Fast Computation of $k$-Runs, Parameterized Squares, and Other Generalised Squares

A $k$-mismatch square is a string of the form $XY$ where $X$ and $Y$ are two equal-length strings that have at most $k$ mismatches. Kolpakov and Kucherov [Theor. Comput. Sci., 2003] defined two notions of $k$-mismatch repeats, called $k$-repetitions and $k$-runs, each representing a sequence of consecutive $k$-mismatch squares of equal length. They proposed algorithms for computing $k$-repetitions and $k$-runs working in $O(nk \log k + output)$ time for a string of length $n$ over an integer alphabet, where $output$ is the number of the reported repeats. We show that $output=O(nk \log k)$, both in case of $k$-repetitions and $k$-runs, which implies that the complexity of their algorithms is actually $O(nk \log k)$. We apply this result to computing parameterized squares. A parameterized square is a string of the form $XY$ such that $X$ and $Y$ parameterized-match, i.e., there exists a bijection $f$ on the alphabet such that $f(X) = Y$. Two parameterized squares $XY$ and $X'Y'$ are equivalent if they parameterized match. Recently Hamai et al. [SPIRE 2024] showed that a string of length $n$ over an alphabet of size $σ$ contains less than $nσ$ non-equivalent parameterized squares, improving an earlier bound by Kociumaka et al. [Theor. Comput. Sci., 2016]. We apply our bound for $k$-mismatch repeats to propose an algorithm that reports all non-equivalent parameterized squares in $O(nσ\log σ)$ time. We also show that the number of non-equivalent parameterized squares can be computed in $O(n \log n)$ time. This last algorithm applies to squares under any substring compatible equivalence relation and also to counting squares that are distinct as strings. In particular, this improves upon the $O(nσ)$-time algorithm of Gawrychowski et al. [CPM 2023] for counting order-preserving squares that are distinct as strings if $σ= ω(\log n)$.