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This approach involves topologically sorting the courses based on their prerequisites and then calculating the minimal time to complete each course using dynamic programming.
(a,b) is a directed edge from a to b.Time Complexity: O(n + E), where n is the number of courses and E is the number of prerequisite relationships, due to traversing all nodes and edges once.
Space Complexity: O(n + E), needed to store the graph and additional arrays.
1```c
2#include <stdio.h>
3#include <stdlib.h>
4#include <string.h>
5
6#define MAX 50001
```
This C solution initializes adjacency lists for the graph and an array for in-degrees. The courses are processed based on their prerequisites using a queue (to implement the topological sort). The dp array keeps track of the minimum time to finish each course as the prerequisites are progressively met. The total time to complete all courses becomes the maximum value in the dp array.
This approach uses Depth-First Search (DFS) with memorization to efficiently compute the completion time for courses.
Time Complexity: O(n + E), visiting each node and edge at least once.
Space Complexity: O(n + E), due to storage for graph data and function call stack.
```The Java solution conducts DFS to determine each course's minimal completion time, storing the results in a dp array to speed up computations. Using this technique, we can compute the maximum time needed to complete all courses efficiently.