Sparse Text Indexing in Small Space

Abstract

In this work we present efficient algorithms for constructing sparse suffix trees, sparse suffix arrays and sparse positions heaps for b arbitrary positions of a text T of length n while using only O(b) words of space during the construction.Attempts at breaking the naive bound of Ω(nb) time for constructing sparse suffix trees in O(b) space can be traced back to the origins of string indexing in 1968. First results were only obtained in 1996, but only for the case where the b suffixes were evenly spaced in T . In this paper there is no constraint on the locations of the suffixes.Our main contribution is to show that the sparse suffix tree (and array) can be constructed in O(n log 2 b) time. To achieve this we develop a technique, that allows to efficiently answer b longest common prefix queries on suffixes of T , using only O(b) space. We expect that this technique will prove useful in many other applications in which space usage is a concern. Our first solution is Monte-Carlo and outputs the correct tree with high probability. We then give a Las-Vegas algorithm which also uses O(b) space and runs in the same time bounds with high probability when b = O( √ n).Furthermore, additional tradeoffs between the space usage and the construction time for the Monte-Carlo algorithm are given. Finally, we show that at the expense of slower pattern queries, it is possible to construct sparse position heaps in O(n + b log b) time and O(b) space.

Locally Consistent Parsing for Text Indexing in Small Space

“…However, we design a new way to combine Karp-Rabin fingerprints together with approximately min-wise hash functions, in order to use this method using only small amount of space. In previous results for the LCE problem [3,4,19,37], a set of positions of size O( n τ ) was also considered by the data structures. In contrast to these algorithms, where the selected positions were dependent only on the length of the text, we introduce the first algorithm that exploit the actual text using local properties in order to decide which positions to select.…”

“…Text indexing is one of the most fundamental problems in the area of string algorithms and information retrieval (e.g. see [40,24,39,30,13,3,15,16,6,17,1,26,23,8,34]). In this paper we consider two closely related problems of text indexing in sub-linear working space.…”

“…We consider the trade-off version, where there is a parameter 1 ≤ τ ≤ n and the goal is to construct a data structure using only O( n τ ) words of space, while achieving fast construction and query time. The trade-off LCE problem is closely tied to the SST problem, and in several papers the LCE was used as the main tool for constructing the SST efficiently [19,3].…”

In-Place Sparse Suffix Sorting

“…The authors of these papers showed that suffix sorting is possible within the same space of the text and the final suffix array, that is, in place. Parallel to the study of techniques to sort all suffixes of a text, several authors started considering the problem of efficiently sorting only a subset of b text's suffixes [2,3,6,11,17,19,20], a fundamental step in the construction of compressed and sparse text indexes [20] and space-efficient compression algorithms. Very recently, Gawrychowski and Kociumaka [11] gave the first optimal time-and-space solution to the problem, showing that O(b) working space and O(n) running time are achievable with a Monte Carlo algorithm (they also consider a Las Vegas algorithm with higher running time).…”

“…It is well known (see, e.g. [2]) that the Longest Common Extension problem (LCE) -that is, to find the length of the longest common prefix between any two text suffixes -is closely related to the suffix sorting problem. In this paper, we exploit this relation and give the first in-place solution to the sparse suffix sorting problem.…”