What is the time complexity of Range Sum Query 2D - Immutable?

The optimal solution has a time complexity of O(m*n).

What is the space complexity of Range Sum Query 2D - Immutable?

The space complexity is O(m*n).

What data structures are used to solve Range Sum Query 2D - Immutable?

This problem uses Array, Design, Matrix, Prefix Sum. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Range Sum Query 2D - Immutable?

Insight 1: Prefix sum technique can be extended to two dimensions to pre-compute sums of regions efficiently. Insight 2: By pre-computing cumulative sums, we can calculate the sum of any sub-rectangle using a constant number of operations in the `sumRegion` method.

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Range Sum Query 2D - Immutable - LeetCode 304 Solution

Q: Which companies ask LeetCode 304. Range Sum Query 2D - Immutable in interviews?

This problem is frequently asked at Applied Intuition, DoorDash, Lyft, Nvidia, Snowflake and 1 more companies.

Range Sum Query 2D - Immutable - Complete Solution Guide

Range Sum Query 2D - Immutable is LeetCode problem 304, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given a 2D matrix matrix , handle multiple queries of the following type: 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 . 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) a

Detailed Explanation

The problem asks us to implement a `NumMatrix` class that can efficiently calculate the sum of elements within a rectangular region of a given 2D matrix. We need to pre-process the matrix in the constructor and then provide a `sumRegion` method that returns the sum of the elements within a given rectangle (defined by its top-left and bottom-right corners) in O(1) time. The main constraint is to make the `sumRegion` method as fast as possible, which suggests pre-computation during initialization.

Solution Approach

The solution uses a 2D prefix sum (or cumulative sum) approach. During the `NumMatrix` constructor, we create a `dp` table of size (m+1) x (n+1), where `dp[i][j]` stores the sum of all elements in the submatrix `matrix[0...i-1][0...j-1]`. The `sumRegion` method calculates the sum of the desired rectangle by using the pre-computed sums in the `dp` table. Specifically, it uses the formula: `sumRegion = dp[row2+1][col2+1] - dp[row1][col2+1] - dp[row2+1][col1] + dp[row1][col1]` This formula effectively calculates the sum of the target rectangle by adding the sum of the entire rectangle from (0,0) to (row2, col2), and then subtracting the sums of the rectangles above and to the left of the desired area. Finally, it adds the overlapping region that was subtracted twice.

Step-by-Step Algorithm

Step 1: Initialize the `dp` table with dimensions (m+1) x (n+1) and fill it with zeros.
Step 2: Iterate through the original matrix `matrix` from row 0 to m-1 and column 0 to n-1.
Step 3: For each element `matrix[r][c]`, calculate the cumulative sum `dp[r+1][c+1]` using the formula: `dp[r+1][c+1] = dp[r][c+1] + dp[r+1][c] - dp[r][c] + matrix[r][c]`.
Step 4: In the `sumRegion` method, given `row1`, `col1`, `row2`, and `col2`, return the sum of the rectangle using the formula: `dp[row2+1][col2+1] - dp[row1][col2+1] - dp[row2+1][col1] + dp[row1][col1]`.

Key Insights

Insight 1: Prefix sum technique can be extended to two dimensions to pre-compute sums of regions efficiently.
Insight 2: By pre-computing cumulative sums, we can calculate the sum of any sub-rectangle using a constant number of operations in the `sumRegion` method.
Insight 3: Using a DP table with an extra row and column (initialized to 0) simplifies the calculation of cumulative sums and avoids boundary checks.

Complexity Analysis

Time Complexity: O(m*n)

Space Complexity: O(m*n)

Topics

This problem involves: Array, Design, Matrix, Prefix Sum.

Companies

Asked at: Applied Intuition, DoorDash, Lyft, Nvidia, Snowflake, Upstart.

Range Sum Query 2D - Immutable | LeetCode 304

Medium

Array Design Matrix Prefix Sum

Time: O(m*n)Space: O(m*n)

Solution Code

Available in:

class NumMatrix {
    private int[][] dp;

    public NumMatrix(int[][] matrix) {
        int rows = matrix.length;
        if (rows == 0) return;
        int cols = matrix[0].length;
        dp = new int[rows + 1][cols + 1];
        for (int r = 0; r < rows; r++) {
            for (int c = 0; c < cols; c++) {
                dp[r + 1][c + 1] = dp[r][c + 1] + dp[r + 1][c] - dp[r][c] + matrix[r][c];
            }
        }
    }
    
    public int sumRegion(int row1, int col1, int row2, int col2) {
        return dp[row2 + 1][col2 + 1] - dp[row1][col2 + 1] - dp[row2 + 1][col1] + dp[row1][col1];
    }
}

/**
 * Your NumMatrix object will be instantiated and called as such:
 * NumMatrix obj = new NumMatrix(matrix);
 * int param_1 = obj.sumRegion(row1,col1,row2,col2);
 */

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Problem Info

Description

Problem 304: Range Sum Query 2D - Immutable - Detailed explanation and solution approaches.

Topics

Array Design Matrix Prefix Sum

Companies

Applied IntuitionDoorDashLyftNvidiaSnowflakeUpstart

Stats

Difficulty:Medium

Languages:4

Companies:6

Problem #:304

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

4 steps

1
Step 1: Initialize the `dp` table with dimensions (m+1) x (n+1) and fill it with zeros.
2
Step 2: Iterate through the original matrix `matrix` from row 0 to m-1 and column 0 to n-1.
3
Step 3: For each element `matrix[r][c]`, calculate the cumulative sum `dp[r+1][c+1]` using the formula: `dp[r+1][c+1] = dp[r][c+1] + dp[r+1][c] - dp[r][c] + matrix[r][c]`.
4
Step 4: In the `sumRegion` method, given `row1`, `col1`, `row2`, and `col2`, return the sum of the rectangle using the formula: `dp[row2+1][col2+1] - dp[row1][col2+1] - dp[row2+1][col1] + dp[row1][col1]`.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(m*n)

💾

Space Complexity

O(m*n)

📝

Why?

The time complexity of the constructor is O(m*n) because we need to iterate through all `m*n` elements of the input matrix to compute the prefix sums. The space complexity is also O(m*n) because we store the prefix sums in a `dp` table of size (m+1)*(n+1). The `sumRegion` method only performs a few constant-time lookups and arithmetic operations on the `dp` table, so its time complexity is O(1).

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Naive approach (without pre-computation)

Calculate the sum of the region each time the `sumRegion` method is called by iterating through all elements in the specified rectangle.

Time: O((row2-row1) * (col2-col1))Space: O(1)

Trade-offs: Simpler to implement, but very inefficient if `sumRegion` is called many times. Only suitable if `sumRegion` is rarely called. This approach avoids the extra memory overhead but becomes very slow for multiple queries.

✨

Key Insights

1Insight 1: Prefix sum technique can be extended to two dimensions to pre-compute sums of regions efficiently.
2Insight 2: By pre-computing cumulative sums, we can calculate the sum of any sub-rectangle using a constant number of operations in the `sumRegion` method.
3Insight 3: Using a DP table with an extra row and column (initialized to 0) simplifies the calculation of cumulative sums and avoids boundary checks.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Edge Cases

Edge case 1: Empty matrix. The Java and C++ solutions handle the case of an empty matrix during initialization, ensuring that no errors occur. The Python code handles it implicitly.
Edge case 2: `row1`, `col1`, `row2`, `col2` being equal. The formula still works correctly, returning the value of the single element in that location, which is the desired behavior.
Edge case 3: Input validation. The problem statement defines the constraints `0 <= row1 <= row2 < m` and `0 <= col1 <= col2 < n`, so the provided solutions don't include extra input validation for these cases. However, in a real-world scenario, it might be beneficial to add these checks to ensure the input values are within the valid range.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Incorrectly calculating the prefix sums. The formula for updating the `dp` table must be correct to ensure accurate results. Forgetting to add `matrix[r][c]` or incorrectly subtracting the overlapping region will lead to incorrect sums.

✗Common mistake 2: Off-by-one errors. The `dp` table is one size larger than the input matrix to simplify the calculations. Failing to adjust the row and column indices correctly when accessing the `dp` table in the `sumRegion` method can lead to incorrect results.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Image processing. Calculating the sum of pixel values within a region of an image can be useful for various tasks like blurring, edge detection, and feature extraction. The prefix sum technique can be used to speed up these calculations.

🏦Real-world scenario 2: Data analysis. Analyzing sales data for a particular region within a larger dataset. For example, quickly calculating the total sales within a specific geographical area (defined by coordinates) over a period of time.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

If stuck, try working through a small example by hand. The pattern often reveals itself.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: October 3, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 14

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 7

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Range Sum Query 2D - Immutable"?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether design techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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