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.
@InProceedings{verbeek_et_al:LIPIcs.CPM.2024.28, author = {Verbeek, Hilde and Ayad, Lorraine A.K. and Loukides, Grigorios and Pissis, Solon P.}, title = {{Minimizing the Minimizers via Alphabet Reordering}}, booktitle = {35th Annual Symposium on Combinatorial Pattern Matching (CPM 2024)}, pages = {28:1--28:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-326-3}, ISSN = {1868-8969}, year = {2024}, volume = {296}, editor = {Inenaga, Shunsuke and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2024.28}, URN = {urn:nbn:de:0030-drops-201383}, doi = {10.4230/LIPIcs.CPM.2024.28}, annote = {Keywords: sequence analysis, minimizers, alphabet reordering, feedback arc set} }
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