Wavelet Tree

A wavelet tree on the string "abracadabra". At each node the symbols of the string are projected onto two partitions of the alphabet, and a bitvector denotes to which partition each symbol belongs. Note that only the bitvectors are stored; the strings in the nodes are only for illustratory purposes.

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

Originally introduced to represent compressed suffix arrays, it has found application in several contexts. 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.

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

Properties

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

If the tree is balanced, the operations $\mathbf {access}$, $\mathbf {rank} _{q}$, and $\mathbf {select} _{q}$ can be supported in $O(\log \sigma )$ 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 i^th position in the string :-

Algorithm access
  Input:
    - The position i in the string of which we want to know the letter, starting at 1.
    - The top node W of the wavelet tree that represents the string
  Output: The letter at position i

    if W.isLeafNode return W.letter
    if W.bitvector[i] = 0 return access(i - rank(W.bitvector, i), W.left)
    else return access(rank(W.bitvector, i), W.right)

• "←" denotes assignment. For instance, "largest ← item" means that the value of largest changes to the value of item.
• "return" terminates the algorithm and outputs the following value.

In this context, the rank of a position $i$ in a bitvector $b$ is the number of ones that appear in the first $i$ positions of $b$. Because the rank can be calculated in O(1) by using succinct dictionaries, any S[i] in string S can be accessed in $O(\log \sigma )$ time, as long as the tree is balanced.

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. 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. This can be further generalized, allowing the update operations to change the underlying alphabet: the Wavelet Trie exploits the trie structure on an alphabet of strings to enable dynamic tree modifications.

Further reading

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

The wavelet Tree has many applications in different domains such as geographical search. Researchers presented a dual indexing technique based on a wavelet tree for Geographical Information Retrieval (GIR). The large amount of geographical information on the internet has made the indexing challenging during the information retrieval which is defined as an activity to retrieve a set of information based on our need from a set of resources. An efficient indexing technique is an essential requirement in processing of the plenty of data that are present in today’s information retrieval systems. This has been the motivation towards the design and development of several different indexing structures with a primary focus towards secondary memory as compared to the main memory which used to consist of a smaller size as well as a higher cost in the beginning, however with a dramatic reduction in the main memory prices, indexes in the main memory has also been introduced which led to the development of primary memory-based indexing structures.

Dual indexing technique uses wavelet tree data structure for both textual and spatial indexing and allows for the use of MBR in dynamic insertion during index construction. It has the advantage of supporting parallel computing when there is an extensive data set involved. It can be used extensively and concurrently on different machines such that textual data can be searched on one machine and spatial data on another machine which in turn results in the reduction of the search time and the storage space, hence making this method way more efficient as compared to the previous proposed methods. Experimental results support this by showing that when pure spatial indexing is used, search time complexity is reduced and takes even less than one third time of that of spatial indexing performed using previously introduced methods namely R-tree or R*-tree. In addition, the search time is decreased by half in the case of dual indexing (textual and spatial) using wavelet tree, as compared to the other techniques like B/R, B/R* when the search query length is larger than 2 keywords.

External links

• Media related to Wavelet Tree at Wikimedia Commons