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The Cover Suffix Tree (CST) of a string $T$ is the suffix tree of $T$ withadditional explicit nodes corresponding to halves of square substrings of $T$.In the CST an explicit node corresponding to a substring $C$ of $T$ isannotated with two numbers: the number of non-overlapping consecutiveoccurrences of $C$ and the total number of positions in $T$ that are covered byoccurrences of $C$ in $T$. Kociumaka et al. (Algorithmica, 2015) have shown howto compute the CST of a length-$n$ string in $O(n \log n)$ time. We show how tocompute the CST in $O(n)$ time assuming that $T$ is over an integer alphabet.  Kociumaka et al. (Algorithmica, 2015; Theor. Comput. Sci., 2018) have shownthat knowing the CST of a length-$n$ string $T$, one can compute a linear-sizedrepresentation of all seeds of $T$ as well as all shortest $\alpha$-partialcovers and seeds in $T$ for a given $\alpha$ in $O(n)$ time. Thus our resultimplies linear-time algorithms computing these notions of quasiperiodicity. Theresulting algorithm computing seeds is substantially different from theprevious one (Kociumaka et al., SODA 2012, ACM Trans. Algorithms, 2020).Kociumaka et al. (Algorithmica, 2015) proposed an $O(n \log n)$-time algorithmfor computing a shortest $\alpha$-partial cover for each $\alpha=1,\ldots,n$;we improve this complexity to $O(n)$.  Our results are based on a new characterization of consecutive overlappingoccurrences of a substring $S$ of $T$ in terms of the set of runs (see Kolpakovand Kucherov, FOCS 1999) in $T$. This new insight also leads to an $O(n)$-sizedindex for reporting overlapping consecutive occurrences of a given pattern $P$of length $m$ in $O(m+output)$ time, where $output$ is the number ofoccurrences reported. In comparison, a general index for reporting bounded-gapconsecutive occurrences of Navarro and Thankachan (Theor. Comput. Sci., 2016)uses $O(n \log n)$ space.

Linear Time Construction of Cover Suffix Tree and Applications