> ## Documentation Index
> Fetch the complete documentation index at: https://leetcode-py.wisl.dev/llms.txt
> Use this file to discover all available pages before exploring further.

> ## Agent Instructions
> leetcode-py is a Python LeetCode practice environment generator with one CLI: lcpy. It is not a service or platform.
> Each problem is a directory under leetcode/ with README.md, solution.py, test_solution.py, helpers.py, and playground.ipynb. lcpy gen creates them from JSON templates bundled with the package.
> Examples are backed by tests; copy them verbatim.

# Range Sum Query 2D - Immutable Python Solution

> Tested Python solution for LeetCode 304 with 15 pytest cases. Generate a practice environment with lcpy.

LeetCode 304, Medium. Topics: Array, Design, Matrix, Prefix Sum. [View on LeetCode](https://leetcode.com/problems/range-sum-query-2d-immutable/description/).

Generate this problem as a practice environment: tested reference solution, 15 [parametrized pytest cases](/practice/testing), and a playground notebook:

```bash theme={"theme":{"light":"github-light","dark":"github-dark"}}
lcpy gen -n 304   # by problem number
lcpy gen -s range_sum_query_2d_immutable   # by problem name
```

## Problem

Given a 2D `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)` and **lower right corner** `(row2, col2)`.

You must design an algorithm where `sumRegion` works on `O(1)` time complexity.

### Examples

![Example 1](https://assets.leetcode.com/uploads/2021/03/14/sum-grid.jpg)

```
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]

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
numMatrix.sumRegion(1, 1, 2, 2); // return 11
numMatrix.sumRegion(1, 2, 2, 4); // return 12
```

### Constraints

* m == matrix.length
* n == matrix\[i].length
* 1 \<= m, n \<= 200
* -10^4 \<= matrix\[i]\[j] \<= 10^4
* 0 \<= row1 \<= row2 \< m
* 0 \<= col1 \<= col2 \< n
* At most 10^4 calls will be made to sumRegion.

## Solution

Reference implementation from [solution.py on GitHub](https://github.com/wislertt/leetcode-py/blob/main/leetcode/range_sum_query_2d_immutable/solution.py), full suite in [test\_solution.py](https://github.com/wislertt/leetcode-py/blob/main/leetcode/range_sum_query_2d_immutable/test_solution.py):

```python theme={"theme":{"light":"github-light","dark":"github-dark"}}
class NumMatrix:
    # Time: O(m * n) precompute, O(1) per query
    # Space: O(m * n)
    def __init__(self, matrix: list[list[int]]) -> None:
        rows = len(matrix)
        cols = len(matrix[0]) if rows else 0
        # prefix[r][c] = sum of matrix[0..r-1][0..c-1] (1-indexed).
        self.prefix = [[0] * (cols + 1) for _ in range(rows + 1)]
        for r in range(rows):
            for c in range(cols):
                self.prefix[r + 1][c + 1] = (
                    matrix[r][c] + self.prefix[r][c + 1] + self.prefix[r + 1][c] - self.prefix[r][c]
                )

    def sum_region(self, row1: int, col1: int, row2: int, col2: int) -> int:
        return (
            self.prefix[row2 + 1][col2 + 1]
            - self.prefix[row1][col2 + 1]
            - self.prefix[row2 + 1][col1]
            + self.prefix[row1][col1]
        )
```

## Complexity

| Time                                 | Space     |
| ------------------------------------ | --------- |
| O(m \* n) precompute, O(1) per query | O(m \* n) |

## Tags

[NeetCode 250](/catalog/neetcode-250), [NeetCode All](/catalog/neetcode).