Solution: Parallel Courses III

Statement  

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:

• $1 \leq n \leq 5 \times 10^4$ ...