A $k$-long sequence $P$ is a ($k$,$s$)-open-syncmer, $s\le k$, if $P[1,s]$ is the smallest among all $s$-mers in $P$. Suppose function $\phi$ is a bijective hash function of $k$-long sequences. $P$ is a random ($k$,$s$)-syncmer if $\phi(P)$ is an open syncmer. Because we often map $k$-mers to integers, $\phi$ can take the form of an invertible integer hash function. In practice, $\phi$ does not have to be a bijection. It can also map a sequence to an integer of a …
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