# Wavelet Tree

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The Wavelet Tree is a succinct data structure to store strings in compressed space. It generalizes the ${\displaystyle \mathbf {rank} _{q}}$ and ${\displaystyle \mathbf {select} _{q}}$ operations defined on bitvectors to arbitrary alphabets.

In computer science, a succinct data structure is a data structure which uses an amount of space that is "close" to the information-theoretic lower bound, but still allows for efficient query operations. The concept was originally introduced by Jacobson to encode bit vectors, (unlabeled) trees, and planar graphs. Unlike general lossless data compression algorithms, succinct data structures retain the ability to use them in-place, without decompressing them first. A related notion is that of a compressed data structure, in which the size of the data structure depends upon the particular data being represented.

## Contents

Originally introduced to represent compressed suffix arrays, [1] it has found application in several contexts. [2] [3] The tree is defined by recursively partitioning the alphabet into pairs of subsets; the leaves correspond to individual symbols of the alphabet, and at each node a bitvector stores whether a symbol of the string belongs to one subset or the other.

In computer science, a compressed suffix array is a compressed data structure for pattern matching. Compressed suffix arrays are a general class of data structure that improve on the suffix array. These data structures enable quick search for an arbitrary string with a comparatively small index.

The name derives from an analogy with the wavelet transform for signals, which recursively decomposes a signal into low-frequency and high-frequency components.

In mathematics, a wavelet series is a representation of a square-integrable function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.

## Properties

Let ${\displaystyle \Sigma }$ be a finite alphabet with ${\displaystyle \sigma =|\Sigma |}$. By using succinct dictionaries in the nodes, a string ${\displaystyle s\in \Sigma ^{*}}$ can be stored in ${\displaystyle nH_{0}(s)+o(|s|\log \sigma )}$, where ${\displaystyle H_{0}(s)}$ is the order-0 empirical entropy of ${\displaystyle s}$.

Information entropy is the average rate at which information is produced by a stochastic source of data.

If the tree is balanced, the operations ${\displaystyle \mathbf {access} }$, ${\displaystyle \mathbf {rank} _{q}}$, and ${\displaystyle \mathbf {select} _{q}}$ can be supported in ${\displaystyle O(\log \sigma )}$ time.

### Access operation

A wavelet tree contains a bitmap representation of a string. If we know the alphabet set, then the exact string can be inferred by tracking bits down the tree. To find the letter at ith position in the string :-

If the ith element at root is 0, we move to the left child, else we move to the right child. Now our index in the child node is the rank of the respective bit in the parent node. This rank can be calculated in O(1) by using succinct dictionaries. Along with moving to a child we also refine our alphabet to the respective subset. These steps are repeated till we reach a leaf, where we are left only with one letter in our alphabet, which is the one we were looking for. Thus for a balanced tree, any S[i] in string S can be accessed in ${\displaystyle O(\log \sigma )}$ [3] time.

## Extensions

Several extensions to the basic structure have been presented in the literature. To reduce the height of the tree, multiary nodes can be used instead of binary. [2] The data structure can be made dynamic, supporting insertions and deletions at arbitrary points of the string; this feature enables the implementation of dynamic FM-indexes. [4] This can be further generalized, allowing the update operations to change the underlying alphabet: the Wavelet Trie [5] exploits the trie structure on an alphabet of strings to enable dynamic tree modifications.

In computer science, an FM-index is a compressed full-text substring index based on the Burrows-Wheeler transform, with some similarities to the suffix array. It was created by Paolo Ferragina and Giovanni Manzini, who describe it as an opportunistic data structure as it allows compression of the input text while still permitting fast substring queries. The name stands for Full-text index in Minute space.

In computer science, a trie, also called digital tree, radix tree or prefix tree, is a kind of search tree—an ordered tree data structure used to store a dynamic set or associative array where the keys are usually strings. Unlike a binary search tree, no node in the tree stores the key associated with that node; instead, its position in the tree defines the key with which it is associated. All the descendants of a node have a common prefix of the string associated with that node, and the root is associated with the empty string. Keys tend to be associated with leaves, though some inner nodes may correspond to keys of interest. Hence, keys are not necessarily associated with every node. For the space-optimized presentation of prefix tree, see compact prefix tree.

• Wavelet Trees. A blog post describing the construction of a wavelet tree, with examples.

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## References

1. R. Grossi, A. Gupta, and J. S. Vitter, High-order entropy-compressed text indexes, Proceedings of the 14th Annual SIAM/ACM Symposium on Discrete Algorithms (SODA), January 2003, 841-850.
2. P. Ferragina, R. Giancarlo, G. Manzini, The myriad virtues of Wavelet Trees, Information and Computation, Volume 207, Issue 8, August 2009, Pages 849-866
3. G. Navarro, Wavelet Trees for All, Proceedings of 23rd Annual Symposium on Combinatorial Pattern Matching (CPM), 2012
4. H.-L. Chan, W.-K. Hon, T.-W. Lam, and K. Sadakane, Compressed Indexes for dynamic text collections, ACM Transactions on Algorithms, 3(2), 2007
5. R. Grossi and G. Ottaviano, The Wavelet Trie: maintaining an indexed sequence of strings in compressed space, In Proceedings of the 31st Symposium on the Principles of Database Systems (PODS), 2012