Tree-like array of data structures

Tag: Cheng represents the smallest cell i+1 tle linear structure worst case structure

1. Overview

The tree array (binary indexed tree) is a novel array structure that efficiently fetches the number of consecutive n in an array. In summary, a tree array is often used to solve the problem that the elements in the array {A} can be constantly modified, and how can the sum of several consecutive numbers be obtained quickly?

2. Basic operation of tree-like array

The complexity of element modification and continuous element summation for a traditional array (n elements) is O (1) and O (n), respectively. The tree-like array transforms the linear structure into a pseudo-tree structure (linear structures can scan the elements individually, and the tree structure allows for a jump scan), which makes the modification and summation complexity both O (LGN), greatly improving the overall efficiency.

Given a sequence (series) A, we set an array C to satisfy

C[i] = a[i–2^k+ 1] + ... + a[i]

where k is I in the binary at the end of the number of 0, I start from 1 count!

Then we call c a tree-like array.

The question below is, given I, how do I ask for 2^k?

The answer is simple:2^k=i& (i^ (i-1)), which is i& (-i)

The following explanation:

Take I=6 as an example (note: a_x indicates that the number a is an X-binary representation):

(i) _10 = (0110) _2

(i-1) _10= (0101) _2

I xor (i-1) = (0011) _2

I and (I xor (i-1)) = (0010) _2

2^k = 2

C[6] = c[6-2+1]+...+a[6]=a[5]+a[6]

The exact meaning of array C is as follows:

When we modify the value of a[i], we can go upstream from c[i] to the root node, adjust all c[on this path], the complexity of this operation in the worst case is the height of the tree is O (logn). In addition, for the first n of the sequence number and, just find all of the largest subtree before n, the root of the node C together. It is not difficult to find that the number of these subtrees is n at the binary 1 number, or the number of the power of the N expansion into 2 and the time, so the complexity of the sum operation is also O (Logn).

The tree array can quickly find any interval of sum: a[i] + a[i+1] + ... + a[j], set sum (k) = A[1]+a[2]+...+a[k], then a[i] + a[i+1] + ... + a[j] = SUM (j)-sum (i-1).

The following is a C-language implementation of a tree-like array:

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849

//求2^kintlowbit(intt){    returnt & ( t ^ ( t - 1 ) );}//求前n项和intsum(intend){   intsum = 0;    while(end > 0)  {     sum += in[end];     end -= lowbit(end);  }  returnsum;}//增加某个元素的大小voidplus(intpos, intnum){   while(pos <= n)  {     in[pos] += num;      pos += lowbit(pos);  }}

3, extension--two-dimensional tree-like array

One-dimensional tree arrays can easily be extended to two-dimensional, two-dimensional tree arrays as follows:

C[x][y] = SUM (A[i][j])

Among them, x-lowbit[x]+1 <= i<=x and y-lowbit[y]+1 <= J <=y

4. Application

(1) One-dimensional tree-like array:

See also: http://hi.baidu.com/lilu03555/blog/item/4118f04429739580b3b7dc74.html

(2) Two-dimensional tree-like array:

A large matrix of numbers that can be manipulated in two ways

1) Add an integer to a number in the matrix (can be positive negative)

2) query all the numbers in a sub-matrix and

Requirements for each query, output results

5. Summary

The tree array was originally discovered when designing a compression algorithm (see Resources 1), and it is now often used to maintain sub-sequences and. It is similar to the line tree (see section Tree of data structure), which is less space-saving and programming complexity than line-segment trees, but smaller in scope than segment trees (such as querying each interval minimum problem).

6. References

(1) Binary Indexed Trees:

Http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=binaryIndexedTrees

(2) Gaurap article "Tree-like array":

Http://www.java3z.com/cwbwebhome/article/article19/zip/treearray.zip

(3) Guo Wei article "segment Tree and tree-like array":

Http://poj.org/summerschool/1_interval_tree.pdf

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Tree-like array of data structures