DSA: Implement a scheduler that executes jobs with dependency constraints

Start by clarifying the exact contract for Dependency Scheduler: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is topological scheduling and cycle reporting. 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 build prerequisite-to-dependent edges and indegrees; enqueue zero-indegree jobs; process and release dependents; report cycle if count is short. A valid order does not by itself handle job retries or failure semantics. 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 O(V + E) time and space. 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.