304. Range Sum Query 2D - Immutable

LeetCode 304. Range Sum Query 2D - Immutable

Description

Given a 2D matrix matrix, find the sum of the elements 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 the matrix array inside the rectangle defined by its upper left corner (row1, col1) and lower right corner (row2, col2).

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

Constraints:

• m == matrix.length

• n == matrix[i].length

• 1 <= m, n <= 200

• -105 <= matrix[i][j] <= 105

• 0 <= row1 <= row2 < m

• 0 <= col1 <= col2 < n

• At most 104 calls will be made to sumRegion.

Tags

Dynamic Programming, Design

Solution

We can pre-compute a cumulative region sum with respect to the origin at (0, 0), with the formula: $f(i,j)=f(i−1,j)+f(i,j−1)−f(i−1,j−1)+matrix[i][j]$

Based on the obtained partial-sum, we can formulate the region sum with given 2 points: $sumRegion(row1,col1,row2,col2)=sums[row2+1][col2+1]−sums[row1][col2+1]−sums[row2+1][col1]+sums[row1][col1]$

Complexity

• Time complexity: $O(mn)$for initializing sum; $O(1)$ for performing search;

• Space complexity: $O(mn)$

Code

type NumMatrix struct {
	sum [][]int
}

func Constructor(matrix [][]int) NumMatrix {
	if len(matrix) == 0 {
		return NumMatrix{}
	}
	sum := make([][]int, len(matrix)+1)
	sum[0] = make([]int, len(matrix[0])+1)
	for i, row := range matrix {
		sum[i+1] = make([]int, len(matrix[0])+1)
		for j, v := range row {
			sum[i+1][j+1] = sum[i+1][j] + sum[i][j+1] - sum[i][j] + v
		}
	}
	return NumMatrix{sum: sum}
}

func (this *NumMatrix) SumRegion(row1 int, col1 int, row2 int, col2 int) int {
	return this.sum[row2+1][col2+1] - this.sum[row2+1][col1] - this.sum[row1][col2+1] + this.sum[row1][col1]
}

Reference

1. 二维区域和检索 - 矩阵不可变