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2050. Parallel Courses III

{leetcode}/problems/parallel-courses-iii/[LeetCode - 2050. 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] = [prevCourse<sub>j</sub>, nextCourse<sub>j</sub>] denotes that course prevCourse<sub>j</sub> has to be completed before course nextCourse<sub>j</sub> (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: <img alt="" src="https://assets.leetcode.com/uploads/2021/10/07/ex1.png" style="width: 392px; height: 232px;" />

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

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 ⇐ prevCourse<sub>j</sub>, nextCourse<sub>j</sub> ⇐ n

  • prevCourse<sub>j</sub> != nextCourse<sub>j</sub>

  • All the pairs [prevCourse<sub>j</sub>, nextCourse<sub>j</sub>] are unique.

  • time.length == n

  • 1 ⇐ time[i] ⇐ 104

  • The given graph is a directed acyclic graph.

思路分析

一刷
link:{sourcedir}/_2050_ParallelCoursesIii.java[role=include]

参考资料