Wavelet Tree Explained

The Wavelet Tree is a succinct data structure to store strings in compressed space. It generalizes the

rank _{q} and 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

access , rank _{q} , and 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 :-

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)

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.

References

^{[1] [2] [3] [4] [5]}

Notes and 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