Question

Formatted question description: https://leetcode.ca/all/304.html

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.

Algorithm

Maintain a two-dimensional array dp, where dp[i][j] represents the sum of all the numbers in the cumulative interval (0, 0) to (i, j).

Then if we want to quickly find (r1 , c1) to (r2, c2) rectangular interval, only dp[r2][c2]-dp[r2][c1-1]-dp[r1-1][c2] + dp[r1-1 ][c1-1].

Code

public class Range_Sum_Query_2D_Immutable {

    public class NumMatrix {
        private int[][] matrix;
        private int[][] sum;

        public NumMatrix(int[][] matrix) {
            this.matrix = matrix;

            if (matrix == null || matrix.length == 0) {
                return;
            }

            int m = matrix.length;
            int n = matrix[0].length;

            sum = new int[m + 1][n + 1];

            for (int i = 1; i <= m; i++) {
                for (int j = 1; j <= n; j++) {
                    sum[i][j] = sum[i - 1][j] + sum[i][j - 1] - sum[i - 1][j - 1] + matrix[i - 1][j - 1];
                }
            }
        }

        public int sumRegion(int row1, int col1, int row2, int col2) {
            return sum[row2 + 1][col2 + 1] - sum[row2 + 1][col1] - sum[row1][col2 + 1] + sum[row1][col1];
        }
    }

// Your NumMatrix object will be instantiated and called as such:
// NumMatrix numMatrix = new NumMatrix(matrix);
// numMatrix.sumRegion(0, 1, 2, 3);
// numMatrix.sumRegion(1, 2, 3, 4);

}


############

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);
 */

class NumMatrix {
public:
    NumMatrix(vector<vector<int> > &matrix) {
        if (matrix.empty() || matrix[0].empty()) return;
        dp.resize(matrix.size() + 1, vector<int>(matrix[0].size() + 1, 0));
        for (int i = 1; i <= matrix.size(); ++i) {
            for (int j = 1; j <= matrix[0].size(); ++j) {
                dp[i][j] = dp[i][j - 1] + dp[i - 1][j] - dp[i - 1][j - 1] + matrix[i - 1][j - 1];
            }
        }
    }
    int sumRegion(int row1, int col1, int row2, int col2) {
        return dp[row2 + 1][col2 + 1] - dp[row1][col2 + 1] - dp[row2 + 1][col1] + dp[row1][col1];
    }
    
private:
    vector<vector<int> > dp;
};

'''
>>> a = [ [1,2,3], [4,5,6] ]
>>> b = a.copy()
>>> b
[[1, 2, 3], [4, 5, 6]]
'''
class NumMatrix:
    def __init__(self, matrix: List[List[int]]):
        m, n = len(matrix), len(matrix[0])
        self.s = [[0] * (n + 1) for _ in range(m + 1)]
        for i, row in enumerate(matrix):
            for j, v in enumerate(row):
                self.s[i + 1][j + 1] = (
                    self.s[i][j + 1] + self.s[i + 1][j] - self.s[i][j] + v
                )

    def sumRegion(self, row1: int, col1: int, row2: int, col2: int) -> int:
        return (
            self.s[row2 + 1][col2 + 1]
            - self.s[row2 + 1][col1]
            - self.s[row1][col2 + 1]
            + self.s[row1][col1]
        )


# Your NumMatrix object will be instantiated and called as such:
# obj = NumMatrix(matrix)
# param_1 = obj.sumRegion(row1,col1,row2,col2)

############

class NumMatrix(object):
  def __init__(self, matrix):
    """
    initialize your data structure here.
    :type matrix: List[List[int]]
    """
    self.dp = [[0] * len(matrix[0]) for i in range(0, len(matrix))]
    for i in range(0, len(matrix)):
      for j in range(0, len(matrix[0])):
        if i == 0:
          self.dp[0][j] = self.dp[0][j - 1] + matrix[i][j]
        elif j == 0:
          self.dp[i][0] = self.dp[i - 1][0] + matrix[i][j]
        else:
          self.dp[i][j] = self.dp[i - 1][j] + self.dp[i][j - 1] - self.dp[i - 1][j - 1] + matrix[i][j]

  def sumRegion(self, row1, col1, row2, col2):
    """
    sum of elements matrix[(row1,col1)..(row2,col2)], inclusive.
    :type row1: int
    :type col1: int
    :type row2: int
    :type col2: int
    :rtype: int
    """
    dp = self.dp

    diagSum = dp[row1 - 1][col1 - 1]
    totalSum = dp[row2][col2]
    leftSum = dp[row2][col1 - 1]
    upSum = dp[row1 - 1][col2]
    if row1 == 0:
      upSum = 0
      diagSum = 0
    if col1 == 0:
      leftSum = 0
      diagSum = 0
    return totalSum - leftSum - upSum + diagSum

type NumMatrix struct {
	s [][]int
}

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

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

/**
 * Your NumMatrix object will be instantiated and called as such:
 * obj := Constructor(matrix);
 * param_1 := obj.SumRegion(row1,col1,row2,col2);
 */

class NumMatrix {
    private s: number[][];

    constructor(matrix: number[][]) {
        const m = matrix.length;
        const n = matrix[0].length;
        this.s = new Array(m + 1).fill(0).map(() => new Array(n + 1).fill(0));
        for (let i = 0; i < m; ++i) {
            for (let j = 0; j < n; ++j) {
                this.s[i + 1][j + 1] =
                    this.s[i + 1][j] +
                    this.s[i][j + 1] -
                    this.s[i][j] +
                    matrix[i][j];
            }
        }
    }

    sumRegion(row1: number, col1: number, row2: number, col2: number): number {
        return (
            this.s[row2 + 1][col2 + 1] -
            this.s[row2 + 1][col1] -
            this.s[row1][col2 + 1] +
            this.s[row1][col1]
        );
    }
}

/**
 * Your NumMatrix object will be instantiated and called as such:
 * var obj = new NumMatrix(matrix)
 * var param_1 = obj.sumRegion(row1,col1,row2,col2)
 */

/**
 * @param {number[][]} matrix
 */
var NumMatrix = function (matrix) {
    const m = matrix.length;
    const n = matrix[0].length;
    this.s = new Array(m + 1).fill(0).map(() => new Array(n + 1).fill(0));
    for (let i = 0; i < m; ++i) {
        for (let j = 0; j < n; ++j) {
            this.s[i + 1][j + 1] =
                this.s[i + 1][j] +
                this.s[i][j + 1] -
                this.s[i][j] +
                matrix[i][j];
        }
    }
};

/**
 * @param {number} row1
 * @param {number} col1
 * @param {number} row2
 * @param {number} col2
 * @return {number}
 */
NumMatrix.prototype.sumRegion = function (row1, col1, row2, col2) {
    return (
        this.s[row2 + 1][col2 + 1] -
        this.s[row2 + 1][col1] -
        this.s[row1][col2 + 1] +
        this.s[row1][col1]
    );
};

/**
 * Your NumMatrix object will be instantiated and called as such:
 * var obj = new NumMatrix(matrix)
 * var param_1 = obj.sumRegion(row1,col1,row2,col2)
 */

304 - Range Sum Query 2D - Immutable

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