Parallel Courses III

Parallel Courses III 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.

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

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:

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