Minimizing the Minimizers via Alphabet Reordering

TL;DR

This paper investigates the problem of optimizing alphabet orderings to minimize the number of minimizers in string sampling, proving it to be NP-hard and explaining the difficulty of finding exact solutions.

Contribution

It introduces the problem of alphabet reordering for minimizer sampling and proves its NP-hardness, providing theoretical insight into the challenge of minimizing minimizers.

Findings

Proves the NP-hardness of alphabet reordering for minimizer minimization.

Provides theoretical justification for the lack of exact algorithms.

Highlights the complexity of optimizing minimizer sampling in bioinformatics.

Abstract

Minimizers sampling is one of the most widely-used mechanisms for sampling strings [Roberts et al., Bioinformatics 2004]. Let $S=S[1]\ldots S[n]$ be a string over a totally ordered alphabet $\Sigma$. Further let $w\geq 2$ and $k\geq 1$ be two integers. The minimizer of $S[i\mathinner{.\,.} i+w+k-2]$ is the smallest position in $[i,i+w-1]$ where the lexicographically smallest length-$k$ substring of $S[i\mathinner{.\,.} i+w+k-2]$ starts. The set of minimizers over all $i\in[1,n-w-k+2]$ is the set $\mathcal{M}_{w,k}(S)$ of the minimizers of $S$. We consider the following basic problem: Given $S$, $w$, and $k$, can we efficiently compute a total order on $\Sigma$ that minimizes $|\mathcal{M}_{w,k}(S)|$? We show that this is unlikely by proving that the problem is NP-hard for any $w\geq 2$ and $k\geq 1$. 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.