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On Strings Having the Same Length- k Substrings

Authors Giulia Bernardini , Alessio Conte, Esteban Gabory, Roberto Grossi, Grigorios Loukides , Solon P. Pissis , Giulia Punzi, Michelle Sweering

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Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • string algorithms
  • combinatorics on words
  • de Bruijn graph
  • Chinese Postman

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Abstract

Let Substr_k(X) denote the set of length-k substrings of a given string X for a given integer k > 0. We study the following basic string problem, called z-Shortest 𝒮_k-Equivalent Strings: Given a set 𝒮_k of n length-k strings and an integer z > 0, list z shortest distinct strings T₁,…,T_z such that Substr_k(T_i) = 𝒮_k, for all i ∈ [1,z]. The z-Shortest 𝒮_k-Equivalent Strings problem arises naturally as an encoding problem in many real-world applications; e.g., in data privacy, in data compression, and in bioinformatics. The 1-Shortest 𝒮_k-Equivalent Strings, referred to as Shortest 𝒮_k-Equivalent String, asks for a shortest string X such that Substr_k(X) = 𝒮_k. 
Our main contributions are summarized below:  
- Given a directed graph G(V,E), the Directed Chinese Postman (DCP) problem asks for a shortest closed walk that visits every edge of G at least once. DCP can be solved in 𝒪̃(|E||V|) time using an algorithm for min-cost flow. We show, via a non-trivial reduction, that if Shortest 𝒮_k-Equivalent String over a binary alphabet has a near-linear-time solution then so does DCP. 
- We show that the length of a shortest string output by Shortest 𝒮_k-Equivalent String is in 𝒪(k+n²). We generalize this bound by showing that the total length of z shortest strings is in 𝒪(zk+zn²+z²n). We derive these upper bounds by showing (asymptotically tight) bounds on the total length of z shortest Eulerian walks in general directed graphs. 
- We present an algorithm for solving z-Shortest 𝒮_k-Equivalent Strings in 𝒪(nk+n²log²n+zn²log n+|output|) time. If z = 1, the time becomes 𝒪(nk+n²log²n) by the fact that the size of the input is Θ(nk) and the size of the output is 𝒪(k+n²).

Cite As Get BibTex

Giulia Bernardini, Alessio Conte, Esteban Gabory, Roberto Grossi, Grigorios Loukides, Solon P. Pissis, Giulia Punzi, and Michelle Sweering. On Strings Having the Same Length- k Substrings. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CPM.2022.16

Author Details

Giulia Bernardini
  • University of Trieste, Italy
  • CWI, Amsterdam, The Netherlands
Alessio Conte
  • University of Pisa, Italy
Esteban Gabory
  • CWI, Amsterdam, The Netherlands
Roberto Grossi
  • University of Pisa, Italy
Grigorios Loukides
  • King’s College London, UK
Solon P. Pissis
  • CWI, Amsterdam, The Netherlands
  • Vrije Universiteit, Amsterdam, The Netherlands
Giulia Punzi
  • University of Pisa, Italy
Michelle Sweering
  • CWI, Amsterdam, The Netherlands

Funding

The work in this paper is supported in part by: the MIUR project Algorithms for HArnessing networked Data (AHeAD); and 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.
  • Bernardini, Giulia: Supported by the Netherlands Organisation for Scientific Research (NWO) under project OCENW.GROOT.2019.015 "Optimization for and with Machine Learning (OPTIMAL)".
  • Loukides, Grigorios: Supported by the Leverhulme Trust RPG-2019-399 project.
  • Sweering, Michelle: Supported by the Netherlands Organisation for Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003.

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