License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SEA.2023.4
URN: urn:nbn:de:0030-drops-183549
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Alanko, Jarno N. ; Biagi, Elena ; Puglisi, Simon J. ; Vuohtoniemi, Jaakko

Subset Wavelet Trees

LIPIcs-SEA-2023-4.pdf (1 MB)


Given an alphabet Σ of σ = |Σ| symbols, a degenerate (or indeterminate) string X is a sequence X = X[0],X[1]…, X[n-1] of n subsets of Σ. Since their introduction in the mid 70s, degenerate strings have been widely studied, with applications driven by their being a natural model for sequences in which there is a degree of uncertainty about the precise symbol at a given position, such as those arising in genomics and proteomics. In this paper we introduce a new data structural tool for degenerate strings, called the subset wavelet tree (SubsetWT). A SubsetWT supports two basic operations on degenerate strings: subset-rank(i,c), which returns the number of subsets up to the i-th subset in the degenerate string that contain the symbol c; and subset-select(i,c), which returns the index in the degenerate string of the i-th subset that contains symbol c. These queries are analogs of rank and select queries that have been widely studied for ordinary strings. Via experiments in a real genomics application in which degenerate strings are fundamental, we show that subset wavelet trees are practical data structures, and in particular offer an attractive space-time tradeoff. Along the way we investigate data structures for supporting (normal) rank queries on base-4 and base-3 sequences, which may be of independent interest. Our C++ implementations of the data structures are available at

BibTeX - Entry

  author =	{Alanko, Jarno N. and Biagi, Elena and Puglisi, Simon J. and Vuohtoniemi, Jaakko},
  title =	{{Subset Wavelet Trees}},
  booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
  pages =	{4:1--4:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-279-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{265},
  editor =	{Georgiadis, Loukas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-183549},
  doi =		{10.4230/LIPIcs.SEA.2023.4},
  annote =	{Keywords: degenerate strings, compressed data structures, succinct data structures, string processing, data structures, efficient algorithms}

Keywords: degenerate strings, compressed data structures, succinct data structures, string processing, data structures, efficient algorithms
Collection: 21st International Symposium on Experimental Algorithms (SEA 2023)
Issue Date: 2023
Date of publication: 19.07.2023
Supplementary Material: Software (Source Code):
Software (Source Code):

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