## Problem There are `numCourses` courses numbered `0` through `numCourses - 1`. Each prerequisite pair `[course, prerequisite]` means the prerequisite must be completed first. Return whether it is possible to finish every course. ### Function Contract Implement `canFinishCourses(numCourses, prerequisites)`. ### Constraints & Assumptions - `1 <= numCourses <= 200,000`. - `0 <= len(prerequisites) <= 500,000`. - Pairs are valid course IDs but may be duplicated. - A self-dependency makes completion impossible. ### Clarifying Questions to Ask - Can duplicate prerequisite pairs appear? Yes; deduplicate edges or count and remove them consistently. - Is an ordering required? No, return only a boolean. - Are disconnected course groups allowed? Yes, all must be acyclic. - What prevents completion? Any directed cycle. ```hint Remove courses with no unmet prerequisites Build outgoing edges and indegrees, enqueue every zero-indegree course, and count how many can be removed. ``` ### Example ```text numCourses = 2 prerequisites = [[1,0]] output = true numCourses = 2 prerequisites = [[1,0], [0,1]] output = false ``` ### Evaluation Focus - Builds edge direction and indegrees correctly. - Handles duplicate edges without corrupting indegree counts. - Processes disconnected nodes and isolated courses. - Runs in `O(v + e)` time and space. ### Extensions to Discuss 1. How would you return one valid course order? 2. How could DFS produce an explicit cycle? 3. How would you update the result after adding one prerequisite edge?