parallel-courses-iii

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Algorithms

You are given an integer <code>n</code>, which indicates that there are <code>n</code> courses labeled from <code>1</code> to <code>n</code>. You are also given a 2D integer array <code>relations</code> where <code>relations[j] = [prevCoursej, nextCoursej]</code> denotes that course <code>prevCoursej</code> has to be completed before course <code>nextCoursej</code> (prerequisite relationship). Furthermore, you are given a 0-indexed integer array <code>time</code> where <code>time[i]</code> denotes how many months it takes to complete the <code>(i+1)th</code> course.

You must find the minimum number of months needed to complete all the courses following these rules:

<ul>
	<li>You may start taking a course at any time if the prerequisites are met.</li>
	<li>Any number of courses can be taken at the same time.</li>
</ul>

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).

&nbsp;
Example 1:
<img alt="" src="https://assets.leetcode.com/uploads/2021/10/07/ex1.png" style="width: 392px; height: 232px;" />

<pre>
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.
</pre>

Example 2:
<img alt="" src="https://assets.leetcode.com/uploads/2021/10/07/ex2.png" style="width: 500px; height: 365px;" />

<pre>
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.
</pre>

&nbsp;
Constraints:

<ul>
	<li><code>1 &lt;= n &lt;= 5 * 104</code></li>
	<li><code>0 &lt;= relations.length &lt;= min(n * (n - 1) / 2, 5 * 104)</code></li>
	<li><code>relations[j].length == 2</code></li>
	<li><code>1 &lt;= prevCoursej, nextCoursej &lt;= n</code></li>
	<li><code>prevCoursej != nextCoursej</code></li>
	<li>All the pairs <code>[prevCoursej, nextCoursej]</code> are unique.</li>
	<li><code>time.length == n</code></li>
	<li><code>1 &lt;= time[i] &lt;= 104</code></li>
	<li>The given graph is a directed acyclic graph.</li>
</ul>