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What is a binary indexed tree?

Educative Answers Team

A binary indexed tree is used to efficiently answer prefix sum queries. It takes O(n)O(n) space, where nn is the size of the input array.

Construction

Take a look at the important facts below:

  1. The binary indexed tree will take n+1n+1 nodes.
  2. Each binary tree node contains an index and a value. The value is a prefix sum.

The following illustration explains how a binary indexed tree is created. An indexed binary tree can be represented as an array or a tree. For simplicity, we will represent it as a tree.

To obtain the parent of a node with index i:

  1. Covert i to binary.
  2. Flip the rightmost bit containing 1 to 0.

For all nodes with an index greater than zero:

  1. Express the index in terms of the sum of powers of 2.
  2. The first element in the expression denotes the index in the original array from which we need to start computing the sum.
  3. The last element in the expression denotes the number of input array elements that need to be included.
Let's create a binary indexed tree from the input array shown above.
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Update

If there is a change made at index ii of the input array, then the binary indexed tree needs to be updated. Suppose that xx is added to the value at ii in the input array. To update the binary indexed tree, go to index i+1i + 1:

  1. Go to the index m+1m+1 in the binary indexed tree. Add xx to the value in this node.

  2. Find the next index to update in the indexed binary tree by finding current index + (current index & (-current index)). Add xx to the value in this node and repeat this step until the new index is greater than the number of nodes in the indexed binary tree.

The initial input array
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Obtaining the prefix sum

To find the sum of a prefix from 00 to mm:

  1. Let answer = 0. Go to the index m+1m+1 in the binary indexed tree.
  2. Traverse from this node to its parent node and add this value to the variable answer.
  3. Keep traversing until you reach the parent node and keep adding the values (to the variable answer) in the nodes encountered.

The following illustration shows how to obtain a prefix sum from index 00 to 22 of the input array.

Go to the node with the index 3. Let answer = 0. Add its value to answer.
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data structures
tree
binary indexed
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