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**Published in:** LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

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 https://github.com/jnalanko/SubsetWT.

Jarno N. Alanko, Elena Biagi, Simon J. Puglisi, and Jaakko Vuohtoniemi. Subset Wavelet Trees. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{alanko_et_al:LIPIcs.SEA.2023.4, 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 = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.4}, 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} }

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**Published in:** LIPIcs, Volume 242, 22nd International Workshop on Algorithms in Bioinformatics (WABI 2022)

A fundamental operation in computational genomics is to reduce the input sequences to their constituent k-mers. For maximum performance of downstream applications it is important to store the k-mers in small space, while keeping the representation easy and efficient to use (i.e. without k-mer repetitions and in plain text). Recently, heuristics were presented to compute a near-minimum such representation. We present an algorithm to compute a minimum representation in optimal (linear) time and use it to evaluate the existing heuristics. For that, we present a formalisation of arc-centric bidirected de Bruijn graphs and carefully prove that it accurately models the k-mer spectrum of the input. Our algorithm first constructs the de Bruijn graph in linear time in the length of the input strings (for a fixed-size alphabet). Then it uses a Eulerian-cycle-based algorithm to compute the minimum representation, in time linear in the size of the output.

Sebastian Schmidt and Jarno N. Alanko. Eulertigs: Minimum Plain Text Representation of k-mer Sets Without Repetitions in Linear Time. In 22nd International Workshop on Algorithms in Bioinformatics (WABI 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 242, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{schmidt_et_al:LIPIcs.WABI.2022.2, author = {Schmidt, Sebastian and Alanko, Jarno N.}, title = {{Eulertigs: Minimum Plain Text Representation of k-mer Sets Without Repetitions in Linear Time}}, booktitle = {22nd International Workshop on Algorithms in Bioinformatics (WABI 2022)}, pages = {2:1--2:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-243-3}, ISSN = {1868-8969}, year = {2022}, volume = {242}, editor = {Boucher, Christina and Rahmann, Sven}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2022.2}, URN = {urn:nbn:de:0030-drops-170361}, doi = {10.4230/LIPIcs.WABI.2022.2}, annote = {Keywords: Spectrum preserving string sets, Eulerian cycle, Suffix tree, Bidirected arc-centric de Bruijn graph, k-mer based methods} }

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