304. Range Sum Query 2D - Immutable

题目描述：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).



The above rectangle (with the red border) is defined by (row1, col1) = (2, 1) and (row2, col2) = (4, 3), which contains sum = 8.
Example:

[code]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

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.

解题思路：因为要多次调用，所以肯定需要一个数组来保存中间结果，想到用动态规划。但是如何设计动态规划的数组是个问题。我一开始用一个四维数组来表示，严重超时。
后来查看其他人的解法发现只需保存0,0,到行，列的和，通过加减得出想要的结果。

public class NumMatrix {
int[,] matrix;
int length0=0;
int length1=0;
int[][] sum = null;
public NumMatrix(int[,] matrix) {
this.matrix=matrix;
this.length0=matrix.GetLength(0);
this.length1=matrix.GetLength(1);
}

public int SumRegion(int row1, int col1, int row2, int col2) {
if (this.sum == null)
{
this.sum = new int[length0][];
for (int i = 0; i < length0; i++)
sum[i] = new int[length1];

int x = matrix[0, 0];
for (int i = 1; i < length1; i++)
{
x += matrix[0, i];
sum[0][i] = x;
}

int y = matrix[0, 0];
for (int i = 1; i < length0; i++)
{
y += matrix[i, 0];
sum[i][0] = y;
}

sum[0][0] = matrix[0, 0];

for (int i = 1; i < length0; i++)
{
for (int j = 1; j < length1; j++)
{
sum[i][j] = sum[i][j - 1] + sum[i - 1][j] - sum[i - 1][j - 1] + matrix[i,j];
}
}

}

if (row1 >= 1 && col1 >= 1)
return sum[row2][col2] - sum[row1 - 1][col2] - sum[row2][col1 - 1] + sum[row1 - 1][col1 - 1];
else if (row1 == 0 && col1 >= 1)
return sum[row2][col2] - sum[row2][col1 - 1];
else if (row1 >= 1 && col1 == 0)
return sum[row2][col2] - sum[row1 - 1][col2];
else
return sum[row2][col2];

}
}

// 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);
[/code]