Minimizing the Minimizers via Alphabet Reordering

Authors Hilde Verbeek , Lorraine A.K. Ayad , Grigorios Loukides , Solon P. Pissis

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Author Details

Hilde Verbeek
  • CWI, Amsterdam, The Netherlands
Lorraine A.K. Ayad
  • Brunel University London, London, UK
Grigorios Loukides
  • King’s College London, London, UK
Solon P. Pissis
  • CWI, Amsterdam, The Netherlands
  • Vrije Universiteit, Amsterdam, The Netherlands

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Hilde Verbeek, Lorraine A.K. Ayad, Grigorios Loukides, and Solon P. Pissis. Minimizing the Minimizers via Alphabet Reordering. In 35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 296, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Minimizers sampling is one of the most widely-used mechanisms for sampling strings [Roberts et al., Bioinformatics 2004]. Let S = S[1]… S[n] be a string over a totally ordered alphabet Σ. Further let w ≥ 2 and k ≥ 1 be two integers. The minimizer of S[i..i+w+k-2] is the smallest position in [i,i+w-1] where the lexicographically smallest length-k substring of S[i..i+w+k-2] starts. The set of minimizers over all i ∈ [1,n-w-k+2] is the set ℳ_{w,k}(S) of the minimizers of S. We consider the following basic problem: Given S, w, and k, can we efficiently compute a total order on Σ that minimizes |ℳ_{w,k}(S)|? We show that this is unlikely by proving that the problem is NP-hard for any w ≥ 3 and k ≥ 1. Our result provides theoretical justification as to why there exist no exact algorithms for minimizing the minimizers samples, while there exists a plethora of heuristics for the same purpose.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
  • sequence analysis
  • minimizers
  • alphabet reordering
  • feedback arc set


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