Below is a structured article addressing the problem, including intuition, algorithm, a dry run, code implementation in C++, and complexity analysis.

Inserting an Interval

Intuition

The problem is about merging intervals with a twist: we are adding a new interval to an already sorted list of intervals. Intervals may overlap with the new one, and the goal is to ensure the final list is sorted and does not contain any overlapping intervals.

Think of the intervals on a timeline. Inserting the new interval is like identifying where it belongs on the timeline and merging any overlaps that result.

The key insight:

• Before the new interval's starting position, existing intervals remain unchanged.
• Overlapping intervals need merging.
• Once the overlapping ends, the remaining intervals are appended as-is.

Algorithm

1. Initialize Variables:
   • Create a new result list to hold the merged intervals.
2. Iterate Through Intervals:
   • Add intervals before the overlap directly to the result if their end is less than the start of the new interval.
   • Merge intervals that overlap with the new interval. Update the start and end of the new interval to cover the merged range.
3. Add Remaining Intervals:
   • After processing all overlaps, add the remaining intervals from the original list directly to the result.
4. Return the Result.

Dry Run

Input: intervals = [[1,2],[3,5],[6,7],[8,10],[12,16]], newInterval = [4,8]

Output: [[1,2],[3,10],[12,16]]