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>

minimumTime

Parallel Courses III

Course Schedule III

course-schedule-iii

课程表 III

Parallel Courses

parallel-courses

并行课程

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单线程 CPU

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使用服务器处理任务

Maximum Employees to Be Invited to a Meeting

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参加会议的最多员工数

Graph

Topological Sort

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Dynamic Programming

给你一个整数&nbsp;<code>n</code>&nbsp;，表示有&nbsp;<code>n</code>&nbsp;节课，课程编号从&nbsp;<code>1</code>&nbsp;到&nbsp;<code>n</code>&nbsp;。同时给你一个二维整数数组&nbsp;<code>relations</code>&nbsp;，其中&nbsp;<code>relations[j] = [prevCoursej, nextCoursej]</code>&nbsp;，表示课程&nbsp;<code>prevCoursej</code>&nbsp;必须在课程&nbsp;<code>nextCoursej</code>&nbsp;之前&nbsp;完成（先修课的关系）。同时给你一个下标从 0&nbsp;开始的整数数组&nbsp;<code>time</code>&nbsp;，其中&nbsp;<code>time[i]</code>&nbsp;表示完成第&nbsp;<code>(i+1)</code>&nbsp;门课程需要花费的 月份&nbsp;数。

请你根据以下规则算出完成所有课程所需要的 最少&nbsp;月份数：

<ul>
	<li>如果一门课的所有先修课都已经完成，你可以在 任意&nbsp;时间开始这门课程。</li>
	<li>你可以&nbsp;同时&nbsp;上&nbsp;任意门课程&nbsp;。</li>
</ul>

请你返回完成所有课程所需要的 最少&nbsp;月份数。

注意：测试数据保证一定可以完成所有课程（也就是先修课的关系构成一个有向无环图）。

&nbsp;

示例&nbsp;1:

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

<pre>输入：n = 3, relations = [[1,3],[2,3]], time = [3,2,5]
输出：8
解释：上图展示了输入数据所表示的先修关系图，以及完成每门课程需要花费的时间。
你可以在月份 0 同时开始课程 1 和 2 。
课程 1 花费 3 个月，课程 2 花费 2 个月。
所以，最早开始课程 3 的时间是月份 3 ，完成所有课程所需时间为 3 + 5 = 8 个月。
</pre>

示例 2：

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

<pre>输入：n = 5, relations = [[1,5],[2,5],[3,5],[3,4],[4,5]], time = [1,2,3,4,5]
输出：12
解释：上图展示了输入数据所表示的先修关系图，以及完成每门课程需要花费的时间。
你可以在月份 0 同时开始课程 1 ，2 和 3 。
在月份 1，2 和 3 分别完成这三门课程。
课程 4 需在课程 3 之后开始，也就是 3 个月后。课程 4 在 3 + 4 = 7 月完成。
课程 5 需在课程 1，2，3 和 4 之后开始，也就是在 max(1,2,3,7) = 7 月开始。
所以完成所有课程所需的最少时间为 7 + 5 = 12 个月。
</pre>

&nbsp;

提示：

<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>所有的先修课程对&nbsp;<code>[prevCoursej, nextCoursej]</code>&nbsp;都是 互不相同&nbsp;的。</li>
	<li><code>time.length == n</code></li>
	<li><code>1 &lt;= time[i] &lt;= 104</code></li>
	<li>先修课程图是一个有向无环图。</li>
</ul>