Near-Optimal Computation of Runs over General Alphabet via Non-Crossing LCE Queries

Abstract

Longest common extension queries (LCE queries) and runs are ubiquitous in algorithmic stringology. Linear-time algorithms computing runs and preprocessing for constant-time LCE queries have been known for over a decade. However, these algorithms assume a linearly-sortable integer alphabet. A recent breakthrough paper by Bannai et al. (SODA 2015) showed a link between the two notions: all the runs in a string can be computed via a linear number of LCE queries. The first to consider these problems over a general ordered alphabet was Kosolobov (Inf. Process. Lett., 2016), who presented an $O(n(\log n{)}^2/3)$-time algorithm for answering $O(n)$ LCE queries. This result was improved by Gawrychowski et al. (CPM 2016) to $O(n\log \log n)$ time. In this work we note a special non-crossing property of LCE queries asked in the runs computation. We show that any n such non-crossing queries can be answered on-line in $O(n\alpha (n))$ time, where $\alpha (n)$ is the inverse Ackermann function, which yields an $O(n\alpha (n))$-time algorithm for computing runs.