2050\. Parallel Courses III

Hard

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 <code>relations[j] = [prevCourse<sub>j</sub>, nextCourse<sub>j</sub>]</code> denotes that course <code>prevCourse<sub>j</sub></code> has to be completed **before** course <code>nextCourse<sub>j</sub></code> (prerequisite relationship). Furthermore, you are given a **0-indexed** integer array `time` where `time[i]` denotes how many **months** it takes to complete the <code>(i+1)<sup>th</sup></code> 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:**

**![](https://assets.leetcode.com/uploads/2021/10/07/ex1.png)**

**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:**

**![](https://assets.leetcode.com/uploads/2021/10/07/ex2.png)**

**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:**

*   <code>1 <= n <= 5 * 10<sup>4</sup></code>
*   <code>0 <= relations.length <= min(n * (n - 1) / 2, 5 * 10<sup>4</sup>)</code>
*   `relations[j].length == 2`
*   <code>1 <= prevCourse<sub>j</sub>, nextCourse<sub>j</sub> <= n</code>
*   <code>prevCourse<sub>j</sub> != nextCourse<sub>j</sub></code>
*   All the pairs <code>[prevCourse<sub>j</sub>, nextCourse<sub>j</sub>]</code> are **unique**.
*   `time.length == n`
*   <code>1 <= time[i] <= 10<sup>4</sup></code>
*   The given graph is a directed acyclic graph.