The Anti-Lexicographic SUS-Anchor: An Empirically Optimal Selection Scheme

Abstract

In recent years, there has been a renewed interest in the search for low density minimizer schemes. These schemes take a window of $w$ consecutive $k$-mers, and sample one of them: the smallest under some specific order. Schemes such as the mod-minimizer provide a low density (fraction of sampled $k$-mers) when $k \gg w$, while schemes such as the greedy minimizer work well for explicit small parameters roughly in the regime $k \leq 2w$, for $k$ and $w$ up to $15$ or so. When $k < \log_\sigma w$ is very small, minimizer schemes cannot do well, and more general sampling schemes are needed that can be richer than just comparing $k$-mers. Bidirectional-string anchors (bd-anchors) form one such scheme. Inspired by bd-anchors, we introduce the smallest unique substring or SUS-anchor: Given a window, this considers all suffixes that do not occur as a substring elsewhere in the window. It then samples the start position of the smallest suffix according to the new anti-lexicographic order that minimizes the first character and maximizes the remaining characters. We give a linear-time and $O(w)$ space streaming algorithm to compute all SUS-anchors of a string. For alphabet size $\sigma=4$ and $k=1$, the parameter-free anti-lexicographic SUS-anchor empirically has density $<1\%$ away from the density lower bound and is at least $4\times$ closer to the lower bound than all other tested schemes. For alphabet size $\sigma=2$, the density is at most $10\%$ above the lower bound, which still improves $2\times$ to $3\times$ the overhead of the random minimizer and is consistently better than the greedy minimizer. Likewise, the anti-lexicographic minimizer performs better than all other schemes apart from the greedy minimizer.