Binary Indexed Tree/Fenwick Tree meaning in DSA

Last Updated : 21 Feb, 2023

Binary Indexed Tree (BIT), also known as Fenwick Tree, is a data structure used for efficiently querying and updating cumulative frequency tables, or prefix sums.

A Fenwick Tree is a complete binary tree, where each node represents a range of elements in an array and stores the sum of the elements in that range. Below is a simple structure of a Binary indexed tree.

Characteristics of Fenwick Tree:

Space Efficiency: Fenwick Trees are space efficient as they use a binary representation of indices to store only the required elements of the tree, making the overall memory usage of the tree much less compared to other data structures like segment trees.
Dynamic: Fenwick Trees are dynamic data structures, which means they can handle changes in the input data and can efficiently update the cumulative frequencies.
No Propagation: Fenwick Trees do not require any propagation of updates, which makes them more efficient in terms of time complexity.
Binary Representation: Fenwick Trees use a binary representation of indices to store and manipulate the cumulative frequencies. This binary representation allows for efficient calculation of cumulative frequencies as well as efficient updates to the tree.

Advantages of Fenwick Tree:

Efficient updates: Fenwick Trees allow for O(log n) time complexity when updating elements in the tree, which is much faster compared to other data structures like Segment Trees, which take O(n) time for updates.
Space Efficiency: Fenwick Trees use a compact representation, as it only stores the cumulative frequency values in the tree and uses a formula to derive the actual values in the array. This makes Fenwick Trees more memory efficient compared to other data structures.
Efficient Range Queries: Fenwick Trees allow for efficient range queries, as they support cumulative frequency queries in O(log n) time, which is much faster than the linear time complexity of other data structures like arrays.

Disadvantages of Fenwick Tree:

Space complexity: Fenwick trees require extra memory space for storing the tree structure, which can be an issue for large datasets.
Limited updates: Fenwick trees can only perform two types of updates: adding a value to an element and increasing the value of an element. This can make them less flexible than other data structures like segment trees which can form a wider range of updates.
Not suited for all problems: Fenwick trees are specifically designed for solving problems that involve prefix sums. If a different type of problem is being solved, it may be more appropriate to use a different data structure.

Application of Fenwick Tree:

Image Processing: In image processing, BITs can be used for calculating the sum of pixel intensities in a given rectangular region of an image, which is useful for many applications such as image filtering and segmentation.
Game Development: BITs can be used in game development for efficiently updating and querying the game state, such as checking for collisions or computing line-of-sight between objects.
Computational Biology: BITs can be used in computational biology for solving problems such as sequence alignment and gene expression analysis, which involve querying and updating large arrays of data.
Data Compression: BITs can be used in data compression algorithms, such as the Burrows-Wheeler Transform (BWT), to efficiently compute the frequency of each symbol in a given string.