Suffix Tree for a Sliding Window: An Overview

The suffix tree is a very powerful data structure developed originally for string matching and string searching. It has found many applications over the time and some of them belong into the data compression field. Many of these applications need a suffix tree built for a sliding window and there exist two clever algorithms by Fiala and Greene and by Larsson that make this possible. However, as we show both approaches have flawed proofs. We remedy this situation both by explaining a simple alternative algorithm and giving a correct proof. Introduction In 1973 Weiner introduced a new powerful data structure for string matching and searching [Weiner, 1973]. This data structure is called suffix tree and has found many applications over the time. Despite loosing some ground to CDAWG [Crochemore and Vérin, 1997] and the suffix array [Manber and Mayers, 1993] lately, the suffix tree is still a very interesting data structure. Particularly interesting is the application of the suffix tree to data compression [Fiala and Greene, 1989; Larsson, 1999; Senft, 2005]. Many of them require the suffix tree to be maintained over so called sliding window [Ziv and Lempel, 1977]. Fiala and Greene developed a clever method to adapt suffix tree for a sliding window [Fiala and Greene, 1989] that was later modified by Larsson [Larsson, 1999]. These methods are very similar and have a common weakness: their correctness proofs are flawed as we will show later. We remedy this situation both by giving a correct proof and also describing a simpler working method. This paper is organised as follows: The next section reviews some necessary notation and terminology, leading to the definition of two main concepts: the suffix tree and the sliding window. The third section describes a suffix tree adaptation for a sliding window and also contains our original results. First the suffix tree construction and symbol deletion algorithms are reviewed, then a suffix tree implementation is described and edge label maintenance addressed. Two well known algorithms for edge label maintenance [Fiala and Greene, 1989; Larsson, 1999] are described and analysed and our own simple replacement given. Weaknesses in correctness proofs are shown for Fiala’s and Greene’s as well as Larsson’s algorithm and a new sound proof is given. We conclude this paper with final remarks in the last section. Concepts and Notation We omit basic string and graph-related definitions like the definition of an alphabet, symbol and prefix or root, edge and parent. We also give only informal definitions for most non-basic concepts used in this paper and refer the reader to e.g. [Senft, 2005] for details. Strings The following string-related definitions will simplify the suffix tree definition as well as the description of algorithms in further sections. String α is said to occur in string δ if there exists a position i such that the sequence of symbols beginning at position i and ending at position i+ |α| − 1 equals to string α. This sequence of characters is called an occurrence of α in δ at position i. A string is unique in δ if it occurs in δ exactly once. A right branching substring of string δ occurs at least twice in δ and at least two of these occurrences are followed by two different characters. A proper substring of string δ is either the empty string or a right branching substring of δ or a unique suffix of δ. Suffix Tree To give a suffix tree definition a standard graph terminology (cf. [Harary, 1969]) will be used. However, to simplify things a bit vertices of a tree that are not leaves will be called nodes. The suffix tree for a string δ is a rooted tree with edges labelled by nonempty substrings of δ (see Fig. 1). The strings represented in the suffix tree are exactly all substrings of δ. The root represents WDS'05 Proceedings of Contributed Papers, Part I, 41–46, 2005. ISBN 80-86732-59-2 © MATFYZPRESS