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] = [ai, bi] indicates that you must take course ai first if you want to take course bi.
[0, 1] indicates that you have to take course 0 before you can take course 1.Prerequisites can also be indirect. If course a is a prerequisite of course b, and course b is a prerequisite of course c, then course a is a prerequisite of course c.
You are also given an array queries where queries[j] = [uj, vj]. For the jth query, you should answer whether course uj is a prerequisite of course vj or not.
Return a boolean array answer, where answer[j] is the answer to the jth query.
Example 1:
Input: numCourses = 2, prerequisites = [[1,0]], queries = [[0,1],[1,0]] Output: [false,true] Explanation: The pair [1, 0] indicates that you have to take course 1 before you can take course 0. Course 0 is not a prerequisite of course 1, but the opposite is true.
Example 2:
Input: numCourses = 2, prerequisites = [], queries = [[1,0],[0,1]] Output: [false,false] Explanation: There are no prerequisites, and each course is independent.
Example 3:
Input: numCourses = 3, prerequisites = [[1,2],[1,0],[2,0]], queries = [[1,0],[1,2]] Output: [true,true]
Constraints:
2 <= numCourses <= 1000 <= prerequisites.length <= (numCourses * (numCourses - 1) / 2)prerequisites[i].length == 20 <= ai, bi <= numCourses - 1ai != bi[ai, bi] are unique.1 <= queries.length <= 1040 <= ui, vi <= numCourses - 1ui != viWhen you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:
For each question about whether one class is a prerequisite for another, the brute force approach explores every possible path between the two classes to see if a path exists. This means considering all possible combinations of classes to check if one leads to the other. It's like manually checking every possible route on a map to see if you can get from point A to point B.
Here's how the algorithm would work step-by-step:
def course_schedule_brute_force(number_of_courses, prerequisites, queries):
results = []
for query in queries:
course_a = query[0]
course_b = query[1]
if can_take_course(number_of_courses, prerequisites, course_a, course_b):
results.append(True)
else:
results.append(False)
return results
def can_take_course(number_of_courses, prerequisites, course_a, course_b):
# Use a recursive depth-first search to find a path
return has_path(prerequisites, course_a, course_b, set())
def has_path(prerequisites, current_course, target_course, visited):
if current_course == target_course:
return True
if current_course in visited:
return False
visited.add(current_course)
# Iterate through prerequisites to explore neighbors.
for prerequisite, course in prerequisites:
if prerequisite == current_course:
# Recursively search for path to target
if has_path(prerequisites, course, target_course, visited):
return True
# Mark as unvisited for other paths.
visited.remove(current_course)
# If no path found, return false
return FalseThe problem asks us to determine if one course is a prerequisite of another. We can efficiently solve this by pre-computing all possible prerequisite relationships using a technique similar to finding the shortest path between all pairs of courses, and then answering queries based on this pre-computed information.
Here's how the algorithm would work step-by-step:
def course_schedule_iv(number_of_courses, prerequisites, queries):
adjacency_list = [[] for _ in range(number_of_courses)]
for source, destination in prerequisites:
adjacency_list[source].append(destination)
reachable = [[False] * number_of_courses for _ in range(number_of_courses)]
# Iterate over all courses to find reachable nodes.
for start_course in range(number_of_courses):
visited = [False] * number_of_courses
queue = [start_course]
visited[start_course] = True
while queue:
current_course = queue.pop(0)
reachable[start_course][current_course] = True
for neighbor in adjacency_list[current_course]:
if not visited[neighbor]:
visited[neighbor] = True
queue.append(neighbor)
result = []
# Evaluate each query to determine prerequisite relationship.
for course_a, course_b in queries:
result.append(reachable[course_a][course_b])
return result| Case | How to Handle |
|---|---|
| Empty prerequisites list | Return a list of 'true' values, indicating all queries are reachable since no prerequisites exist. |
| Empty queries list | Return an empty list as there are no queries to evaluate. |
| Course dependency cycle exists | The topological sort should detect cycles and return an empty ordering, resulting in all queries being 'false'. |
| Queries involving non-existent courses (out of bounds) | Treat non-existent courses as unreachable, resulting in a 'false' result for those queries. |
| Maximum number of courses (n is large) and prerequisites (array length large) | Ensure the graph representation and transitive closure algorithm are space and time efficient (e.g., using adjacency list and optimized DFS or Floyd-Warshall). |
| All courses depend on the first course | The transitive closure will correctly identify reachability from the first course to all others. |
| All courses are independent (no prerequisites) | The transitive closure matrix will have 'true' only on the diagonal, representing reachability from a course to itself. |
| Large number of queries | Precompute the transitive closure for efficiency; answering each query then becomes an O(1) lookup. |