Order Courses by Prerequisites, Returning the Smallest Valid Order

There are `num_courses` courses, numbered `0` to `num_courses - 1`. Some courses must be completed before others: each pair `[a, b]` in `prerequisites` means course `b` has to be finished before course `a` can be taken. Return an order for taking all of the courses in which every course appears after all of its prerequisites. When more than one order is valid, return the lexicographically smallest one: comparing two valid orders position by position, the first position at which they differ must hold the smaller course number in your answer. If no valid order exists because the prerequisites form a cycle, return an empty list. ### Function Signature ```python def order_courses(num_courses: int, prerequisites: List[List[int]]) -> List[int]: ``` ### Rules - A pair `[a, b]` means `b` comes somewhere before `a`; it says nothing about how far apart they are. - A course with no prerequisites may appear anywhere the ordering rule allows. - A non-empty answer contains every course exactly once. ### Constraints - `1 <= num_courses <= 2000` - `0 <= len(prerequisites) <= 5000` - Every pair has exactly two elements `[a, b]` with `0 <= a, b < num_courses` and `a != b`. - No pair appears twice. ### Examples **Example 1** ``` Input: num_courses = 2, prerequisites = [[1, 0]] Output: [0, 1] ``` Course 1 requires course 0, so 0 must come first. **Example 2** ``` Input: num_courses = 4, prerequisites = [[0, 3], [1, 3]] Output: [2, 3, 0, 1] ``` Courses 0 and 1 both require course 3, so a valid order starts with 2 or 3. Of all valid orders, `[2, 3, 0, 1]` is the smallest. **Example 3** ``` Input: num_courses = 3, prerequisites = [[0, 1], [1, 2], [2, 0]] Output: [] ``` Course 0 needs 1, course 1 needs 2 and course 2 needs 0, so no course can be taken first.