How do you solve Range Sum Query 2D - Immutable on LeetCode?

Range Sum Query 2D - Immutable (LeetCode #304) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(1) time complexity and O(m * n) space complexity. Key insight: Using a prefix sum matrix allows for efficient range queries.

What is the brute force solution for Range Sum Query 2D - Immutable?

The brute force approach for Range Sum Query 2D - Immutable is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach involves directly summing the elements within the specified rectangle for each query. This is straightforward but inefficient for large matrices or multiple queries. The optimal Optimal Solution approach improves this to O(1).

What algorithm or data structure is used to solve Range Sum Query 2D - Immutable?

Range Sum Query 2D - Immutable uses the following concepts: Array, Design, Matrix, Prefix Sum. The recommended patterns to study are: Prefix Sum, Dynamic Programming.

What are the interview tips for Range Sum Query 2D - Immutable?

Always clarify the constraints and expected time complexity before diving into coding. Explain your thought process clearly while coding to demonstrate your understanding. Practice explaining the inclusion-exclusion principle as it's a common technique in range queries.

What are common mistakes when solving Range Sum Query 2D - Immutable?

Not initializing the prefix sum matrix correctly. Forgetting to apply the inclusion-exclusion principle when calculating sums.

#304

Range Sum Query 2D - Immutable

Medium

ArrayDesignMatrixPrefix SumPrefix SumDynamic Programming

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(1)
Space	O(1)	O(m * n)

💡

Intuition

Time O(1)Space O(m * n)

By using a prefix sum matrix, we can preprocess the input matrix to allow for O(1) sum queries. This is like having a cheat sheet that tells you the sum of any rectangle in constant time.

⚙️

Algorithm

3 steps

1Step 1: Create a prefix sum matrix where each cell (i, j) contains the sum of all elements from (0, 0) to (i, j).
2Step 2: For each query, use the prefix sum matrix to calculate the sum of the specified rectangle using the inclusion-exclusion principle.
3Step 3: Return the calculated sum.

solution.py12 lines

1class NumMatrix:
2    def __init__(self, matrix):
3        self.matrix = matrix
4        self.rows = len(matrix)
5        self.cols = len(matrix[0]) if self.rows > 0 else 0
6        self.prefix_sum = [[0] * (self.cols + 1) for _ in range(self.rows + 1)]
7        for i in range(1, self.rows + 1):
8            for j in range(1, self.cols + 1):
9                self.prefix_sum[i][j] = (self.matrix[i-1][j-1] + self.prefix_sum[i-1][j] + self.prefix_sum[i][j-1] - self.prefix_sum[i-1][j-1])
10
11    def sumRegion(self, row1, col1, row2, col2):
12        return (self.prefix_sum[row2 + 1][col2 + 1] - self.prefix_sum[row1][col2 + 1] - self.prefix_sum[row2 + 1][col1] + self.prefix_sum[row1][col1])

ℹ

Complexity note: The time complexity is O(1) for each query because we are using precomputed sums. The space complexity is O(m * n) due to the prefix sum matrix.

1Using a prefix sum matrix allows for efficient range queries.
2Understanding the inclusion-exclusion principle is crucial for calculating sums in submatrices.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.