Parallel Courses II

Problem Description

You are given n courses labeled from 1 to n along with prerequisite relations. In a single semester you can take at most k courses if you have finished all their prerequisites in earlier semesters. The objective is to determine the minimum number of semesters required to complete all courses.

Key Insights

• Use bitmasking to represent the set of courses taken. Since n is at most 15, a bitmask is an ideal representation for subsets.
• For each course, precompute a bitmask that indicates its prerequisite courses.
• Use Dynamic Programming (DP) where the state is a bitmask of completed courses. The DP transition involves selecting a valid subset of the available courses (with prerequisites met) that can be taken in the next semester.
• When the number of available courses exceeds k, iterate through all submasks (subsets) of the available courses and consider those with at most k courses.
• This problem involves a combination of bit manipulation and DP with state space of size 2^n, which is manageable given the constraint n <= 15.

Space and Time Complexity

Time Complexity: O(2^n * n * 2^k) in worst-case since for each state (2^n) we might enumerate subsets (up to 2^k) and check prerequisites which is O(n) per state. Space Complexity: O(2^n) for the DP table.

Solution

We use a bitmask DP approach where each state represents the set of courses taken so far. First, for every course, we compute a prerequisite mask showing which courses must be completed before taking that course. The DP state dp[mask] stores the minimum semesters required to reach that state.

For each state, we calculate the available courses which are not taken yet and whose prerequisites are all met in the current mask. If the number of available courses is less than or equal to k, we can take them all; otherwise, we iterate through all submasks of the available set that contain at most k courses and update the DP state accordingly.

Data structures used:

• Integer bitmask for set representation.
• Array dp of size 2^n for storing the minimum number of semesters for each state.

The algorithm leverages bit manipulation techniques for checking prerequisites and enumerating submasks.

Code Solutions