Two, the tree-like array can be extended to two-dimensional.
Problem: A large matrix made up of numbers that can perform two kinds of operations
1 to a number in the Matrix plus an integer (can be positive and negative)
2 Query a matrix of all the numbers and requirements for each query, output results.
A one-dimensional tree array can easily be extended to two-dimensional, in two-dimensional cases: array a[][] The tree array is defined as:
C[x][y] =∑a[i][j], of which,
X-lowbit (x) + 1 <= i <= x,
Y-lowbit (y) + 1 <= J <= y.
For example, look at the composition of c[][].
Set the original two-dimensional array as:
A[][]={{a11,a12,a13,a14,a15,a16,a17,a18,a19},
{A21,a22,a23,a24,a25,a26,a27,a28,a29},
{A31,a32,a33,a34,a35,a36,a37,a38,a39},
{a41,a42,a43,a44,a45,a46,a47,a48,a49}};
Then it corresponds to the two-dimensional tree-like array c[][].
Remember:
B[1]={a11,a11+a12,a13,a11+a12+a13+a14,a15,a15+a16,...} This is a one-dimensional tree array of the first row
B[2]={a21,a21+a22,a23,a21+a22+a23+a24,a25,a25+a26,...} This is a one-dimensional, tree-like array of the second row
B[3]={a31,a31+a32,a33,a31+a32+a33+a34,a35,a35+a36,...} This is a one-dimensional, tree-like array of the third row
B[4]={a41,a41+a42,a43,a41+a42+a43+a44,a45,a45+a46,...} This is a one-dimensional, tree-like array of line fourth
So:
C[1][1]=a11,c[1][2]=a11+a12,c[1][3]=a13,c[1][4]=a11+a12+a13+a14,c[1][5]=a15,c[1][6]=a15+a16,...
This is a[][] the first row of the one-dimensional tree array
C[2][1]=a11+a21,c[2][2]=a11+a12+a21+a22,c[2][3]=a13+a23,c[2][4]=a11+a12+a13+a14+a21+a22+a23+a24,
C[2][5]=a15+a25,c[2][6]=a15+a16+a25+a26,...
This is the a[][] array of the first row and the second row after adding the tree-like array
C[3][1]=a31,c[3][2]=a31+a32,c[3][3]=a33,c[3][4]=a31+a32+a33+a34,c[3][5]=a35,c[3][6]=a35+a36,...
This is a[][] the third row of one-dimensional tree-like array
C[4][1]=a11+a21+a31+a41,c[4][2]=a11+a12+a21+a22+a31+a32+a41+a42,c[4][3]=a13+a23+a33+a43,...
This is the a[][] array of the first row + the second line + the third row and the fourth row after the tree array
Have you figured out the rule of a two-dimensional tree-like array c[][]? With a closer look, you will find that:
(1) In two-dimensional cases, if the A[i][j]=delta is modified, the corresponding two-dimensional tree-like array update function is:
[Java] View plain copy print? private void Modify (int i, int j, int delta) {A[i][j]+=delta; for (int x = i; x< a.length x + + lowbit (x)) for (int y = j; y <A[i].length; y = = lowbit (y)) {c[x] [y] + = Delta; } }
(2) In a two-dimensional case, the function of finding the sum of the elements of the ∑a[i][j] (the first I row and the first J column) is
[java] View plain copy print? int sum (int i, int j) { int result = 0; for (int x = i; x > 0; x -= Lowbit (x)) { for (int y = j; y > 0; y -= lowbit (y)) { result += C[x][y]; } } return result; } sun (1,1) =c[1][1]; Sun (1,2) =c[1][2]; sun (1,3) =c[1][3]+c[1][2];... sun (2,1) =c[2][1]; sun (2,2) =C[2 ][2]; sun (2,3) =c[2][3]+c[2][2];... sun (3,1) =c[3][1]+c[2][1]; sun (3,2) =C[3][2]+C[2][ 2];
