The anti-lexicographic SUS-anchor: a near-optimal k=1 sampling scheme
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arXiv
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| Format: | Preprint |
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2026
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| author | Koerkamp, Groot Ragnar |
| author_facet | Koerkamp, Groot Ragnar |
| contents | 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 $k < \log_σw$ 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 $σ=4$ and $k=1$, the anti-lexicographic SUS-anchor empirically has density $<1\%$ away from the density lower bound, significantly improving over bd-anchors that are often $>15\%$ above it. For alphabet size $σ=2$, the density is at most $10\%$ above the lower bound, which again improves over the $>50\%$ overhead of bd-anchors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01190 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The anti-lexicographic SUS-anchor: a near-optimal k=1 sampling scheme Koerkamp, Groot Ragnar Data Structures and Algorithms 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 $k < \log_σw$ 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 $σ=4$ and $k=1$, the anti-lexicographic SUS-anchor empirically has density $<1\%$ away from the density lower bound, significantly improving over bd-anchors that are often $>15\%$ above it. For alphabet size $σ=2$, the density is at most $10\%$ above the lower bound, which again improves over the $>50\%$ overhead of bd-anchors. |
| title | The anti-lexicographic SUS-anchor: a near-optimal k=1 sampling scheme |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2606.01190 |