Parallel Courses III

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.