Course Schedule IV

Problem Description

There are a total of numCourses courses you have to take, labeled from 0 to numCourses - 1. You are given an array prerequisites where prerequisites[i] = [a_i, b_i] indicates that you must take course a_i first if you want to take course b_i. You are also given an array queries where queries[j] = [u_j, v_j]. For the jth query, you should answer whether course u_j is a prerequisite of course v_j or not. Return a boolean array answer, where answer[j] is the answer to the jth query.

Key Insights

• The problem can be modeled as a directed graph where courses are nodes and prerequisites are directed edges.
• We need to determine if there is a path from node u to node v for each query, which can be efficiently solved using graph traversal techniques like Depth-First Search (DFS) or Breadth-First Search (BFS).
• Transitive relationships must be considered, meaning if course a is a prerequisite of course b, and b is a prerequisite of c, then a is a prerequisite of c.

Space and Time Complexity

Time Complexity: O(numCourses^2 + queries.length)
Space Complexity: O(numCourses^2)

Solution

To solve this problem, we will use graph traversal techniques. We will build an adjacency list to represent the directed graph of courses and their prerequisites. For each query, we will perform a graph traversal (either DFS or BFS) to check if there is a path from course u to course v.

1. Construct the graph using an adjacency list from the prerequisites array.
2. For each query (u, v), perform a DFS starting from u to see if we can reach v.
3. Collect the results of each query in a boolean array and return it.

Code Solutions