FIG. 1.

(a) Example of selecting minimizers with $k=3$, $w=5$, and the lexicographic order (i.e., $AAA<AAC<AAG<\dots <TTT$). The top line is the input sequence, each subsequent line is a 7-bases long window (the number of bases in a window is $w+k-1=7$) with the minimum 3-mer highlighted. The positions {1, 2, 5, 9, 10, 11} are selected for a density $d=6∕\left(18-3+1\right)=0.375$. (b) On the same sequence, an example of a selection scheme for $w=7$ (and $k=1$ because it is a selection scheme, hence the number of bases in a window is also w). The set of positions selected is {1, 6, 7, 8, 11, 13, 14}. This is not a forward scheme as the sequence of selected position is not decreasing. (c) A forward selection scheme for $w=7$ with selected positions {1, 7, 8, 12, 13}. Like the minimizer scheme, the sequence of selected positions is nondecreasing.