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]
]
sumRegion(2, 1, 4, 3) -> 8
sumRegion(1, 1, 2, 2) -> 11
sumRegion(1, 2, 2, 4) -> 12
这道题目属于二维的矩阵运算,题目中假设矩阵没有变化,对比前面的一道题 Range Sum Query - Mutable来看,我们也可以用树状数组来做,这里我们就需要构建一个二维的树状数组。具体如何构建一个树状数组请看连接中的那道题。这里先给出一个AC的代码:
public class NumMatrix { int[][] matrix; int[][] BIT; public NumMatrix(int[][] matrix) { this.matrix = matrix; if(matrix == null || matrix.length == 0 ) return; BIT = new int[matrix.length + 1][matrix[0].length + 1]; for(int i = 0; i < matrix.length; i++) for(int j = 0; j < matrix[0].length; j++) init(i, j, matrix[i][j]); } public void init(int i, int j, int val) { j ++; while(j <= matrix[0].length) { BIT[i][j] += val; j += (j & -j); } } /* implement mutable public void update(int i, int j, int val) { int differ = val - matrix[i][j]; matrix[i][j] = val; init(i, j, differ); } */ public int sumRegion(int row1, int col1, int row2, int col2) { int result = 0; for(int i = row1; i <= row2; i++) { result += getSumRange(i, col1, col2); } return result; } public int getSumRange(int i, int start, int end) { return getSum(i, end) - getSum(i, start - 1); } public int getSum(int i, int point) { point ++; int sum = 0; while(point > 0) { sum += BIT[i][point]; point -= (point & -point); } return sum; } } // 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);
代码中注释的部分是update方法,对应mutable的情况。
因为树状数组的优势在于解决对变化的数据求和,在不变的情况下,优势就没有那么明显,因为构建一个树状数组也需要消耗时间和空间。这道题我们采用动态规划的思想解决比较合适。
代码如下:
public class NumMatrix { int[][] dp; public NumMatrix(int[][] matrix) { if(matrix == null || matrix.length == 0) return; dp = new int[matrix.length + 1][matrix[0].length + 1]; for(int i = 1; i <= matrix.length; i++) { for(int j = 1; j <= matrix[0].length; j++) { dp[i][j] = dp[i - 1][j] + dp[i][j - 1] + matrix[i - 1][j - 1] - dp[i - 1][j - 1]; } } } public int sumRegion(int row1, int col1, int row2, int col2) { return dp[row2 + 1][col2 + 1] - dp[row2 + 1][col1] + dp[row1][col1] - dp[row1][col2 + 1]; } } // 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);