An α-gapped repeat (α ≥ 1) in a word w is a factor uvu of w such that |u v| ≤ α|u|; the two occurrences of u are called arms of this α-gapped repeat. An α-gapped repeat is called maximal if its arms cannot be extended simultaneously with the same character to the right nor to the left. We show that the number of all maximal α-gapped repeats occurring in words of length n is upper bounded by 18α n. In the case of α-gapped palindromes, i.e., factors \(uv{{u}^{\intercal }}\) with |u v|≤ α|u|, we show that the number of all maximal α-gapped palindromes occurring in words of length n is upper bounded by 28α n + 7n. Both upper bounds allow us to construct algorithms finding all maximal α-gapped repeats and/or all maximal α-gapped palindromes of a word of length n on an integer alphabet of size \(n^{\mathcal {O}(1)}\) in \({\mathcal {O}(\alpha n)}\) time. The presented running times are optimal since there are words that have Θ(α n) maximal α-gapped repeats/palindromes.

Computing the matching statistics of patterns with respect to a text is a fundamental task in bioinformatics, but a formidable one when the text is a highly compressed genomic database. Bannai et al. gave an efficient solution for this case, which Rossi et al. recently implemented, but it uses two passes over the patterns and buffers a pointer for each character during the first pass. In this paper, we simplify their solution and make it streaming, at the cost of slowing it down slightly. This means that, first, we can compute the matching statistics of several long patterns (such as whole human chromosomes) in parallel while still using a reasonable amount of RAM; second, we can compute matching statistics online with low latency and thus quickly recognize when a pattern becomes incompressible relative to the database. Our code is available at https://github.com/koeppl/phoni.

We present algorithms computing the non-overlapping Lempel–Ziv-77 factorization and the longest previous non-overlapping factor table within small space in linear or near-linear time with the help of modern suffix tree representations fitting into limited space. With similar techniques, we show how to answer substring compression queries for the Lempel–Ziv-78 factorization with a possible logarithmic multiplicative slowdown depending on the used suffix tree representation.