Search a 2D Matrix

The problem involves searching a target integer in an m x n matrix that is sorted row-wise and the first element of each row is greater than the last element of the previous row. The challenge here is to do it in O(log(m * n)) time complexity.

Given the sorted nature of the matrix, we can apply a binary search. But since it’s a 2D matrix, we’ll need to convert the 2D indices to a 1D form and then apply a standard binary search algorithm.

Here’s a step-by-step explanation:

  1. Determine Rows and Columns: Determine the number of rows and columns in the matrix.

  2. Binary Search: Conduct a binary search where the middle element is found using a 1D index and then converted to 2D indices in the matrix.

  3. Check the Target: Compare the target with the middle element. If it matches, return true. If the target is less than the middle element, search in the left half, otherwise in the right half.

  4. Final Decision: If the binary search doesn’t find the element, return false.

Here’s the code:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17

class Solution:
    def searchMatrix(self, matrix: List[List[int]], target: int) -> bool:
        rows, cols = len(matrix), len(matrix[0])
        left, right = 0, rows * cols - 1

        while left <= right:
            mid = (left + right) // 2
            mid_value = matrix[mid // cols][mid % cols]  # Convert 1D index to 2D

            if mid_value == target:
                return True
            elif mid_value < target:
                left = mid + 1
            else:
                right = mid - 1

        return False

Example Explained

Consider the matrix:

1

matrix = [[1,3,5,7],[10,11,16,20],[23,30,34,60]]

If we’re looking for target = 3, the binary search will find this element in the matrix and return true. If we’re looking for target = 13, since this element doesn’t exist in the matrix, the function will return false.

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20

class Solution:
    def searchMatrix(self, matrix: List[List[int]], target: int) -> bool:
        if not matrix or not matrix[0]:
            return False

        rows, cols = len(matrix), len(matrix[0])
        row, col = 0, cols - 1

        # Start from the top-right corner
        while row < rows and col >= 0:
            if matrix[row][col] == target:
                return True
            elif matrix[row][col] > target:
                # Move to the left if the current element is greater than the target
                col -= 1
            else:
                # Move down if the current element is less than the target
                row += 1

        return False

Identifying Problem Isomorphism

“Search a 2D Matrix” can be mapped to “Search in Rotated Sorted Array”.

Here’s the reasoning:

Both problems deal with finding a target value in a sorted data structure. In “Search a 2D Matrix”, you’re given a 2D matrix where each row is sorted, and the first element of each row is greater than the last element of the previous row. You need to find if a target number exists in this matrix. In “Search in Rotated Sorted Array”, you’re given a sorted array that has been rotated at an unknown pivot. You need to determine if a target number is in the array.

Although one problem deals with a 2D matrix and the other with a 1D array, both share a similar concept. They both require a modified binary search technique to solve efficiently due to the sorted but slightly twisted nature of the data structures.

“Search in Rotated Sorted Array” is a more challenging because it involves a rotated sorted array, which adds a layer of complexity to the binary search. The mapping is based on the similar searching techniques required by both problems.

1
2
3
4

class Solution:
    def searchMatrix(self, matrix: List[List[int]], target: int) -> bool:
        r = bisect_left(matrix, target, key=lambda row: row[-1])  # or key=itemgetter(-1)
        return r < len(matrix) and matrix[r][bisect_left(matrix[r], target)] == target

Problem Classification

You are given an m x n integer matrix matrix with the following two properties:

Each row is sorted in non-decreasing order. The first integer of each row is greater than the last integer of the previous row. Given an integer target, return true if target is in matrix or false otherwise.

You must write a solution in O(log(m * n)) time complexity.

Example 1:

Input: matrix = [[1,3,5,7],[10,11,16,20],[23,30,34,60]], target = 3 Output: true Example 2:

Input: matrix = [[1,3,5,7],[10,11,16,20],[23,30,34,60]], target = 13 Output: false

Constraints:

m == matrix.length n == matrix[i].length 1 <= m, n <= 100 -10^4 <= matrix[i][j], target <= 10^4

Language Agnostic Coding Drills

  1. Variable Declaration and Initialization

    • Declare a variable and assign it an integer value, then print the variable.

  2. Arrays and Lists

    • Create a list of integers and print each integer in the list using a loop.

  3. Tuples and the zip function

    • Create two lists of integers. Use the zip() function to combine these lists into a list of tuples.

  4. List Sorting

    • Create a list of tuples where each tuple contains two integers. Sort this list by the second integer in each tuple.

  5. Heap Operations

    • Import the heapq module and create a min-heap. Add integers to the heap and pop integers from the heap.

  6. Prefix Sum Computation

    • Create a list of integers. Compute the prefix sum array for this list and print it.

  7. Conditional Statements and Loops

    • Create a loop that iterates over a list of integers and prints each integer if it is even, else it prints a message saying the integer is odd.

  8. Functions and Methods

    • Define a function that takes two integers as parameters, adds them, and returns the result. Call this function with two integers and print the result.

  9. Lambda Functions

    • Use the itemgetter function from the operator module to sort a list of tuples by the second element.

  10. List Slicing and Accessing Elements

    • Create a list of integers. Print the first 5 elements, the last 5 elements, and every other element in the list.

Targeted Drills in Python

1
2

x = 5
print(x)

1
2
3

nums = [1, 2, 3, 4, 5]
for num in nums:
    print(num)

1
2
3
4

list1 = [1, 2, 3]
list2 = [4, 5, 6]
zipped = list(zip(list1, list2))
print(zipped)

1
2
3
4

from operator import itemgetter
tuples = [(1, 2), (3, 1), (2, 3)]
sorted_tuples = sorted(tuples, key=itemgetter(1))
print(sorted_tuples)

1
2
3
4
5
6

import heapq
heap = []
heapq.heappush(heap, 3)
heapq.heappush(heap, 1)
heapq.heappush(heap, 2)
print(heapq.heappop(heap))

1
2
3

nums = [1, 2, 3, 4, 5]
prefix_sums = [sum(nums[:i+1]) for i in range(len(nums))]
print(prefix_sums)

1
2
3
4
5
6

nums = [1, 2, 3, 4, 5]
for num in nums:
    if num % 2 == 0:
        print(num)
    else:
        print(f"{num} is odd")

1
2
3

def add(x, y):
    return x + y
print(add(3, 4))

1
2
3
4

from operator import itemgetter
tuples = [(1, 2), (3, 1), (2, 3)]
sorted_tuples = sorted(tuples, key=itemgetter(1))
print(sorted_tuples)

1
2
3
4
5
6

nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 

10]
print(nums[:5])
print(nums[-5:])
print(nums[::2])

10 Prerequisite LeetCode Problems

The problem “74. Search a 2D Matrix” involves binary search within a sorted matrix. Here are 10 problems of lesser complexity that can help you prepare for this problem:

  1. 278. First Bad Version: It’s a simple application of binary search.

  2. 704. Binary Search: It’s the classic binary search problem on a 1D array.

  3. 35. Search Insert Position: This problem is another variation of binary search where you have to find the insert position of a target in a sorted array.

  4. 153. Find Minimum in Rotated Sorted Array: This problem involves binary search in a slightly modified sorted array.

  5. 33. Search in Rotated Sorted Array: This problem also deals with searching in a modified sorted array.

  6. 349. Intersection of Two Arrays: This problem requires you to find the intersection of two arrays. One of the ways to solve it is by using binary search.

  7. 367. Valid Perfect Square: This problem asks you to determine if a number is a perfect square, where binary search can be used to optimize the process.

  8. 441. Arranging Coins: In this problem, binary search can be used to find the maximum number of complete rows of coins that can be formed.

  9. 658. Find K Closest Elements: This problem involves finding the closest elements to a target in a sorted array, where binary search can be used to find the starting point efficiently.

  10. 475. Heaters: This problem asks for the minimum radius of heaters to warm all the houses. One of the approaches to solve this problem involves sorting and binary search.

By solving these problems, you’ll be more familiar with binary search and its variations which would be helpful for problem “74. Search a 2D Matrix”.

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:You are given an m x n integer matrix matrix with the following two properties:

Each row is sorted in non-decreasing order. The first integer of each row is greater than the last integer of the previous row. Given an integer target, return true if target is in matrix or false otherwise.

You must write a solution in O(log(m * n)) time complexity.

Example 1:

Input: matrix = [[1,3,5,7],[10,11,16,20],[23,30,34,60]], target = 3 Output: true

Example 2:

Input: matrix = [[1,3,5,7],[10,11,16,20],[23,30,34,60]], target = 13 Output: false

Constraints:

m == matrix.length n == matrix[i].length 1 <= m, n <= 100 -104 <= matrix[i][j], target <= 104

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?

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.