Parallel Courses III

arraydynamic programminggraph theory

Problem Statement

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 _j , nextCourse _j ] denotes that course prevCourse _j has to be completed before course nextCourse _j (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).

Examples

Example 1

Input: n = 3, relations = [[1,3],[2,3]], time = [3,2,5]
Output: 8
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

Input: n = 5, relations = [[1,5],[2,5],[3,5],[3,4],[4,5]], time = [1,2,3,4,5]
Output: 12
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 * 10 ^4

0 <= relations.length <= min(n * (n - 1) / 2, 5 * 10 ^4 )

relations[j].length == 2

1 <= prevCourse _j , nextCourse _j <= n

prevCourse _j != nextCourse _j

All the pairs [prevCourse _j , nextCourse _j ] are unique .

time.length == n

1 <= time[i] <= 10 ^4

The given graph is a directed acyclic graph.

😤

Hard

Difficulty

Topic Info

ModuleDSA

CategoryArray

Sub-topicDynamic Programming