What is the optimal time complexity of solving Course Schedule II?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Course Schedule II?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Course Schedule II?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Course Schedule II?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Course Schedule II

Learn how to solve the 'Course Schedule II' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

There are a total of numCourses courses you have to take, labeled from 0 to numCourses - 1. You are given an array prerequisites where prerequisites[i] = [ai, bi] indicates that you must take course bi first if you want to take course ai.

Return the ordering of courses you should take to finish all courses. If there are many valid answers, return any of them. If it is impossible to finish all courses, return an empty array.

Write a function findOrder(numCourses: int, prerequisites: List[List[int]]) -> List[int].

Constraints

•1 <= numCourses <= 2000
•0 <= len(prerequisites) <= 5000
•prerequisites[i].length == 2
•0 <= ai, bi < numCourses

Examples

Example 1

Input

numCourses = 2, prerequisites = [[1,0]]

Output

[0,1]

Explanation

There are a total of 2 courses to take. To take course 1 you should have finished course 0. So the correct course order is [0,1].

Example 2

Input

numCourses = 4, prerequisites = [[1,0],[2,0],[3,1],[3,2]]

Output

[0,2,1,3]

Explanation

There are a total of 4 courses to take. To take course 3 you should have finished both courses 1 and 2. Both courses 1 and 2 should be taken after you finished course 0. So one correct course order is [0,1,2,3]. Another correct ordering is [0,2,1,3].

Need a Hint?

Represent graph node connections as an adjacency list/matrix, then use standard BFS or DFS graph traversal.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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