Maximum Side Length of a Square with Sum Less than or Equal to Threshold
To solve this problem, we will use the concept of prefix sums to efficiently check the sum of any submatrix (square in our case).
Prefix Sum:
The prefix sum can be represented by another matrix prefixSum
where prefixSum[i][j]
is the sum of all elements of mat
in the rectangle from topleft (0,0)
to (i,j)
.
Algorithm:
 Calculate the prefix sum matrix.
 For each side length
k
starting from 1, check if there exists a square of sidek
with a sum less than or equal to the threshold.  If such a square exists for side
k
, try the next lengthk+1
. If not, returnk1
as the maximum side length.
Steps:
 First, calculate the
prefixSum
matrix.  To check for a square with side length
k
, traverse through the matrix and for every cell(i,j)
, check the sum for the square topleft at(ik+1, jk+1)
and bottomright at(i,j)
.  Use the prefix sum to get the sum for this square in O(1) time.
 If this sum is within the threshold and the side length is greater than our current maximum, update our answer.
 Repeat steps 24 for each possible square side length from 1 to
min(m, n)
.
Here’s how you can implement it:


Key Takeaways:
 Prefix sums allow for efficient calculation of the sum of any submatrix.
 Binary search helps in optimizing the search for the maximum side length of the square.
Identifying Problem Isomorphism
“Maximum Side Length of a Square with Sum Less than or Equal to Threshold” is isomorphic to “Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts”.
In “Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts” problem, you’re given a cake of size h x w, and you can make several horizontal and vertical cuts. The aim is to find the maximum area of a piece that can be obtained after making the cuts, considering the restriction that the area of a piece must not exceed a certain threshold.
The similarity is in the concept of a grid (matrix/cake) and the task of finding a maximal square/piece under certain constraints (sum/area less than or equal to threshold). In both problems, the values of the cells contribute to the sum or area which is required to be less than or equal to a threshold.
However, the method to calculate the sum or area is different. In “Maximum Side Length of a Square with Sum Less than or Equal to Threshold”, the sum is obtained by summing up the cell values, while in the “Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts” problem, the area is calculated based on the dimensions of the cake piece.
“Maximum Side Length of a Square with Sum Less than or Equal to Threshold” involves more complexity in calculating the sum of the cells of a square. In “Maximum Area of a Piece of Cake After Horizontal and Vertical Cuts”, calculating the area is more straightforward, as it involves the simple multiplication of the dimensions. However, the challenge of choosing where to make the cuts adds a layer of difficulty to this problem.
10 Prerequisite LeetCode Problems
This is an application of dynamic programming and binary search on a 2D grid. Here are 10 problems to prepare for this problem:
Easy Difficulty:
 303. Range Sum Query  Immutable: This problem is about prefix sums, which is a key concept used in the given problem.
 485. Max Consecutive Ones: This problem will help you get comfortable with calculating maximum lengths of sequences, though it’s not on a 2D grid.
 661. Image Smoother: A simple problem on 2D grid operations.
Medium Difficulty:
 221. Maximal Square: This problem is a simpler version of the problem at hand and will help in understanding the approach of dynamic programming on a 2D grid.
 304. Range Sum Query 2D  Immutable: This problem introduces 2D prefix sums, a technique you’ll need for the given problem.
 463. Island Perimeter: This problem will help in understanding how to deal with 2D grid problems.
 792. Number of Matching Subsequences: This problem requires binary search in an array, which is similar to how you’d need to perform binary search on the side length in the problem at hand.
 1277. Count Square Submatrices with All Ones: This problem is a similar, but slightly more complex dynamic programming problem on a 2D grid.
 1290. Convert Binary Number in a Linked List to Integer: This problem is about binary operations, which are often used in similar dynamic programming and binary search problems.
Hard Difficulty:
 410. Split Array Largest Sum: This problem involves binary search on the maximum sum, which is similar to how you’d need to perform binary search on the side length in the problem at hand.
These cover dynamic programming and binary search on a 2D grid, as well as handling edge cases and doing efficient computations on a grid.
The problem “1292. Maximum Side Length of a Square with Sum Less than or Equal to Threshold” is a binary search problem on the prefix sum matrix. Here are 10 problems that can help you prepare:
303. Range Sum Query  Immutable: This problem introduces the concept of prefix sum, which is crucial in solving problem 1292.
304. Range Sum Query 2D  Immutable: This problem is a 2D version of problem 303. It will help you get familiar with computing prefix sums in a 2D matrix.
1074. Number of Submatrices That Sum to Target: Although it’s more challenging, this problem could give you some insights into how to handle prefix sums of a 2D array, which is very useful for problem 1292.
221. Maximal Square: This problem shares the same concept of finding the largest square but in terms of binary values instead of sums. It can help you understand the general approach of finding the maximum side length of a square in a matrix.
240. Search a 2D Matrix II: This problem could give you some hints about how to handle matrixrelated problems efficiently, using binary search.
209. Minimum Size Subarray Sum: This problem is a simplified 1D version of problem 1292. It involves finding a subarray with a sum greater than or equal to a given value.
363. Max Sum of Rectangle No Larger Than K: This problem involves the same concept of finding a submatrix that sums to a target value. It will help you understand how to use prefix sums to solve this kind of problems efficiently.
74. Search a 2D Matrix: This problem is a basic problem to understand how to efficiently handle 2D arrays or matrices.
410. Split Array Largest Sum: This problem is not about matrices, but it will help you get more familiar with binary search related to the sum.
1011. Capacity To Ship Packages Within D Days: This problem is a binary search problem related to the sum. Solving this problem will help you understand how to implement binary search in problems related to the sum or total values.
Problem Analysis and Key Insights
What are the key insights from analyzing the problem statement?
Problem Boundary
What is the scope of this problem?
How to establish the boundary of this problem?
Problem Classification
Problem Statement:Given a m x n matrix mat and an integer threshold, return the maximum sidelength of a square with a sum less than or equal to threshold or return 0 if there is no such square.
Example 1:
Input: mat = [[1,1,3,2,4,3,2],[1,1,3,2,4,3,2],[1,1,3,2,4,3,2]], threshold = 4 Output: 2 Explanation: The maximum side length of square with sum less than 4 is 2 as shown.
Example 2:
Input: mat = [[2,2,2,2,2],[2,2,2,2,2],[2,2,2,2,2],[2,2,2,2,2],[2,2,2,2,2]], threshold = 1 Output: 0
Constraints:
m == mat.length n == mat[i].length 1 <= m, n <= 300 0 <= mat[i][j] <= 104 0 <= threshold <= 105
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
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?
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 realworld 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 edge 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?
Simple Explanation
Can you explain this problem in simple terms or like you would explain to a nontechnical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?
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 ProblemSolving Approach from the Problem Statement
Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?
How did you infer from the problem statement that this problem can be solved using ?
Simple Explanation of the Proof
I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?
Stepwise Refinement
Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the highlevel 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
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.
Identify Invariant
What is the invariant in this problem?
Identify Loop Invariant
What is the loop invariant in this problem?
Thought Process
Can you explain the basic thought process and steps involved in solving this type of problem?
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.
Establishing Preconditions and Postconditions
Parameters:
 What are the inputs to the method?
 What types are these parameters?
 What do these parameters represent in the context of the problem?
Preconditions:
 Before this method is called, what must be true about the state of the program or the values of the parameters?
 Are there any constraints on the input parameters?
 Is there a specific state that the program or some part of it must be in?
Method Functionality:
 What is this method expected to do?
 How does it interact with the inputs and the current state of the program?
Postconditions:
 After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
 What does the return value represent or indicate?
 What side effects, if any, does the method have?
Error Handling:
 How does the method respond if the preconditions are not met?
 Does it throw an exception, return a special value, or do something else?
Problem Decomposition
Problem Understanding:
 Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
 Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
 For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
 Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
 For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
 Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
 How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?
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 highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, 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 languageagnostic 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 problemsolving 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 stepbystep 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 Pythonbased 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 problemspecific 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.
Q&A
Similar Problems
Can you suggest 10 problems from LeetCode that require similar problemsolving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem.