/**
* Given a 2D matrix <i>matrix</>, find the sum of the elements inside the rectangle defined by its upper left corner
* (row1,
* col1) and lower right corner (row2, col2).
* <p>
* Range Sum Query 2D
* The above rectangle (with the red border) is defined by (row1, col1) = (2, 1) and (row2, col2) = (4, 3), which
* contains sum = 8.
* <p>
* Example:
* Given matrix = [
* [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]
* ]
* <p>
* sumRegion(2, 1, 4, 3) -> 8
* sumRegion(1, 1, 2, 2) -> 11
* sumRegion(1, 2, 2, 4) -> 12
* <p>
* Note:
* You may assume that the matrix does not change.
* There are many calls to sumRegion function.
* You may assume that row1 ≤ row2 and col1 ≤ col2.
* <p>
* Tags: Dynamic Programming
* Similar Problems: (E) Range Sum Query - Immutable, (H) Range Sum Query 2D - Mutable
*/
public class RangeSumQuery2DImmutable {
public class NumMatrix {
/**
* Pre-compute sum from (0, 0) to current point in another matrix
* Sum from (0, 0) to (row, col) is A + B - C + D
* 1. A: sum from (0, 0) to (row - 1, col)
* 2. B: sum from (0, 0) to (row, col - 1)
* 3. C: sum from (0, 0) to (row - 1, col - 1)
* 4. D: sum from (row ,col) to (row, col), which is matrix[row][col] itself
* C is added twice in A and B
*/
private int[][] dp;
public NumMatrix(int[][] matrix) {
if (matrix == null || matrix.length == 0 || matrix[0].length == 0) return;
// Make dp one row and one column bigger than matrix to avoid trivial range validation
dp = new int[matrix.length + 1][matrix[0].length + 1];
for (int r = 0; r < matrix.length; r++) {
for (int c = 0; c < matrix[0].length; c++) {
dp[r + 1][c + 1] = dp[r + 1][c] + dp[r][c + 1] + matrix[r][c] - dp[r][c];
}
}
}
/**
* A: Sum of (0, 0) to (row2, col2) is dp[row2 + 1][col2 + 1]
* B: Sum of (0, 0) to (row1 - 1, col2) is dp[row1][col2 + 1]
* C: Sum of (0, 0) to (row2, col1 - 1) is dp[row2 + 1][col1]
* D: Sum of (0, 0) to (row1 - 1, col1 - 1) is dp[row1][col1]
* Range sum is A - B - C + D (D is subtracted twice in B and C)
*/
public 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];
}
}
// 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);
}