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 linearlysortable 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(nα(n)) time, where α(n) is the inverse Ackermann function, which yields an O(nα(n))-time algorithm for computing runs.