You are tasked with determining the minimum time required to complete a set of courses, given their prerequisite relationships and individual durations.

There are n courses labeled from 1 to n. The prerequisite relationships between these courses are provided as a 2D integer array relations, where each entry $\text{relations}[j] = [\text{prevCourse}_j, \text{nextCourse}_j]$ indicates that course $\text{prevCourse}_j$ must be completed before course $\text{nextCourse}_j$. An array, time, is also given, where time[i] specifies the number of months required to complete the course labeled i + 1.

You may begin any course as soon as you have completed all its prerequisites. If all the prerequisites are satisfied, multiple courses may be taken simultaneously. Calculate the minimum months needed to finish all the courses.

Note: The given prerequisite structure is guaranteed to form a directed acyclic graph (DAG), ensuring that all courses can be completed.

Constraints:

• $1 \leq n \leq 5 \times 10^4$

• $0 \leq$ relations.length $\leq \min\left(n \cdot (n - 1) / 2, 5 \times 10^4\right)$

• relations[j].length $= 2$

• $1 \leq \text{prevCourse}_j$, $\text{nextCourse}_j \leq n$

• $\text{prevCourse}_j \neq \text{nextCourse}_j$

• All pairs $\text{prevCourse}_j$, $\text{nextCourse}_j$ are unique.

• time.length $= n$

• $1 \leq$ time[i] $\leq 10^4$

• The prerequisite graph is a directed acyclic graph (DAG).

Let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

We have a game for you to play. Rearrange the logical building blocks to develop a clearer understanding on how to solve this problem.

Implement your solution in the following coding playground.