Binary Indexed Tree

A Fenwick tree or binary indexed tree is a data structure that can efficiently update elements and calculate prefix sums in a table of numbers. It's actually an array rather than a real tree. Note that its index begins from 1 rather than 0.

Given an array arr[1...n], there are mainly two methods:

• prefixSum(idx)

  Compute the sum of the first i-th elements.

• update(idx,delta)

  Modify the value of a specified element of the array arr[i] = x

1.prefixSum

• 1. Initialize the output sum as 0, the current index as x+1.
  2. Do following while the current index is greater than 0.
     ...a) Add BITree[index] to sum
     ...b) Go to the parent of BITree[index]. The parent can be obtained by removing
     the last set bit from the current index, i.e., index = index - (index & (-index))
  3. Return sum.

• int prefixSum(int idx) {
      idx += 1;
      int result = 0;
      while (idx > 0) {
          result += bitArr[idx];
          idx = idx - (idx & -idx);
      }
  
      return result;
  }
  

2.update

• 1. Initialize the current index as x+1.
  2. Do the following while the current index is smaller than or equal to n.
     ...a) Add the val to BITree[index]
     ...b) Go to parent of BITree[index]. The parent can be obtained by incrementing
     the last set bit of the current index, i.e., index = index + (index & (-index))

• void update(int idx, int delta) {
      idx += 1;
      while (idx < bitArr.length) {
          this.bitArr[idx] += delta;
          idx = idx + (idx & -idx);
      }
  }
  

Visualization

Reference

1. https://en.wikipedia.org/wiki/Fenwick_tree
2. https://blog.csdn.net/Yaokai_AssultMaster/article/details/79492190
3. https://www.geeksforgeeks.org/binary-indexed-tree-or-fenwick-tree-2/
4. https://leetcode.com/problems/reverse-pairs/discuss/97268/general-principles-behind-problems-similar-to-reverse-pairs