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

• Published in: String Processing and Information Retrieval Vol. 9954; pp. 22 - 34
• Main Authors: Crochemore, Maxime, Iliopoulos, Costas S., Kociumaka, Tomasz, Kundu, Ritu, Pissis, Solon P., Radoszewski, Jakub, Rytter, Wojciech, Waleń, Tomasz
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 01.01.2016; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Information retrieval; Membership Query; Outerplanar Graph; Query Algorithm; Similar Query; Suffix Array
• ISBN: 9783319460482; 331946048X
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-46049-9_3

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 $$\mathcal {O}(n (\log n)^{2/3})$$ -time algorithm for answering $$\mathcal {O}(n)$$ LCE queries. This result was improved by Gawrychowski et al. (CPM 2016) to $$\mathcal {O}(n \log \log n)$$ 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 $$\mathcal {O}(n \alpha (n))$$ time, where $$\alpha (n)$$ is the inverse Ackermann function, which yields an $$\mathcal {O}(n \alpha (n))$$ -time algorithm for computing runs.