorting algorithm used, it represents a balanced approach to handling array manipulation tasks. This problem not only highlights key concepts in algorithm design but also is a practical application in various fields like calendar event scheduling, resource allocation, and more. Understanding this solution is valuable for programmers in both algorithmic problem-solving and real-world application development....