## C# || Parallel Courses III – How To Find Minimum Number Months To Complete All Courses Using C#

The following is a module with functions which demonstrates how to find the minimum number of months needed to complete all courses using C#.

1. Minimum Time – 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**has to be completed

_{j}**before**course

**nextCourse**(prerequisite relationship). Furthermore, you are given a

_{j}**0-indexed**integer array

**time**where

**time[i]**denotes how many

**months**it takes to complete the

**(i+1)**course.

^{th}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:**

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:**

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.

2. Minimum Time – Solution

The following is a solution which demonstrates how to find the minimum number of months needed to complete all courses.

```
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// ============================================================================// Author: Kenneth Perkins// Date: Nov 1, 2023// Taken From: http://programmingnotes.org/// File: Solution.cs// Description: Demonstrates how to find the minimum months complete courses// ============================================================================public class Solution { public int MinimumTime(int n, int[][] relations, int[] time) { var graph = new Dictionary<int, List<int>>(); for (int i = 0; i < n; i++) { graph[i] = new List<int>(); } int[] indegree = new int[n]; foreach (int[] edge in relations) { int x = edge[0] - 1; int y = edge[1] - 1; graph[x].Add(y); indegree[y]++; } var queue = new Queue<int>(); int[] maxTime = new int[n]; for (int node = 0; node < n; node++) { if (indegree[node] == 0) { queue.Enqueue(node); maxTime[node] = time[node]; } } while (queue.Count > 0) { int node = queue.Dequeue(); foreach (int neighbor in graph[node]) { maxTime[neighbor] = Math.Max(maxTime[neighbor], maxTime[node] + time[neighbor]); indegree[neighbor]--; if (indegree[neighbor] == 0) { queue.Enqueue(neighbor); } } } int ans = 0; for (int node = 0; node < n; node++) { ans = Math.Max(ans, maxTime[node]); } return ans; }}// http://programmingnotes.org/

**QUICK NOTES**:

The highlighted lines are sections of interest to look out for.

The code is heavily commented, so no further insight is necessary. If you have any questions, feel free to leave a comment below.

Once compiled, you should get this as your output for the example cases:

8

12

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