Course Schedule IV

class Solution {
public:
    vector<bool> prerequisitesCheck(int coursesCount,
                                     vector<vector<int>>& preconditions,
                                     vector<vector<int>>& courseQueries) {
        vector<vector<bool>> prerequisiteMatrix(coursesCount,
                                                vector<bool>(coursesCount, false));
        for (auto pre : preconditions) {
            prerequisiteMatrix[pre[0]][pre[1]] = true;
        }
    
        for (int k = 0; k < coursesCount; k++) {
            for (int i = 0; i < coursesCount; i++) {
                for (int j = 0; j < coursesCount; j++) {
                    // Transitive closure update.
                    prerequisiteMatrix[i][j] = prerequisiteMatrix[i][j] ||
                                               (prerequisiteMatrix[i][k] && prerequisiteMatrix[k][j]);
                }
            }
        }
    
        vector<bool> results;
        for (auto query : courseQueries) {
            results.push_back(prerequisiteMatrix[query[0]][query[1]]);
        }
    
        return results;
    }
};

The provided C++ code defines a function within a class Solution that determines if specific courses are prerequisites for others based on given preconditions. The function prerequisitesCheck involves the following steps:

1. Initialize a 2D boolean matrix prerequisiteMatrix sized according to the coursesCount, setting all values initially to false. This matrix represents whether a course at index i is a prerequisite for a course at index j.

2. Populate this matrix using a given list of preconditions. Each precondition is a vector where the first element is a prerequisite for the second element; this relationship is marked as true in the matrix.

3. Utilize the Floyd-Warshall algorithm to compute the transitive closure of the prerequisites matrix. This determines whether a sequence of prerequisite relationships exists between any two courses, updating the matrix to reflect these extended relationships.

4. Process each query in courseQueries to determine if one course is a prerequisite for another by checking the final state of prerequisiteMatrix.

5. Push the result (true or false) into a results vector based on the queried relationships from courseQueries.

6. Return the results vector, which contains boolean values corresponding to each query, indicating whether the first course in the query is a prerequisite for the second course based on both direct and indirect preconditions.

public class Solution {
    
    public List<Boolean> canTakeCourses(
        int totalCourses,
        int[][] courseDependencies,
        int[][] courseQueries
    ) {
        boolean[][] directPrerequisite = new boolean[totalCourses][totalCourses];
    
        for (int[] coursePair : courseDependencies) {
            directPrerequisite[coursePair[0]][coursePair[1]] = true;
        }
    
        for (int middleCourse = 0; middleCourse < totalCourses; middleCourse++) {
            for (int startCourse = 0; startCourse < totalCourses; startCourse++) {
                for (int endCourse = 0; endCourse < totalCourses; endCourse++) {
                    directPrerequisite[startCourse][endCourse] =
                        directPrerequisite[startCourse][endCourse] ||
                        (directPrerequisite[startCourse][middleCourse] &&
                            directPrerequisite[middleCourse][endCourse]);
                }
            }
        }
    
        List<Boolean> results = new ArrayList<>();
        for (int[] query : courseQueries) {
            results.add(directPrerequisite[query[0]][query[1]]);
        }
    
        return results;
    }
}

This solution addresses the problem of determining if specific courses can be taken given a list of course prerequisites. The Java implementation utilizes a graph represented by a 2D Boolean array (directPrerequisite) to track direct and indirect prerequisites among courses.

• Initialize a 2D array directPrerequisite where each element [i][j] is set to true if course i is a direct prerequisite for course j.
• Iterate over the provided courseDependencies array to populate the directPrerequisite array based on the direct relationships given.
• Use the Floyd-Warshall algorithm to compute transitive closures within the graph. This loop:
  • Goes through each course as a possible intermediate step (middleCourse).
  • Then iterates for each possible starting (startCourse) and ending course (endCourse).
  • Updates directPrerequisite[startCourse][endCourse] to true if there exists a path through middleCourse that makes startCourse a prerequisite for endCourse.

After building the complete transitive closure:

• Initialize an ArrayList results to store Boolean values.
• For each pair of courses in courseQueries, check directPrerequisite to determine if the second course can be taken after completing the first course.
• Return the results list indicating for each query pair whether the second course is accessible given the prerequisites.

This approach effectively queries prerequisite relationships in constant time after an initial computation cost, making it efficient for multiple queries.

class Solution:
    def isPrerequisiteExists(
        self,
        totalCourses: int,
        prereq: List[List[int]],
        courseQueries: List[List[int]],
    ) -> List[bool]:
        directPrereq = [[False] * totalCourses for _ in range(totalCourses)]
    
        for p in prereq:
            directPrereq[p[0]][p[1]] = True
    
        for k in range(totalCourses):
            for i in range(totalCourses):
                for j in range(totalCourses):
                    directPrereq[i][j] = (
                        directPrereq[i][j] or (directPrereq[i][k] and directPrereq[k][j])
                    )
    
        results = []
        for query in courseQueries:
            results.append(directPrereq[query[0]][query[1]])
    
        return results

The Python3 solution provided facilitates determining if certain courses are prerequisites for others within a set schedule. The code addresses the task through the following steps:

1. Initialize a two-dimensional list called directPrereq to store boolean values, set to False by default, which represent whether a course is a direct prerequisite of another course.

2. Update the directPrereq matrix based on the given list of prerequisites. If a specific course a is a prerequisite for course b, then the respective cell in the matrix for these courses is set to True.

3. Use the Floyd-Warshall algorithm to extend the direct prerequisite relations to indirect relationships. This involves three nested loops which iteratively update the directPrereq matrix to reflect whether a transition through an intermediate course establishes a prerequisite relationship.

4. For each query in courseQueries, check the directPrereq matrix to see if the first course is a prerequisite for the second course in the query list. The result for each query is then appended to the results list.

5. Return the list of boolean results corresponding to each query, indicating whether the first course in each query is a prerequisite for the second one.

This solution effectively transforms the initial list of direct prerequisites into a comprehensive map of all (direct and indirect) prerequisite relationships, allowing quick look-up for the queries.