Range Sum Query 2D - Mutable

We need to create a class that can efficiently update individual elements of a matrix and calculate the sum of a region within the matrix. A good approach for this is to use a 2D Fenwick Tree (also known as a Binary Indexed Tree).

Approach:

Initialize: Initialize the Fenwick Tree with the given matrix.
Update: When updating a value at (row, col), update the corresponding value in the Fenwick Tree.
Sum Region: To find the sum of a region, use the Fenwick Tree to quickly calculate the sum of values within the rectangle.

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class NumMatrix:

    def __init__(self, matrix: List[List[int]]):
        self.m, self.n = len(matrix), len(matrix[0])
        self.matrix = matrix
        self.fenwick_tree = [[0] * (self.n + 1) for _ in range(self.m + 1)]
        for row in range(self.m):
            for col in range(self.n):
                self.update_fenwick_tree(row, col, matrix[row][col])

    def update(self, row: int, col: int, val: int) -> None:
        diff = val - self.matrix[row][col]
        self.matrix[row][col] = val
        self.update_fenwick_tree(row, col, diff)

    def sumRegion(self, row1: int, col1: int, row2: int, col2: int) -> int:
        return self.query(row2, col2) - self.query(row2, col1 - 1) - self.query(row1 - 1, col2) + self.query(row1 - 1, col1 - 1)

    def update_fenwick_tree(self, row: int, col: int, diff: int):
        row += 1
        while row <= self.m:
            col_idx = col + 1
            while col_idx <= self.n:
                self.fenwick_tree[row][col_idx] += diff
                col_idx += col_idx & -col_idx
            row += row & -row

    def query(self, row: int, col: int) -> int:
        row += 1
        result = 0
        while row > 0:
            col_idx = col + 1
            while col_idx > 0:
                result += self.fenwick_tree[row][col_idx]
                col_idx -= col_idx & -col_idx
            row -= row & -row
        return result

Key Takeaways:

The 2D Fenwick Tree is an efficient data structure for this problem, allowing for fast updates and range sum queries.
The initialization step builds the Fenwick Tree from the given matrix.
The update method alters a single value, and the sumRegion method calculates the sum of values within a specific rectangle.
The time complexity for both update and sumRegion is (O(\log m \cdot \log n)), where (m) and (n) are the dimensions of the matrix.

10 Prerequisite LeetCode Problems

“308. Range Sum Query 2D - Mutable” involves 2D arrays, range queries, and data structures like Binary Indexed Trees (BIT) or Segment Trees. Here are 10 problems:

303. Range Sum Query - Immutable: A simpler version with 1D array. Understanding this will help you to tackle 2D array problems.
304. Range Sum Query 2D - Immutable: This is the immutable version of the same problem, which is a good starting point to understand the mutable version.
307. Range Sum Query - Mutable: This is a simpler version of the problem where you deal with 1D arrays.
70. Climbing Stairs: This problem will help you understand the basics of dynamic programming.
200. Number of Islands: This problem will give you experience with handling 2D arrays.
121. Best Time to Buy and Sell Stock: This will introduce you to the concept of range queries in a simpler context.
53. Maximum Subarray: Another problem involving range queries.
198. House Robber: A dynamic programming problem that can help you understand how to tackle subproblems and build up to a solution.
322. Coin Change: This problem involves dynamic programming and can provide insight into handling different cases for the problem at hand.
155. Min Stack: Although it involves a different data structure, this problem is a good exercise for understanding the mutable data structures.

Problem Classification

Problem Statement: Given a 2D matrix matrix, handle multiple queries of the following types: Update the value of a cell in matrix. Calculate the sum of the elements of matrix inside the rectangle defined by its upper left corner (row1, col1) and lower right corner (row2, col2). Implement the NumMatrix class: NumMatrix(int[][] matrix) Initializes the object with the integer matrix matrix. void update(int row, int col, int val) Updates the value of matrix[row][col] to be val. int sumRegion(int row1, int col1, int row2, int col2) Returns the sum of the elements of matrix inside the rectangle defined by its upper left corner (row1, col1) and lower right corner (row2, col2).

Example 1:

Input [“NumMatrix”, “sumRegion”, “update”, “sumRegion”] [[[[3, 0, 1, 4, 2], [5, 6, 3, 2, 1], [1, 2, 0, 1, 5], [4, 1, 0, 1, 7], [1, 0, 3, 0, 5]]], [2, 1, 4, 3], [3, 2, 2], [2, 1, 4, 3]] Output [null, 8, null, 10]

Explanation NumMatrix numMatrix = new NumMatrix([[3, 0, 1, 4, 2], [5, 6, 3, 2, 1], [1, 2, 0, 1, 5], [4, 1, 0, 1, 7], [1, 0, 3, 0, 5]]); numMatrix.sumRegion(2, 1, 4, 3); // return 8 (i.e. sum of the left red rectangle) numMatrix.update(3, 2, 2); // matrix changes from left image to right image numMatrix.sumRegion(2, 1, 4, 3); // return 10 (i.e. sum of the right red rectangle)

Constraints:

m == matrix.length n == matrix[i].length 1 <= m, n <= 200 -1000 <= matrix[i][j] <= 1000 0 <= row < m 0 <= col < n -1000 <= val <= 1000 0 <= row1 <= row2 < m 0 <= col1 <= col2 < n At most 5000 calls will be made to sumRegion and update.

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.