The anti-lexicographic SUS-anchor: a near-optimal k=1 sampling scheme

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 $k\le 2w$, 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 anti-lexicographic SUS-anchor empirically has density $<1\mathrm{\%}$ away from the density lower bound, significantly improving over bd-anchors that are often $>15\mathrm{\%}$ above it. For alphabet size $\sigma =2$, the density is at most $10\mathrm{\%}$ above the lower bound, which again improves over the $>50\mathrm{\%}$ overhead of bd-anchors.