Dynamic interval Data Structure

Start by clarifying the exact contract for Dynamic Interval Data Structure: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is matching interval data structure to exact query and update workload. Work a tiny example aloud and name the invariant after each step, because that is the proof that the approach stays correct as the input grows.

A dependable implementation plan is to clarify overlap stabbing or aggregate query; define endpoint convention; choose augmented tree or segment structure; describe updates deletion and output cost. There is no universal interval structure without a query contract. Keep the code aligned with that invariant instead of adding special cases that silently change the contract. Test the smallest input, a normal case, duplicates or ties where relevant, an adversarial boundary, and a case that exercises the failure path. The expected complexity is often O(log n) plus output, depending on structure. Explain both time and auxiliary memory, including recursion or helper structures; do not hide sorting, copying, or repeated scans. If the title omits critical constraints, say what you need to know before committing to a more specialized solution.