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Write a C# program to determine if it's possible to finish all the courses. You are given a total of numCourses and an array prerequisites where prerequisites[i] = [ai, bi] indicates that you must take course bi first if you want to take course ai. Return true if you can finish all courses, or false otherwise.

Example 1
    
  
  • Input: numCourses = 2, prerequisites = [[1,0]]
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  • Output: true
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  • Explanation:
    
      
      
    • There are 2 courses to take.
      • 
      
    • To take course 1, you need to have finished course 0.
      • 
      
    • So it's possible.
      • 
    
    
  
    •
Example 2
    
  
  • Input: numCourses = 2, prerequisites = [[1,0],[0,1]]
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  • Output: false
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  • Explanation:
    
      
      
    • There are 2 courses to take.
      • 
      
    • To take course 1, you need to finish course 0.
      • 
      
    • To take course 0, you need to finish course 1.
      • 
      
    • This creates a cycle, making it impossible to finish all courses.
      • 
    
    
  
    •
Constraints
    
  
  • 1 <= numCourses <= 2000
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  • 0 <= prerequisites.length <= 5000
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  • prerequisites[i].length == 2
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  • 0 <= ai, bi < numCourses
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  • All the pairs [ai, bi] are distinct
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  • Time Complexity: O(V + E) where V is the number of vertices and E is the number of edges
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  • Space Complexity: O(V + E)
    •
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Hint
Solution

Solution Hints

    
  
  • Model the problem as a directed graph where each course is a vertex and prerequisites are edges
    • 
  
  • Use Depth-First Search (DFS) or Breadth-First Search (BFS) to detect cycles in the graph
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  • If a cycle exists, it's impossible to complete all courses
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  • Use a visited array to keep track of visited vertices
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  • Use another array to keep track of vertices in the current DFS path
    •

Steps to solve by this approach:

 Step 1: Create an adjacency list representation of the directed graph.
 Step 2: Initialize a visited array to track nodes' states (0: not visited, 1: being visited, 2: completely visited).
 Step 3: For each course, perform DFS if it hasn't been visited yet.
 Step 4: During DFS, mark current node as "being visited" (1).
 Step 5: If any neighbor is marked as "being visited", a cycle exists, return false.
 Step 6: After visiting all neighbors, mark current node as "completely visited" (2).
 Step 7: If no cycles are detected, return true.

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