2050. Parallel Courses III

Parallel Courses III (LeetCode 2050) is a Hard Data Structures and Algorithms interview problem you can solve, run and submit on Thita.ai. It belongs to the Graph Traversal Patterns (DFS & BFS) pattern, in the Graph BFS - Topological Sort (Kahn's Algorithm) subpattern.

How to solve Parallel Courses III

Parallel Courses III is a Topological Sort problem, part of Graph Traversal Patterns (DFS & BFS). Master topological sorting using Kahn's BFS algorithm. Solve course schedule and build order problems.

Work towards the solution in these steps:

1. What is the earliest time a course can be taken?
2. How would you solve the problem if all courses take equal time?
3. How would you generalize this approach?

The reference solution runs in O(|V| + |E|) time and O(|E|) space.

The Parallel Courses III solution and editorial walks through the full approach with reference code.

Problem statement

You are given an integer n, which indicates that there are n courses labeled from 1 to n. You are also given a 2D integer array relations where relations[j] = [prevCoursej, nextCoursej] denotes that course prevCoursej has to be completed before course nextCoursej (prerequisite relationship). Furthermore, you are given a 0-indexed integer array time where time[i] denotes how many months it takes to complete the (i+1)th course.

You must find the minimum number of months needed to complete all the courses following these rules:

- You may start taking a course at any time if the prerequisites are met.

- Any number of courses can be taken at the same time.

Return the minimum number of months needed to complete all the courses.

Note: The test cases are generated such that it is possible to complete every course (i.e., the graph is a directed acyclic graph).

Example 1:

Input: n = 3, relations = [[1,3],[2,3]], time = [3,2,5] Output: 8 Explanation: The figure above represents the given graph and the time required to complete each course. We start course 1 and course 2 simultaneously at month 0. Course 1 takes 3 months and course 2 takes 2 months to complete respectively. Thus, the earliest time we can start course 3 is at month 3, and the total time required is 3 + 5 = 8 months.

Example 2:

Input: n = 5, relations = [[1,5],[2,5],[3,5],[3,4],[4,5]], time = [1,2,3,4,5] Output: 12 Explanation: The figure above represents the given graph and the time required to complete each course. You can start courses 1, 2, and 3 at month 0. You can complete them after 1, 2, and 3 months respectively. Course 4 can be taken only after course 3 is completed, i.e., after 3 months. It is completed after 3 + 4 = 7 months. Course 5 can be taken only after courses 1, 2, 3, and 4 have been completed, i.e., after max(1,2,3,7) = 7 months. Thus, the minimum time needed to complete all the courses is 7 + 5 = 12 months.

Constraints:

- 1 <= n <= 5 104

- 0 <= relations.length <= min(n (n - 1) / 2, 5 104)

- relations[j].length == 2

- 1 <= prevCoursej, nextCoursej <= n

- prevCoursej != nextCoursej

- All the pairs [prevCoursej, nextCoursej] are unique.

- time.length == n

- 1 <= time[i] <= 104

- The given graph is a directed acyclic graph.

Topics and companies

Topics: Array, Dynamic Programming, Graph, Topological Sort.

How to practise Parallel Courses III on Thita.ai

Starter code is provided in C, C#, C++, Go, Java, JavaScript and Python. Your solution runs against 5 test cases for this problem, with the AI coach available for a hint when you are stuck rather than a finished answer. The reference solution runs in O(|V| + |E|) time and O(|E|) space.

Open Parallel Courses III in the code editor, or read the Parallel Courses III editorial for a worked solution with its approach and complexity analysis.

Where Parallel Courses III sits in the DSA pattern sheet

• Graph Traversal Patterns (DFS & BFS) pattern guide
• Graph BFS - Topological Sort (Kahn's Algorithm) subpattern
• The full DSA patterns sheet

Problems related to Parallel Courses III

Other problems that use the same Graph BFS - Topological Sort (Kahn's Algorithm) and Graph Traversal Patterns (DFS & BFS) techniques:

• Alien Dictionary — Hard, same subpattern
• Build a Matrix With Conditions — Hard, same subpattern
• Course Schedule IV — Medium, same subpattern
• Find All Possible Recipes from Given Supplies — Medium, same subpattern
• Largest Color Value in a Directed Graph — Hard, same subpattern
• Minimum Height Trees — Medium, same subpattern

Parallel Courses III FAQ