Median of a Row Wise Sorted Matrix

Finding the median of a matrix in less than (O(m \times n)) complexity can be achieved using binary search. Since each row is sorted in non-decreasing order, we can take advantage of this property.

Here’s a simple algorithm to achieve the goal:

  1. Find the minimum and maximum elements in the matrix. This can be done in (O(m)) time since the matrix is row-sorted.
  2. Perform binary search between the minimum and maximum elements to find the median. In each step of the binary search:
    • Count the number of elements less than or equal to the mid-value.
    • If the count is less than the required number for the median, update the low value.
    • If the count is greater than or equal to the required number, update the high value.
  3. Repeat until low is less than or equal to high, and the median value is found.

Here’s the code:

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from bisect import bisect_right

class Solution:
    def matrixMedian(self, grid: List[List[int]]) -> int:
        m, n = len(grid), len(grid[0])
        low, high = grid[0][0], grid[-1][-1]

        # Finding the minimum and maximum values in the matrix
        for i in range(m):
            low = min(low, grid[i][0])
            high = max(high, grid[i][-1])

        # Median position
        median_pos = (m * n + 1) // 2

        while low <= high:
            mid = (low + high) // 2
            count = 0

            # Count elements less than or equal to mid
            for i in range(m):
                count += bisect_right(grid[i], mid)

            # Updating the low or high values based on the count
            if count < median_pos:
                low = mid + 1
            else:
                high = mid - 1

        return low

This code finds the median of the matrix in (O(m \log n)) time complexity, meeting the constraint of less than (O(m \times n)).

This problem requires you to find the median of all the numbers in a matrix, where each row of the matrix is sorted.

Identifying Problem Isomorphism

An approximate isomorphic problem for this is “Find Kth Smallest Pair Distance” (719). In this problem, you have to find the kth smallest distance among all the pairs of the given numbers.

The main similarity between these two problems lies in the search space and the binary search algorithm. Both problems require working with sorted arrays (or a sorted matrix, which can be treated as a sorted array) and trying to find a specific element - the median in one case, and the kth smallest element in another.

In both cases, binary search can be applied not on the actual data but rather on the possible answer space, which is a common theme in many complex search problems.

The binary search in “Median of a Row Wise Sorted Matrix” works on the range of numbers from the smallest number to the largest number in the matrix. The mid value of this range is checked to determine the count of numbers less than the mid value. If this count is less than (n*m)/2, where n and m are the number of rows and columns of the matrix, then the search space is reduced to the numbers greater than mid.

Similarly, in “Find Kth Smallest Pair Distance”, binary search is used to search the space of possible answers (the range of distances), not the actual data.

The concept of binary search in the answer space is similar in both problems, but the actual problems and the specific details in the solutions may be different. It’s the abstract algorithmic concept and approach that are closely related.

10 Prerequisite LeetCode Problems

“Median of a Row Wise Sorted Matrix” requires understanding of binary search, arrays, and matrices. Here are 10 problems to build the necessary skills:

  1. “Two Sum” (LeetCode problem #1): This problem introduces basic array manipulation techniques.

  2. “Search a 2D Matrix” (LeetCode problem #74): This is a fundamental problem for understanding how to work with 2D matrices in a binary search context.

  3. “Binary Search” (LeetCode problem #704): Before you can tackle the “Median of a Row Wise Sorted Matrix” problem, you need a solid understanding of the binary search algorithm. This problem provides a good basis for that.

  4. “First Bad Version” (LeetCode problem #278): This problem provides another scenario to apply binary search, strengthening the understanding of its applications.

  5. “Search Insert Position” (LeetCode problem #35): This problem applies binary search in a slightly more complex scenario, which is a good practice after understanding the basic concept.

  6. “Rotate Image” (LeetCode problem #48): This problem helps to understand how to manipulate 2D arrays (or matrices), which is an essential skill for the problem in question.

  7. “Spiral Matrix” (LeetCode problem #54): It is a slightly more complex problem involving matrix manipulation, providing a good stepping stone towards the “Median of a Row Wise Sorted Matrix” problem.

  8. “Set Matrix Zeroes” (LeetCode problem #73): This problem provides good practice in handling and manipulating 2D arrays or matrices.

  9. “Find Smallest Common Element in All Rows” (LeetCode problem #1198): This problem is another example of handling matrices and rows separately, similar to what will be needed for the “Median of a Row Wise Sorted Matrix” problem.

  10. “Kth Smallest Element in a Sorted Matrix” (LeetCode problem #378): This problem is quite similar to “Median of a Row Wise Sorted Matrix” but slightly less complex. You need to find the Kth smallest element in a sorted matrix, which involves understanding of binary search on matrices.

Problem Classification

Problem Statement: Given an m x n matrix grid containing an odd number of integers where each row is sorted in non-decreasing order, return the median of the matrix.

You must solve the problem in less than O(m * n) time complexity.

Example 1:

Input: grid = [[1,1,2],[2,3,3],[1,3,4]] Output: 2 Explanation: The elements of the matrix in sorted order are 1,1,1,2,2,3,3,3,4. The median is 2.

Example 2:

Input: grid = [[1,1,3,3,4]] Output: 3 Explanation: The elements of the matrix in sorted order are 1,1,3,3,4. The median is 3.

Constraints:

m == grid.length n == grid[i].length 1 <= m, n <= 500 m and n are both odd. 1 <= grid[i][j] <= 106 grid[i] is sorted in non-decreasing order.

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?

Alternatively, if you’re working on a specific problem, you might ask something like:

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 ?

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. 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.

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.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. 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.

  1. 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.

  2. 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.

  3. 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.

  1. 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.

  2. 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.

  3. 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

Given the problem [provide the problem], identify and list down 10 similar problems on LeetCode. These should cover similar concepts or require similar problem-solving approaches as the provided problem. Please also give a brief reason as to why you think each problem is similar to the given problem.

10 Prerequisite LeetCode Problems

This problem “2387. Median of a Row Wise Sorted Matrix” combines binary search and counting sort, applying it in the context of 2-dimensional arrays or matrices. Here are 10 LeetCode problems that touch on these concepts and can provide good practice:

  1. 704. Binary Search: This is a basic binary search problem that can serve as an introduction to the concept.

  2. 35. Search Insert Position: This problem is a simple application of binary search where you have to find the position to insert a target element in a sorted array.

  3. 278. First Bad Version: This problem applies the binary search concept to a slightly different context, where you are trying to find the first occurrence of a “bad” version in a series.

  4. 153. Find Minimum in Rotated Sorted Array: This problem requires applying binary search in a more complex context, where the sorted array has been rotated.

  5. 34. Find First and Last Position of Element in Sorted Array: Here you need to find the range of a given element in a sorted array, which requires two applications of binary search.

  6. 75. Sort Colors: This problem involves counting sort, a sorting algorithm that can be useful when dealing with a limited range of integer values.

  7. 1122. Relative Sort Array: This problem also involves a sort of counting sort, where the order of one array should dictate the order of elements in another.

  8. 378. Kth Smallest Element in a Sorted Matrix: This problem involves finding the kth smallest element in a sorted matrix, which is similar to finding the median element. Binary search is applied in a slightly different way here, making it a good practice problem.

  9. 240. Search a 2D Matrix II: This problem involves searching in a sorted 2D matrix, which can provide good practice for handling 2D matrices.

  10. 668. Kth Smallest Number in Multiplication Table: This problem involves a special kind of matrix (a multiplication table), but the application of binary search is similar to this problem.

By working on these problems, you should become comfortable with the concepts of binary search and counting sort, and applying these concepts in the context of 2D matrices, which are key to solving problem 2387.

Clarification Questions

What are the clarification questions we can ask about this problem?

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 an m x n matrix grid containing an odd number of integers where each row is sorted in non-decreasing order, return the median of the matrix. You must solve the problem in less than O(m * n) time complexity.

Example 1:

Input: grid = [[1,1,2],[2,3,3],[1,3,4]] Output: 2 Explanation: The elements of the matrix in sorted order are 1,1,1,2,2,3,3,3,4. The median is 2.

Example 2:

Input: grid = [[1,1,3,3,4]] Output: 3 Explanation: The elements of the matrix in sorted order are 1,1,3,3,4. The median is 3.

Constraints:

m == grid.length n == grid[i].length 1 <= m, n <= 500 m and n are both odd. 1 <= grid[i][j] <= 106 grid[i] is sorted in non-decreasing order.

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 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 edge cases?

How to visualize 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?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical 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 Problem-Solving 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

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. 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

  1. Parameters:

    • What are the inputs to the method?
    • What types are these parameters?
    • What do these parameters represent in the context of the problem?

  2. 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?

  3. Method Functionality:

    • What is this method expected to do?
    • How does it interact with the inputs and the current state of the program?

  4. 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?

  5. 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

  1. Problem Understanding:

    • Can you explain the problem in your own words? What are the key components and requirements?

  2. Initial Breakdown:

    • Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?

  3. 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?

  4. Task Identification:

    • Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?

  5. 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?

  6. Method Naming:

    • Can you give each task a simple, descriptive name that makes its purpose clear?

  7. 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

  1. Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.

  2. 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?

  3. 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.

  4. 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?

  5. Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.

  6. 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.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. 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.

  1. 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.

  2. 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.

  3. 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.

  1. 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.

  2. 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.

  3. 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 problem-solving 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.