Detecting $k$-(Sub-)Cadences and Equidistant Subsequence Occurrences

The equidistant subsequence pattern matching problem is considered. Given a pattern string $P$ and a text string $T$, we say that $P$ is an \emph{equidistant subsequence} of $T$ if $P$ is a subsequence of the text such that consecutive symbols of $P$ in the occurrence are equally spaced. We can consider the problem of equidistant subsequences as generalizations of (sub-)cadences. We give bit-parallel algorithms that yield $o(n^2)$ time algorithms for finding $k$-(sub-)cadences and equidistant subsequences. Furthermore, $O(n{\log }^{2}n)$ and $O(n\log n)$ time algorithms, respectively for equidistant and Abelian equidistant matching for the case $\mathrm{\mid }P\mathrm{\mid }=3$, are shown. The algorithms make use of a technique that was recently introduced which can efficiently compute convolutions with linear constraints.