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Comparing Elastic-Degenerate Strings: Algorithms, Lower Bounds, and Applications

Authors Esteban Gabory , Moses Njagi Mwaniki , Nadia Pisanti , Solon P. Pissis , Jakub Radoszewski , Michelle Sweering , Wiktor Zuba

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ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • elastic-degenerate string
  • sequence comparison
  • languages intersection
  • pangenome
  • acronym identification

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Abstract

An elastic-degenerate (ED) string T is a sequence of n sets T[1],…,T[n] containing m strings in total whose cumulative length is N. We call n, m, and N the length, the cardinality and the size of T, respectively. The language of T is defined as ℒ(T) = {S_1 ⋯ S_n : S_i ∈ T[i] for all i ∈ [1,n]}. ED strings have been introduced to represent a set of closely-related DNA sequences, also known as a pangenome. The basic question we investigate here is: Given two ED strings, how fast can we check whether the two languages they represent have a nonempty intersection? We call the underlying problem the ED String Intersection (EDSI) problem. For two ED strings T₁ and T₂ of lengths n₁ and n₂, cardinalities m₁ and m₂, and sizes N₁ and N₂, respectively, we show the following:  
- There is no 𝒪((N₁N₂)^{1-ε})-time algorithm, thus no 𝒪((N₁m₂+N₂m₁)^{1-ε})-time algorithm and no 𝒪((N₁n₂+N₂n₁)^{1-ε})-time algorithm, for any constant ε > 0, for EDSI even when T₁ and T₂ are over a binary alphabet, unless the Strong Exponential-Time Hypothesis is false. 
- There is no combinatorial 𝒪((N₁+N₂)^{1.2-ε}f(n₁,n₂))-time algorithm, for any constant ε > 0 and any function f, for EDSI even when T₁ and T₂ are over a binary alphabet, unless the Boolean Matrix Multiplication conjecture is false. 
- An 𝒪(N₁log N₁log n₁+N₂log N₂log n₂)-time algorithm for outputting a compact (RLE) representation of the intersection language of two unary ED strings. In the case when T₁ and T₂ are given in a compact representation, we show that the problem is NP-complete. 
- An 𝒪(N₁m₂+N₂m₁)-time algorithm for EDSI. 
- An Õ(N₁^{ω-1}n₂+N₂^{ω-1}n₁)-time algorithm for EDSI, where ω is the exponent of matrix multiplication; the Õ notation suppresses factors that are polylogarithmic in the input size. 
We also show that the techniques we develop have applications outside of ED string comparison.

Cite As Get BibTex

Esteban Gabory, Moses Njagi Mwaniki, Nadia Pisanti, Solon P. Pissis, Jakub Radoszewski, Michelle Sweering, and Wiktor Zuba. Comparing Elastic-Degenerate Strings: Algorithms, Lower Bounds, and Applications. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CPM.2023.11

Author Details

Esteban Gabory
  • CWI, Amsterdam, The Netherlands
Moses Njagi Mwaniki
  • University of Pisa, Italy
Nadia Pisanti
  • University of Pisa, Italy
Solon P. Pissis
  • CWI, Amsterdam, The Netherlands
  • Vrije Universiteit, Amsterdam, The Netherlands
Jakub Radoszewski
  • Institute of Informatics, University of Warsaw, Poland
Michelle Sweering
  • CWI, Amsterdam, The Netherlands
Wiktor Zuba
  • CWI, Amsterdam, The Netherlands

Funding

The work in this paper is supported in part: by the PANGAIA and ALPACA projects that have received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreements No 872539 and 956229, respectively; by the Netherlands Organisation for Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003; and by PNRR ECS00000017 Tuscany Health Ecosystem Spoke 6 "Precision medicine & personalized healthcare", funded by the European Commission under the NextGeneration EU programme.
  • Radoszewski, Jakub: Supported by the Polish National Science Center, grant no. 2018/31/D/ST6/03991.

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