LeetCode 210: Course Schedule II — Python Solution

Solve LeetCode 210: Course Schedule II in Python with a topological ordering approach. The key is to make the state invariant explicit, so the implementation and complexity follow naturally.

This guide paraphrases the task and does not reproduce LeetCode’s prompt. Use the official page for the complete statement, examples, constraints, and submission runner.

What the problem is testing

Use Kahn’s algorithm and append each removed zero-indegree course to the order; return it only if every course is processed.

Algorithm

1. Use Kahn’s algorithm and append each removed zero-indegree course to the order; return it only if every course is processed.
2. Maintain this invariant: Every appended course has all prerequisites earlier in the output.
3. Continue until every input item or reachable state has been resolved, then return the accumulated result.

Python solution

from collections import Counter, defaultdict, deque, OrderedDict
import random

class Solution:
    def findOrder(self, numCourses, prerequisites):
        graph = [[] for _ in range(numCourses)]; indegree = [0] * numCourses
        for course, prerequisite in prerequisites:
            graph[prerequisite].append(course); indegree[course] += 1
        queue = deque(i for i, degree in enumerate(indegree) if degree == 0)
        order = []
        while queue:
            course = queue.popleft(); order.append(course)
            for following in graph[course]:
                indegree[following] -= 1
                if indegree[following] == 0: queue.append(following)
        return order if len(order) == numCourses else []

Why this is correct

The proof follows the maintained state: Every appended course has all prerequisites earlier in the output. Each iteration preserves that claim while permanently resolving at least one position, node, interval, or search state. When the loop or recursion ends, every candidate required by the problem has therefore been included or ruled out, so the returned value is correct.

Complexity

O(V+E) time and O(V+E) space. The stated auxiliary space excludes the returned output unless the output is the data structure being built.

Edge cases

A cycle returns an empty list; isolated courses begin with zero indegree.

Tested reference code

This implementation is included in the site’s downloadable 100-solution Python library. The complete suite compiles every solution and runs a behavioral assertion for every problem before publication.

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