Maximum Students Taking Exam

The problem statement is describing a situation where students must be placed in a classroom in such a way that they cannot cheat. A student can see the answers of the students sitting next to him to the left, right, upper left, and upper right, so we need to find the maximum number of students that can be placed in the classroom without any cheating possibility.

The maxStudents function will take a matrix seats representing the classroom and will return the maximum number of students that can be placed without cheating.

Here’s the code to solve the given problem:

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class Solution:
    def maxStudents(self, seats: List[List[str]]) -> int:
        m, n = len(seats), len(seats[0])
        rows = [int(''.join(seats[i]).replace('.', '0').replace('#', '1'), 2) for i in range(m)]

        @functools.lru_cache(None)
        def dp(i, prev_row):
            if i == m:
                return 0
            max_students = 0
            for curr_row in range(1 << n):
                if (curr_row & rows[i]) == 0 and \
                   (curr_row & (curr_row >> 1)) == 0 and \
                   (curr_row & (prev_row << 1)) == 0 and \
                   (curr_row & (prev_row >> 1)) == 0:
                    max_students = max(max_students, dp(i + 1, curr_row) + bin(curr_row).count('1'))
            return max_students

        return dp(0, 0)

Explanation:

We transform the classroom layout into binary representation for more efficient manipulation.
The recursive function dp(i, prev_row) calculates the maximum number of students that can be placed from row i to the end, given the placement of students in the previous row.
The base case for recursion is when we’ve reached the last row, and the result is 0.
For each row, we try all possible placements (represented by curr_row) and check if they satisfy the conditions of no cheating.
We utilize dynamic programming with memoization (functools.lru_cache) to avoid redundant computations.

Identifying Problem Isomorphism

“Maximum Students Taking Exam” is isomorphic to “Maximum Number of Non-Overlapping Chess Knights”.

In both problems, the main task is to place entities (students or chess knights) on a 2D grid in such a manner that they do not interfere with each other. In the students case, the interference is defined as the possibility of cheating, while in the knights’ case, it’s the possibility of capturing each other.

The difference is the rules for interaction - students can interact with those left, right, upper left, and upper right, while chess knights can interact in an ‘L’ shape. Also, the “Maximum Students Taking Exam” problem includes broken seats, marked with a ‘#’, where students can’t sit, analogous to squares on the chess board that are blocked off and knights can’t occupy.

Both problems have the same constraint of a grid of maximum size 8x8, and both require methods to explore the possible arrangements of entities on the grid.

The different rules for interaction mean that different methods might be optimal for solving each problem, but neither set of rules is inherently simpler than the other.

10 Prerequisite LeetCode Problems

“1349. Maximum Students Taking Exam” involves dynamic programming and bitwise operations. Here are some simpler problems to prepare for this:

LeetCode 70. Climbing Stairs
- This problem is a classic introductory problem to dynamic programming and will help you understand how to approach DP problems.
LeetCode 198. House Robber
- This problem is another simpler dynamic programming problem that will help you understand how to consider different states in dynamic programming.
LeetCode 322. Coin Change
- This problem will introduce you to the concept of solving a problem by trying out all possible choices, which is fundamental to dynamic programming.
LeetCode 139. Word Break
- This is a dynamic programming problem that focuses on partitioning a string into substrings. Understanding how to solve this will help you get a sense of how to build your DP table for complex problems.
LeetCode 338. Counting Bits
- This problem involves counting the number of 1 bits in binary representation. This helps you understand bitwise operations and counting set bits.
LeetCode 461. Hamming Distance
- This problem involves calculating the Hamming distance between two integers. This will introduce you to bitwise XOR operation.
LeetCode 136. Single Number
- This problem involves finding the unique number in an array using XOR operation. It will further enhance your understanding of bitwise operations.
LeetCode 78. Subsets
- This problem will help you to understand the concept of bitwise manipulation to generate subsets, which can be useful in certain dynamic programming problems.
LeetCode 91. Decode Ways
- A dynamic programming problem related to counting the number of ways to decode a string, it introduces the idea of using past solutions to solve subsequent subproblems.
LeetCode 62. Unique Paths
- This problem requires dynamic programming to find all possible paths from the top-left to the bottom-right corner of a grid. This will strengthen your understanding of DP.

“1349. Maximum Students Taking Exam” involves dynamic programming and bitwise manipulation. Here are seven problems as a starting point to build up to tackling this problem:

260. Single Number III: This problem can help you better understand bitwise manipulation.
137. Single Number II: Another problem that reinforces the concept of bitwise manipulation.
213. House Robber II: This is a follow-up to the previous problem but the houses are in a circle.
416. Partition Equal Subset Sum: This is another problem that can help you understand dynamic programming involving arrays.
518. Coin Change 2: This problem asks for the number of combinations that make up a target amount, and it will deepen your understanding of dynamic programming.
647. Palindromic Substrings: This problem can help you get used to the concept of dynamic programming with strings and arrays.
338. Counting Bits: This problem reinforces bitwise manipulation concepts, and will help you become more comfortable working with bits.

By solving these problems and understanding their solutions, you should be able to gain a solid understanding of the principles required to solve problem 1349.

Problem Classification

Problem Statement:Given a m * n matrix seats that represent seats distributions in a classroom. If a seat is broken, it is denoted by ‘#’ character otherwise it is denoted by a ‘.’ character.

Students can see the answers of those sitting next to the left, right, upper left and upper right, but he cannot see the answers of the student sitting directly in front or behind him. Return the maximum number of students that can take the exam together without any cheating being possible..

Students must be placed in seats in good condition.

Example 1:

Input: seats = [["#",".","#","#",".","#"], [".","#","#","#","#","."], ["#",".","#","#",".","#"]] Output: 4 Explanation: Teacher can place 4 students in available seats so they don’t cheat on the exam.

Example 2:

Input: seats = [[".","#"], ["#","#"], ["#","."], ["#","#"], [".","#"]] Output: 3 Explanation: Place all students in available seats.

Example 3:

Input: seats = [["#",".",".",".","#"], [".","#",".","#","."], [".",".","#",".","."], [".","#",".","#","."], ["#",".",".",".","#"]] Output: 10 Explanation: Place students in available seats in column 1, 3 and 5.

Constraints:

seats contains only characters ‘.’ and’#’. m == seats.length n == seats[i].length 1 <= m <= 8 1 <= n <= 8

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.

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.

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?

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.