Range Sum Query 2D - Immutable

MEDIUM

Description

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) and lower right corner (row2, col2).

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

Example 1:

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 (i.e sum of the red rectangle)
numMatrix.sumRegion(1, 1, 2, 2); // return 11 (i.e sum of the green rectangle)
numMatrix.sumRegion(1, 2, 2, 4); // return 12 (i.e sum of the blue rectangle)

Constraints:

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.

Approaches

Checkout 3 different approaches to solve Range Sum Query 2D - Immutable. Click on different approaches to view the approach and algorithm in detail.

Brute Force - Direct Calculation

The most straightforward approach is to calculate the sum for each query by iterating through all elements in the specified rectangle. For each sumRegion call, we traverse the matrix from (row1, col1) to (row2, col2) and accumulate the sum.

Algorithm

Store the original matrix in the constructor
For each sumRegion query:
- Initialize sum to 0
- Iterate through rows from row1 to row2
- For each row, iterate through columns from col1 to col2
- Add each element to the sum
- Return the accumulated sum

This approach directly calculates the sum for each query without any preprocessing. When sumRegion(row1, col1, row2, col2) is called, we iterate through all rows from row1 to row2 and all columns from col1 to col2, adding each element to our running sum.

class NumMatrix {
    private int[][] matrix;
    
    public NumMatrix(int[][] matrix) {
        this.matrix = matrix;
    }
    
    public int sumRegion(int row1, int col1, int row2, int col2) {
        int sum = 0;
        for (int i = row1; i <= row2; i++) {
            for (int j = col1; j <= col2; j++) {
                sum += matrix[i][j];
            }
        }
        return sum;
    }
}

This approach is simple to implement and understand, but it doesn't meet the O(1) requirement for sumRegion operations.

Complexity Analysis

Time Complexity: O(1) for constructor, O(m*n) for sumRegion in worst case where m and n are the dimensions of the query rectangleSpace Complexity: O(1) additional space (only storing reference to original matrix)

Pros and Cons

Pros:

Simple and straightforward implementation
No additional space required beyond storing the original matrix
Easy to understand and debug

Cons:

Does not meet the O(1) requirement for sumRegion
Inefficient for multiple queries as it recalculates the same sums
Performance degrades significantly with larger query rectangles

Code Solutions

Checking out 4 solutions in different languages for Range Sum Query 2D - Immutable. Click on different languages to view the code.

class NumMatrix { private int [][] s ; public NumMatrix ( int [][] matrix ) { int m = matrix . length , n = matrix [ 0 ]. length ; s = new int [ m + 1 ][ n + 1 ]; for ( int i = 0 ; i < m ; ++ i ) { for ( int j = 0 ; j < n ; ++ j ) { s [ i + 1 ][ j + 1 ] = s [ i + 1 ][ j ] + s [ i ][ j + 1 ] - s [ i ][ j ] + matrix [ i ][ j ]; } } } public int sumRegion ( int row1 , int col1 , int row2 , int col2 ) { return s [ row2 + 1 ][ col2 + 1 ] - s [ row2 + 1 ][ col1 ] - s [ row1 ][ col2 + 1 ] + s [ 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); */

Video Solution

Watch the video walkthrough for Range Sum Query 2D - Immutable

Patterns:

DesignPrefix Sum

Data Structures:

ArrayMatrix

Companies:

DoorDash |Nvidia |Snowflake |Lyft |