Efficiently Finding All Maximal $\alpha$-gapped Repeats

TL;DR

This paper introduces an optimal algorithm for finding all maximal $eta$-gapped repeats in a word, providing tight bounds on their number and extending techniques to $eta$-gapped palindromes.

Contribution

The paper establishes an upper bound of $18eta n$ on the number of maximal $eta$-gapped repeats and presents an $O(eta n)$ algorithm, which is proven to be optimal.

Findings

Number of maximal $eta$-gapped repeats is at most $18eta n$.

Developed an $O(eta n)$ algorithm to find all such repeats.

Techniques extend to $eta$-gapped palindromes.

Abstract

For $\alpha\geq 1$, an $\alpha$-gapped repeat in a word $w$ is a factor $uvu$ of $w$ such that $|uv|\leq \alpha |u|$; the two factors $u$ in such a repeat are called arms, while the factor $v$ is called gap. Such a repeat is called maximal if its arms cannot be extended simultaneously with the same symbol to the right or, respectively, to the left. In this paper we show that the number of maximal $\alpha$-gapped repeats that may occur in a word is upper bounded by $18\alpha n$. This allows us to construct an algorithm finding all the maximal $\alpha$-gapped repeats of a word in $O(\alpha n)$; this is optimal, in the worst case, as there are words that have $\Theta(\alpha n)$ maximal $\alpha$-gapped repeats. Our techniques can be extended to get comparable results in the case of $\alpha$-gapped palindromes, i.e., factors $uvu^\mathrm{T}$ with $|uv|\leq \alpha |u|$.