Range Sum Query 2D Immutable Leetcode Problem 304 [Python Solution]

Here’s to another coding adventure on Range Sum Query 2D Immutable Leetcode Problem! Everything you need to know.

This LeetCode problem falls under the “Arrays & Hashing” category and is categorized as a medium-level problem.

We’ll explore the problem’s overview, constraints, an efficient Python solution, and the reasoning behind our approach.

By the end of this guide, you’ll have a solid grasp of how to solve this problem.

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Problem Overview

The problem at hand is as follows:

You are given a 2D matrix called “matrix.” The task is to handle multiple queries of the following type:

1. Calculate the sum of the elements of the matrix inside the rectangle defined by its upper left corner (row1, col1) and lower right corner (row2, col2).

To achieve this, we’ll implement the NumMatrix class with the following methods:

• __init__(self, matrix): Initializes the object with the integer matrix matrix.
• sumRegion(self, row1, col1, row2, col2): Returns the sum of the elements of the matrix inside the rectangle defined by its upper left corner (row1, col1) and lower right corner (row2, col2).

The challenging part is that we need to design an algorithm where the sumRegion method works in O(1) time complexity.

Example 1

Let’s illustrate this with an example:

Input
["NumMatrix", "sumRegion", "sumRegion", "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], [1, 1, 2, 2], [1, 2, 2, 4]]

Output
[null, 8, 11, 12]

In this example, we create a NumMatrix object with the provided 2D matrix.

Then, we call the sumRegion method with different query parameters and expect the given outputs.

• The first query sumRegion(2, 1, 4, 3) returns 8, which is the sum of the red rectangle.
• The second query sumRegion(1, 1, 2, 2) returns 11, the sum of the green rectangle.
• The third query sumRegion(1, 2, 2, 4) returns 12, the sum of the blue rectangle.

Constraints

It’s essential to understand the constraints of this problem:

• m == matrix.length
• n == matrix[i].length
• 1 <= m, n <= 200
• -104 <= matrix[i][j] <= 104
• 0 <= row1 <= row2 < m
• 0 <= col1 <= col2 < n
• At most 104 calls will be made to sumRegion.

Efficient Python Code Solution

Now, let’s dive into the efficient Python code solution for this problem.

We’ll use the concept of prefix sums to optimize our calculations.

class NumMatrix:
    def __init__(self, matrix):
        self.sum_ = [[0] * (len(matrix[0]) + 1) for _ in range(len(matrix) + 1)]
        for i, line in enumerate(matrix):
            previous = 0
            for j, num in enumerate(line):
                previous += num
                above = self.sum_[i][j + 1]
                self.sum_[i + 1][j + 1] = previous + above

    def sumRegion(self, row1, col1, row2, col2):
        sum_col2 = self.sum_[row2 + 1][col2 + 1] - self.sum_[row1][col2 + 1]
        sum_col1 = self.sum_[row2 + 1][col1] - self.sum_[row1][col1]
        return sum_col2 - sum_col1

The provided Python code efficiently solves the Range Sum Query 2D Immutable problem.

Let’s break down how it works:

1. __init__(self, matrix)

In the constructor of the NumMatrix class, we initialize the object with the integer matrix matrix.

Here’s how we achieve this:

• We create a 2D list called sum_ to store the cumulative sums for the input matrix.

This list has dimensions (m + 1) x (n + 1), where m is the number of rows, and n is the number of columns in the input matrix.

The extra row and column are used for the prefix sums to handle edge cases.

• We then iterate through each element in the input matrix while maintaining a running total (previous) for each row and storing it in the corresponding position in the sum_ matrix.
• To calculate the prefix sums, we also consider the value above the current element (above), which is obtained from the sum_ matrix.
• Finally, we update the sum_ matrix with the cumulative sums for the rows and columns.

2. sumRegion(self, row1, col1, row2, col2)

The sumRegion method allows us to calculate the sum of the elements of the matrix inside the specified rectangle defined by its upper left corner (row1, col1) and lower right corner (row2, col2) with O(1) time complexity.

Here’s how it works:

• We start by calculating the sum of the elements in the rectangle’s rightmost column (from col1 to col2).

We do this by subtracting the cumulative sum of the elements in the col1 column from the cumulative sum of the elements in the col2 + 1 column.

This result is stored in sum_col2.

• Next, we calculate the sum of the elements in the rectangle’s leftmost column (again from col1 to col2).

To do this, we subtract the cumulative sum of the elements in the col1 column from the cumulative sum of the elements in the col1 column in row row2 + 1.

This result is stored in sum_col1.

• Finally, we subtract sum_col1 from sum_col2 to obtain the sum of the elements within the specified rectangle, excluding the overlapping area.

The result is returned as the final answer.

This efficient approach allows us to perform range sum queries in constant time, making it a highly optimized solution for this problem.

Time and Space Complexity

Now, let’s analyze the time and space complexity of our solution:

–

Time Complexity:

• The __init__ method has a time complexity of O(m * n), where m and n are the dimensions of the input matrix.

This is because we iterate through the entire matrix once to compute the prefix sums.

• The sumRegion method works in O(1) time.

Regardless of the size of the matrix, the calculations for each query are constant time operations.

• Space Complexity:
• The space complexity is O(m * n) for the sum_ matrix, where m and n are the dimensions of the input matrix.

This additional space is required to store the prefix sums.

Reasoning Behind Our Approach

The key to solving this problem efficiently lies in the use of prefix sums.

By precomputing and storing the cumulative sums for each position in the matrix, we can quickly calculate the sum of any rectangular submatrix in constant time.

The initialization step sets up the prefix sum matrix, and the sumRegion method takes advantage of this precomputed information to provide rapid responses to queries.

In the __init__ method, we iterate through the input matrix and calculate the cumulative sums while considering the element’s value and the cumulative sum above it.

The additional rows and columns in the sum_ matrix serve as placeholders to handle edge cases and simplify the sum calculations.

The sumRegion method then leverages the precomputed prefix sums to calculate the sum of the specified rectangular region.

It avoids redundant calculations by subtracting the sum of the elements in the overlapping areas of the rectangle.

Overall, our approach optimizes both time and space complexity to efficiently address the problem’s requirements.

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Conclusion

In this guide, we’ve explored the Range Sum Query 2D Immutable problem on LeetCode.

We’ve provided a detailed explanation of the problem’s overview, constraints, and an efficient Python solution that leverages the concept of prefix sums to achieve O(1) time complexity for range sum queries.

By understanding the principles behind this solution, you’ve gained valuable insights into how to approach similar problems that involve cumulative sum calculations.

If you have any questions, suggestions, or comments, please feel free to engage with us in the comments section.

Happy coding and keep exploring the fascinating world of algorithms and data structures!

For more details and to practice this problem, you can visit the LeetCode problem page.

Stay curious and keep coding!