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"title": "Parallel Courses III",
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"content": "<p>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] = [prevCourse<sub>j</sub>, nextCourse<sub>j</sub>]</code> denotes that course <code>prevCourse<sub>j</sub></code> has to be completed <strong>before</strong> course <code>nextCourse<sub>j</sub></code> (prerequisite relationship). Furthermore, you are given a <strong>0-indexed</strong> integer array <code>time</code> where <code>time[i]</code> denotes how many <strong>months</strong> it takes to complete the <code>(i+1)<sup>th</sup></code> course.</p>\n\n<p>You must find the <strong>minimum</strong> number of months needed to complete all the courses following these rules:</p>\n\n<ul>\n\t<li>You may start taking a course at <strong>any time</strong> if the prerequisites are met.</li>\n\t<li><strong>Any number of courses</strong> can be taken at the <strong>same time</strong>.</li>\n</ul>\n\n<p>Return <em>the <strong>minimum</strong> number of months needed to complete all the courses</em>.</p>\n\n<p><strong>Note:</strong> The test cases are generated such that it is possible to complete every course (i.e., the graph is a directed acyclic graph).</p>\n\n<p> </p>\n<p><strong>Example 1:</strong></p>\n<strong><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2021/10/07/ex1.png\" style=\"width: 392px; height: 232px;\" /></strong>\n\n<pre>\n<strong>Input:</strong> n = 3, relations = [[1,3],[2,3]], time = [3,2,5]\n<strong>Output:</strong> 8\n<strong>Explanation:</strong> The figure above represents the given graph and the time required to complete each course. \nWe start course 1 and course 2 simultaneously at month 0.\nCourse 1 takes 3 months and course 2 takes 2 months to complete respectively.\nThus, the earliest time we can start course 3 is at month 3, and the total time required is 3 + 5 = 8 months.\n</pre>\n\n<p><strong>Example 2:</strong></p>\n<strong><img alt=\"\" src=\"https://assets.leetcode.com/uploads/2021/10/07/ex2.png\" style=\"width: 500px; height: 365px;\" /></strong>\n\n<pre>\n<strong>Input:</strong> n = 5, relations = [[1,5],[2,5],[3,5],[3,4],[4,5]], time = [1,2,3,4,5]\n<strong>Output:</strong> 12\n<strong>Explanation:</strong> The figure above represents the given graph and the time required to complete each course.\nYou can start courses 1, 2, and 3 at month 0.\nYou can complete them after 1, 2, and 3 months respectively.\nCourse 4 can be taken only after course 3 is completed, i.e., after 3 months. It is completed after 3 + 4 = 7 months.\nCourse 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.\nThus, the minimum time needed to complete all the courses is 7 + 5 = 12 months.\n</pre>\n\n<p> </p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>1 <= n <= 5 * 10<sup>4</sup></code></li>\n\t<li><code>0 <= relations.length <= min(n * (n - 1) / 2, 5 * 10<sup>4</sup>)</code></li>\n\t<li><code>relations[j].length == 2</code></li>\n\t<li><code>1 <= prevCourse<sub>j</sub>, nextCourse<sub>j</sub> <= n</code></li>\n\t<li><code>prevCourse<sub>j</sub> != nextCourse<sub>j</sub></code></li>\n\t<li>All the pairs <code>[prevCourse<sub>j</sub>, nextCourse<sub>j</sub>]</code> are <strong>unique</strong>.</li>\n\t<li><code>time.length == n</code></li>\n\t<li><code>1 <= time[i] <= 10<sup>4</sup></code></li>\n\t<li>The given graph is a directed acyclic graph.</li>\n</ul>\n",
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