Parallel Courses II

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from itertools import combinations # for picking k of n
class Solution:
    @lru_cache(None) # caching for faster lookups
    def recurse(self, mask, in_degrees):
        # if all the bits are 0, we have taken all the courses
        if not mask: return 0

        # all the nodes that *can* be taken now, following both the properties
        nodes = [i for i in range(self.n) if mask & 1 << i and in_degrees[i] == 0]

        ans = float('inf')
        # enumerating all the possible combinations
        for k_nodes in combinations(nodes, min(self.k, len(nodes))):
            new_mask, new_in_degrees = mask, list(in_degrees)

            # updating what would happen to new_mask and new_in_degrees 
            # if we considered the nodes in k_nodes
            for node in k_nodes:
                # since we know the bit is set, we un-set this bit, to mark it "considered"
                new_mask ^= 1 << node
                # updating each of the in-degrees, since the "parents" have been taken away
                for child in self.graph[node]:
                    new_in_degrees[child] -= 1

            # the heart of recursion
            # note the +1!
            ans = min(ans, 1+self.recurse(new_mask, tuple(new_in_degrees)))
        return ans

    def minNumberOfSemesters(self, n: int, relations: List[List[int]], k: int) -> int:
        # saving n and k for later use
        self.n = n
        self.k = k
        in_degrees = [0]*self.n
        # graph layout remains the same, although the in_degrees change. 
        # This allows us to keep graph as self.graph 
        # instead of passing it over and over.
        self.graph = defaultdict(list)
        for prev_course, next_course in relations:
            # remember, its 0-indexed now!
            in_degrees[next_course - 1] += 1
            self.graph[prev_course - 1].append(next_course - 1)

        # start with all the bits set
        return self.recurse((1 << self.n) - 1, tuple(in_degrees))

10 Prerequisite LeetCode Problems

“1494. Parallel Courses II” involves graph traversal and dynamic programming. You are given a directed graph that represents prerequisite courses that must be taken before other courses, a number of courses you can take at the same time, and the task is to find the minimum number of semesters to finish all courses.

Here are 10 problems to prepare for “Parallel Courses II”:

207. Course Schedule: This problem also deals with course prerequisites, but it only asks whether it’s possible to finish all courses. It’s a simpler problem that introduces you to the concept of a cycle in a directed graph.
210. Course Schedule II: This problem builds on the previous one by not just asking whether it’s possible to finish all courses, but in which order. It introduces the concept of topological sort.
630. Course Schedule III: This problem adds a new twist to the course schedule problems by adding a deadline to each course.
1462. Course Schedule IV: This is another variation of the course schedule problem where you’re asked whether certain queries are possible.
1203. Sort Items by Groups Respecting Dependencies: This problem is more complex. You need to sort items in certain groups, while respecting both group and item dependencies.
269. Alien Dictionary: This problem asks you to find a valid order of characters given a list of alien words. This problem can be seen as a variation of topological sort.
444. Sequence Reconstruction: This problem is about reconstructing a sequence from a list of subsequences. It’s essentially a problem of finding a path in a graph.
802. Find Eventual Safe States: In this problem, you’re asked to find the “safe” nodes in a graph, those which are not part of a cycle.
787. Cheapest Flights Within K Stops: This problem is about finding the cheapest flight plan that uses no more than a given number of stops. It combines graph traversal with dynamic programming.
743. Network Delay Time: This problem asks you to find the time it takes for a signal to reach all nodes in a network. It involves finding shortest paths in a graph.

These cover how to traverse graphs and how to apply dynamic programming to graph problems.

Here are 5 problems that would provide useful groundwork for tackling problem 1494:

1. Smallest String With Swaps: This problem introduces the concept of connected components in a graph and how to use them to solve a problem. The main problem also requires an understanding of connected components.
1. Clone Graph: This problem involves cloning a graph. It provides practice with graph traversal and understanding graph representation, which is important for the main problem.
1. Redundant Connection: This problem requires you to find an edge in an undirected graph that, when removed, leaves the remaining graph as a tree. It involves concepts of graph cycles and union-find, which might be helpful for the main problem.
1. Possible Bipartition: This problem asks for a way to divide a set of people (represented as nodes in a graph) into two groups such that no one is in the same group as their enemy. It’s about graph coloring and can provide a solid understanding of working with graphs.
1. Regions Cut By Slashes: This problem also deals with the concept of connected components in a graph.
1. Check if There is a Valid Path in a Grid: This problem is about finding a valid path in a grid, which involves graph traversal techniques similar to those needed for the main problem.

Problem Classification

Problem Statement:You are given an integer n, which indicates that there are n courses labeled from 1 to n. You are also given an array relations where relations[i] = [prevCoursei, nextCoursei], representing a prerequisite relationship between course prevCoursei and course nextCoursei: course prevCoursei has to be taken before course nextCoursei. Also, you are given the integer k.

In one semester, you can take at most k courses as long as you have taken all the prerequisites in the previous semesters for the courses you are taking.

Return the minimum number of semesters needed to take all courses. The testcases will be generated such that it is possible to take every course.

Example 1:

Input: n = 4, relations = [[2,1],[3,1],[1,4]], k = 2 Output: 3 Explanation: The figure above represents the given graph. In the first semester, you can take courses 2 and 3. In the second semester, you can take course 1. In the third semester, you can take course 4. Example 2:

Input: n = 5, relations = [[2,1],[3,1],[4,1],[1,5]], k = 2 Output: 4 Explanation: The figure above represents the given graph. In the first semester, you can only take courses 2 and 3 since you cannot take more than two per semester. In the second semester, you can take course 4. In the third semester, you can take course 1. In the fourth semester, you can take course 5.

Constraints:

1 <= n <= 15 1 <= k <= n 0 <= relations.length <= n * (n-1) / 2 relations[i].length == 2 1 <= prevCoursei, nextCoursei <= n prevCoursei != nextCoursei All the pairs [prevCoursei, nextCoursei] are unique. The given graph is a directed acyclic graph.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Distilling the Problem to Its Core Elements

In order to have me distill a problem to its core, you could ask questions that prompt for a deeper analysis of the problem, understanding of the underlying concepts, and simplification of the problem’s essence. Here are some examples of such prompts:

Can you identify the fundamental concept or principle this problem is based upon? Please explain.
What is the simplest way you would describe this problem to someone unfamiliar with the subject?
What is the core problem we are trying to solve? Can we simplify the problem statement?
Can you break down the problem into its key components?
What is the minimal set of operations we need to perform to solve this problem?

These prompts guide the discussion towards simplifying the problem, stripping it down to its essential elements, and understanding the core problem to be solved.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.