Tree-like array-magical binary

The tree array is to solve the fast update and the sum of the interval of the statistic array, set an array a[1-n], need to calculate the sum of a[m-k], the brute force solution needs O (k-m), if we find sum (1-k) and sum (1-m), then the answer is sum (1-m)-sum (1-k) ;

So how to quickly find sum (1-n), you can consider the direct request, but if we add a condition, need to update immediately, then the use of brute force solution will be problematic, assuming that update a[k], the sum (1-k) to sum (1-n) to update, this is a very time-consuming process.

Here we use a binary idea, and we'll convert decimal to binary for example, like,

C[0001] = a[0001];

C[0010] = a[0010] + a[0001];

C[0011] = a[0011];

C[0100] = a[0100] + a[0011] + a[0010] + a[0001];

C[0101] = a[0101];

C[0110] = a[0110] + a[0101];

C[0111] = a[0111];

C[1000] = a[1000] + a[0111] + a[0110] + a[0101] + a[0100] + a[0011] + a[0010] + a[0001];

Define k as the end of 0 consecutive numbers, can be understood as c[i] = A[i] + ... + a[i-2^k+1].

The process of updating is to update the C[J] related to A[i], for example: i = 0001,j = 0001, 0010, 0100, 1000,i = 0101, j = 0110, 1000; it is not difficult to see the regularity of continuous rounding.

The process of finding sum is to find the c[i of each digit of 1) and, for example:

SUM[1-1111] = c[1111] + c[1110] + c[1100] + c[1000];

C[1111] = a[1111];

C[1110] = a[1110] + a[1101];

C[1100] = a[1100] + a[1011] + a[1010] + a[1001];

C[1000] = a[1000] + a[0111] + a[0110] + a[0101] + a[0100] + a[0011] + a[0010] + a[0001];

Knowing the binary rules of a tree array, consider how to implement it in code.

int lowbit (x) {    return x&-x;} void Update (int k,int  x) {     for (int i=k;i<=n;i+= lowbit (i)) {        C[i]+ =x     ;

0001, 0010, i.e. 0001 + 0001 = 0010;

0010, 0100, i.e. 0010 + 0010 = 0100;

0101, 0110, i.e. 0101 + 0001 = 0110;

0110, 1000, i.e. 0110 + 0010 = 1000;

The Lowbit function is to find the minimum number of carry, X&-x can understand:-X for the complement, for example, the original code & complement is equal to the minimum number of digits.

The update process is a for loop that updates the relevant elements in all C arrays, starting with the updated K.

int getsum (int  x) {    int ans=0;      for (int i=x;i;i-=lowbit (i)) // I to be greater than 0        ans+=c[i];     return ans;}

1111, 1110, i.e. 1111- 0001 = 1110;

1110, 1100, i.e. 1110- 0010 = 1100;

1100, 1000, i.e. 1100- 0100 = 1000;

0000------ --0000;

By the same token, you can see that the sum is the next value to be added by subtracting the minimum number of digits .

Tree-like array-magical binary