Basic Binary Indexed Tree (English version) - Codeforces

Advantages of BIT:

            - Use less memory than RMQ

            - Easy to code

            - Can be used in many problems about number sequence

            - Runtime: O(logN)

 

Disadvantages : ='= BIT is hard to understand  (so that's why I'm writing :D)

 

---

 

Main content:

            

BIT consists of two operations in an array A[1..N] of numbers

            SET (index, value) : To add value to A[index] (or A[index] += value)

            GET (index) :  To sum up A[1]..A[index ] (or Get ≔ A[1] + … + A[index])

 

---

          

Details:

            We know that a natural number can be expressed as a sum of the powers of 2, for example:

        

            Example 1:

                         22       =          16        +          4          +          2

                                    =          2^4      +          2^2      +          2 ^1

 

            Applying this idea for BIT, we're going to express the sum of A[1]..A[n] as a sum of sub arrays, each of them has 2^k elements

 

---

How to code:

 

S(i,j) is the sum of A[i]..A[j]  (or S[i,j] = A[i] + A[i+1] +…+ A[j]). So with number 22 in Example 1, expressed as a sum of the powers of 2, is gonna be like this:

 

            S(1,22) = S(1,16) + S(17,20) + S(21,22)

To get the positions of sub arrays, use this formula:  i - i AND (-i) + 1.

Demo.:

            22 - (22 AND (-22)) = 20

            20 - (20 AND (-20)) = 16 

            16 - (16 AND (-16)) = 0

Thus, the structure of BIT T[] will be:

             T[i]: Sum of  S(i - i AND (-i) + 1 , i )

---

How does GET work?

 

Notice: GET (index) sums up A[1] → A[index].

 

Idea: To get the sum of A[1].. A[index], here we start with A[index] first. Parent of A[index] can be reached by using the following formula: i – i AND (-i)

Pseudo-code:

GET (T,i)

1 s ← 0

2 while i > 0 do

3          s ← s + T[i]

4          i ← i – i and (-i)

5 return s

---

 

How does SET work?

 

Notice: SET (index,value) adds value units to A[index]

Idea: To increase the value of A[index] with value, is to increase sub arrays which CONTAINS A[index]

Example : We want to add 1 to A[9]. So we also have to add 1 to sub arrays which contains A[9]. Look at the image, here we have A[10], A[12] and A[16] both consist of A[9] J

So how do we find which array contains A[9]?

Use this formula: i + i and (-i)! 

With this relationship, the demonstration of BIT will be opposed to the first one in GET procedure:

 Pseudo-code

 

SET (T,i,v)

1 while i ≤ size[T] do

2          T[i] ← T[i] + v

3          i ← i + i AND (-i) 

Read full article from Basic Binary Indexed Tree (English version) - Codeforces