Tree-like array

Tree-like array

The modification and summation of the tree array are O (Logn), which is very efficient.

The tree-like array--lowbit (x) For example 212 binary is where 10101,1 is located 0,2,4, which can be decomposed into 2^4 + 2 ^ 2 + ^ 0. Further [1,x] can be decomposed into an O (LOGX) Small interval:

1. Small interval of length 2^4 [1,2^4]

2. Small interval of length 2^2 [2^4 + 1,2^4+2^2]

3. Small interval of length 2^0 [2^4+2^2+1,2^4+2^2+2^0]

Tree-like array

C[1] = a[1];

C[2] = a[1] + a[2];

C[3] = a[3];

C[4] = a[1] + a[2]+a[3] + a[4];

C[5] = a[5];

C[6] = a[6] + a[5];

C[7] = a[7];

C[8]=A[1] + a[2]+a[3] + a[4]+a[5] + a[6]+a[7] + a[8]

A tree-like array satisfies the following properties:

1. Each internal node C[x] holds all the leaf nodes of the subtree with which it is rooted.

2. The number of child nodes of each internal node c[x] is equal to the size of Lowbit (x).

3. The parent node of each internal node C[x] is C[x + lowbit (x) except for the root tree)

4. The depth of the tree is O (logn).

I. Seeking Lowbit (n)

　Lowbit (n) is the value of the non-negative integer n in the binary representation of the lowest bit 1 and the 0 behind it.

{The complement expression ~n = -1-n} so lowbit (n) = n& (~n+1) = n& (-N).

1 int lowbit (n) 2 {3     return n& (-n); 4 }

Second, add an element to the operation

　The ancestors of any one node were at most only Logn, and we updated each of their array C values. The following code performs a single point increment operation in O (Logn) time.

1 void Update (int x,int  y)2{3for     1 ; I <= n;x + = x & (-x))4    {5         c[x] = c[x] + y; 6     }7 }

Third, the query prefix and

As written at the beginning of this article, divided into O (logn) arrays, each array of intervals and all have been saved in the array C, the following code in O (logn) time query prefix and.

1 intSumintx)2 {3     intAns =0;4      while(X >0)5     {6Ans = ans +C[x];7x = x-(X & (-x));8         returnans;9     }Ten}

Iv. Statistics A[x] ... The value of A[y].

Call the sum of the above operation: Sum (y)-sum (x-1)

V. Extensions (Multidimensional tree arrays)

Complexity into O (LOGN) ^m

There is a n*m two-dimensional array, the tree array is C, then the single-point modification operation is:

1 intUpdateintXintYintZ//Add (x, y) value to Z2 {3     intI=x;4      while(I <=N)5     {6         intj =y;7          while(J <=m)8         {9C[I][J] + =Z;TenJ + =Lowbit (j); One          }  Ai + =lowbit (i); -      }  -}

1 intSumintXinty)2 {3     intres =0, i =x;4      while(I >0)5     {6         intj =y;7          while(J >0)8         {9res = res +C[i][j];TenJ-=Lowbit (j); One         } AI-=lowbit (i); -     } -     returnRes; the}

VI. matters of caution

The subscript of a tree array cannot be 0, otherwise lowbit (0) = 0 will fall into the dead loop.

Tree-like array