The Anti-Lexicographic SUS-anchor

Minimizer schemes

Input: a window \(W\in \Sigma^{w+k-1}\)

Minimizer scheme: Sample the \(k\)-mer with the one with smallest hash: \[f(W) = \mathrm{argmin}_{0\leq i\lt w\ \ } h(W_{i\dots i+k}).\]

Sampling scheme: Arbitrary function \(f: \Sigma^{w+k-1}\to \{0, \dots, w-1\}.\)

Density: The expected fraction of sampled \(k\)-mers in a random string:

• E​A​DCAE.....

• .​A​DCAEB....

• ..DC​A​EBE...

• ...C​A​EBEC..

• ....​A​EBECD.

• .....E​B​ECDC

• E​A​DC​A​E​B​ECDC

Density bounds

Conjecture: Optimal schemes exist for \(k\equiv 1\pmod w\)?