Finimizers: Variable-Length Bounded-Frequency Minimizers for $k$-mer Sets

The minimizer of a $k$-mer is the smallest $m$-mer inside the $k$-mer according to some total order $< $ of the $m$-mers. Minimizers are often used as keys in hash tables in indexing tasks in metagenomics and pangenomics. The main weakness of minimizer-based indexing is the possibility of very frequently occurring minimizers, which can slow query times down significantly. Popular minimizer alignment tools employ various and often wild heuristics as workarounds, typically by ignoring frequent minimizers or blacklisting commonly occurring patterns, to the detriment of other metrics (e.g., alignment recall, space usage, or code complexity). In this paper, we introduce frequency-bounded minimizers, which we call finimizers, for indexing sets of $k$-mers. The idea is to use an order relation $< $ for minimizer comparison that depends on the frequency of the minimizers within the indexed $k$-mers. With finimizers, the length $m$ of the $m$-mers is not fixed, but is allowed to vary depending on the context, so that the length can increase to bring the frequency down below a user-specified threshold $t$. Setting a maximum frequency solves the issue of very frequent minimizers and gives us a worst-case guarantee for the query time. We show how to implement a particular finimizer scheme efficiently using the Spectral Burrows-Wheeler transform ($SBWT$) (Alanko et al. Proc. SIAM ACDA, 2023) augmented with longest common suffix information. In experiments, we explore in detail the special case in which we set $t = 1$. This choice simplifies the index structure and makes the scheme completely parameter-free apart from the choice of $k$. A prototype implementation of this scheme exhibits $k$-mer localization times close to, and often faster than, state-of-the-art minimizer-based schemes.