Chapter

SOFSEM 2010: Theory and Practice of Computer Science

Volume 5901 of the series Lecture Notes in Computer Science pp 296-307

# Efficient Algorithms for Two Extensions of LPF Table: The Power of Suffix Arrays

• Maxime CrochemoreAffiliated withDept. of Computer Science, King’s College LondonUniversité Paris-Est
• , Costas S. IliopoulosAffiliated withDept. of Computer Science, King’s College LondonDigital Ecosystems & Business Intelligence Institute, Curtin University of Technology
• , Marcin KubicaAffiliated withInstitute of Informatics, University of Warsaw
• , Wojciech RytterAffiliated withInstitute of Informatics, University of WarsawFaculty of Math. and Informatics, Copernicus University
• , Tomasz WaleńAffiliated withInstitute of Informatics, University of Warsaw

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## Abstract

Suffix arrays provide a powerful data structure to solve several questions related to the structure of all the factors of a string. We show how they can be used to compute efficiently two new tables storing different types of previous factors (past segments) of a string. The concept of a longest previous factor is inherent to Ziv-Lempel factorization of strings in text compression, as well as in statistics of repetitions and symmetries. The longest previous reverse factor for a given position i is the longest factor starting at i, such that its reverse copy occurs before, while the longest previous non-overlapping factor is the longest factor v starting at i which has an exact copy occurring before. The previous copies of the factors are required to occur in the prefix ending at position i − 1. We design algorithms computing the table of longest previous reverse factors (LPrF table) and the table of longest previous non-overlapping factors (LPnF table). The latter table is useful to compute repetitions while the former is a useful tool for extracting symmetries. These tables are computed, using two previously computed read-only arrays (SUF and LCP) composing the suffix array, in linear time on any integer alphabet. The tables have not been explicitly considered before, but they have several applications and they are natural extensions of the LPF table which has been studied thoroughly before. Our results improve on the previous ones in several ways. The running time of the computation no longer depends on the size of the alphabet, which drops a log factor. Moreover the newly introduced tables store additional information on the structure of the string, helpful to improve, for example, gapped palindrome detection and text compression using reverse factors.

### Keywords

Longest previous reverse factor longest previous non-overlapping factor longest previous factor palindrome runs Suffix Array text compression