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

TL;DR

This paper introduces an efficient method for computing string runs over general ordered alphabets by leveraging a non-crossing property of LCE queries, achieving near-linear time complexity.

Contribution

It demonstrates that non-crossing LCE queries can be answered online in near-linear time, leading to a faster algorithm for computing runs over general alphabets.

Findings

Answering non-crossing LCE queries online takes O(n α(n)) time.

The new algorithm computes all runs in near-linear time.

This approach generalizes previous results to arbitrary ordered alphabets.

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 (\emph{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.\ (accepted to CPM 2016) to $O(n \log \log n)$ time. In this work we note a special \emph{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, which yields an $O(n \alpha(n))$-time algorithm for computing runs.