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headlines 3 +#+date: <2024-09-12 Thu> + +This post introduces some new practical sampling schemes. It builds on: +- The [[../mod-minimizers/mod-minimizers.org][post]] and paper [cite:@modmini] introducing the mod-minimizer. +- The [[../minimizer-lower-bound][post]] and paper [cite:@sampling-lower-bound] introducing a + lower bound on the density of sampling schemes. + +* Sampling schemes +** Definitions and background +A *sampling scheme* is simply a function $f: \Sigma^{w+k-1} \to [w]$. For a +*window* +of length $w+k-1$, containing $w$ k-mers, it samples the starting position +of one of the k-mers. + +The *density* of a sampling scheme is the expected fraction of distinct positions +sampled over an infinitely long random string. + +A sampling scheme is *forward* when the absolute sampled position never +decreases as we slide the window over a sequence. + +The *random minimizer* has a density of $2/(w+1)$. + +A *trivial lower bound* on the density is $1/w$. + +Previous sampling schemes are *miniception* and the *double-decycling based minimizer*. + +#+caption: Some previous sampling schemes, and the $1/w$ lower bound. We use alphabet size $4$, and window size $24$. ($t$ mentioned in the title is unused here.) +#+attr_html: :class inset +[[./figs/1-background.svg]] + +** Mod-minimizer +The mod-minimizer is a scheme that asymptotically achieves optimal density $1/w$, +and generally outperforms other schemes when $k>w$. + +It works by setting a small parameter $t = (k\bmod w)$ (with the restriction that +$t\geq r:=4$, so that there are few duplicate t-mers). Then, the smallest t-mer in the window is found at position +$0\leq x< w+k-t$. Using this, the k-mer at position $p = (x\bmod w)$ is sampled. + +#+caption: The mod-minimizer is much better for large $k$, and asymptotically optimal. +#+attr_html: :class inset +[[./figs/2-modmini.svg]] + +** Forward scheme lower bound +Together with Bryce, we significantly improved the lower bound on the best +possible density of forward schemes. In fact, we prove that when $k\equiv 1\pmod +w$ and $\sigma\to\infty$, the mod-minimizer is optimal. + +#+caption: The lower bound is quite close to the mod-minimizer! +#+attr_html: :class inset +[[./figs/3-lower-bound.svg]] + +** Open syncmer minimizer +Daniel Liu came up with the idea to build minimizer schemes on top of syncmers. +Given a parameter $t$ (we use $t=4$ when $\sigma=4$), we can hash all t-mers +inside a k-mer. The k-mer is an /open syncmer/ when the smallest t-mer inside it +is in the middle (at position $\lfloor(k-t)/2\rfloor$). +Open syncmers have the property that they are never close to each other. +In each window, the /open syncmer minimizer/ simply takes the smallest +k-mer[fn::Smallest with respect to the hash of the central t-mer.] that is an open +syncmer (or the smallest kmer if there is no open syncmer). +#+caption: For small alphabet $\sigma=4$, the open syncmer minimizer performs nearly as good as decycling minimizer (not shown), and slightly worse than double decycling minimizers. For large alphabet, the open syncmer minimizer performs very similar to (single) decycling. +#+attr_html: :class inset +[[./figs/4-open-syncmer.svg]] +** Open-closed syncmer minimizer +Then Daniel extended this to the /open-closed syncmer minimizer/: If there is an +open syncmer inside the window, prefer the . Otherwise, take a closed syncmer, +i.e., a k-mer whose smallest contained t-mer is at the start or end. Otherwise, +just take the smallest k-mer. +#+caption: The /open-closed syncmer minimizer/ improves the open syncmer minimizer, and (for large alphabets) performs very similar to double decycling for $kw$, it outperforms double decycling. +#+attr_html: :class inset +[[./figs/5-open-closed-syncmer.svg]] +** New: Open-closed mod-minimizer +Looking at the figure above, one wonders if the smoothness of the methods that +perform well for $k