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## Problem

Given a 2D matrix matrix, handle multiple queries of the following types:

1. Update the value of a cell in matrix.
2. 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.
• void update(int row, int col, int val) Updates the value of matrix[row][col] to be val.
• 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).

Example 1:

Input
["NumMatrix", "sumRegion", "update", "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], [3, 2, 2], [2, 1, 4, 3]]
Output
[null, 8, null, 10]

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 left red rectangle)
numMatrix.update(3, 2, 2);       // matrix changes from left image to right image
numMatrix.sumRegion(2, 1, 4, 3); // return 10 (i.e. sum of the right red rectangle)


## Code

class NumMatrix {
class IndexTree{
int[][] sums;
public IndexTree(int row, int col){
sums = new int[row + 1][col + 1];
}

public void update(int row, int col, int delta){
for(int i = row; i < sums.length; i += i & -i){
for(int j = col; j < sums[0].length; j += j & -j){
sums[i][j] += delta;
}
}
}

public int query(int row, int col){
int sum = 0;
for(int i = row; i > 0; i -= i & -i){
for(int j = col; j > 0; j -= j & -j){
sum += sums[i][j];
}
}

return sum;
}
}

IndexTree root;
int[][] matrix_;
public NumMatrix(int[][] matrix) {
if(matrix == null || matrix.length == 0) return;

matrix_ = matrix;
root = new IndexTree(matrix.length, matrix[0].length);
for(int i = 0; i < matrix.length; i++){
for(int j = 0; j < matrix[0].length; j++){
root.update(i + 1, j + 1, matrix_[i][j]);
}
}
}

public void update(int row, int col, int val) {
root.update(row + 1, col + 1, val - matrix_[row][col]);
matrix_[row][col] = val;
}

public int sumRegion(int row1, int col1, int row2, int col2) {
return root.query(row2 + 1, col2 + 1)
- root.query(row2 + 1, col1) - root.query(row1, col2 + 1)
+ root.query(row1, col1);
}
}

/**
* Your NumMatrix object will be instantiated and called as such:
* NumMatrix obj = new NumMatrix(matrix);
* obj.update(row,col,val);
* int param_2 = obj.sumRegion(row1,col1,row2,col2);
*/