Computing covers using prefix tables

Alatabbi, Ali; Rahman, M. Sohel; Smyth, W. Franklin

doi:10.1016/j.dam.2015.05.019

Cited by 8 publications

(7 citation statements)

References 17 publications

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“…This result only holds for z = O(1) and σ = O(1); for other values of z and σ little is known. For indeterminate strings a simple O(nσ k/2 k)-time algorithm and a fixed parameter tractable 2 O(k log k) +nk O(1)time algorithm was presented in [14], where k is the number of non-solid positions and 1 < σ ≤ n. In [1], an O(n 2 )-time algorithm was presented for some, but not all, covers of an indeterminate string.…”

Section: Computing Covers and Seedsmentioning

confidence: 99%

“…To improve readability all singletons have been omitted, and we assume a unique terminating letter for each extended maximal factor.3 Applications to Weighted and Indeterminate StringsAn indeterminate string w of length n on an alphabet Σ is a finite sequence of n sets, such that w[i] ⊆ Σ, w[i] = ∅, for all 0 ≤ i < n. If |w[i]| = 1, that is, w[i]represents a single letter of Σ, we say that w[i] is a solid position. Any indeterminate string w of length n can be represented by a weighted string x of length n such that (a,1 |w…”

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confidence: 99%

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Crochemore’s Partitioning on Weighted Strings and Applications

Barton

Pissis

2017

Algorithmica

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Given a string on alphabet Σ the partitioning problem is to compute classes of equivalences on the set of positions of the input string. These classes implicitly memorise identical factors of the string and, hence, their efficient computation is essential for a wide range of string processing applications. We study this problem for a weighted string: for every position of the weighted string and every letter of the alphabet a probability of occurrence of this letter at this position is given. Thus a weighted string may represent many different strings, each with probability of occurrence equal to the product of probabilities of its letters at subsequent positions. In this article, we present a non-trivial generalisation of Crochemore's partitioning algorithm (IPL, 1981) that works on weighted strings requiring time O(υn log υn), where n is the length of the string, υ = min{z 2 , zn, σ n }, σ is the size of Σ, and 1/z is a cumulative weight threshold, defined as the minimal probability of occurrence of factors in the string. Our contributions can be summarised as follows: (a) we design the first algorithm to solve the partitioning problem on weighted strings for arbitrary z and σ in time O(υn log υn) and space O(υn) improving the state of the art for z = O(1); (b) we improve the state of the art for numerous other string processing problems; and (c) we show further combinatorial insight into the relation between weighted and indeterminate strings, that is, sequences of alphabet subsets without associated occurrence probabilities.

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Section: Computing Covers and Seedsmentioning

confidence: 99%

mentioning

confidence: 99%

Crochemore’s Partitioning on Weighted Strings and Applications

Barton

Pissis

2017

Algorithmica

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“…In [2] two definitions of "cover" for an indeterminate string are proposed: a sliding cover where adjacent or overlapping covering substrings of x must match, and a rooted cover where each covering substring is constrained only to match a prefix of x. The nontransitivity of matching (see Section 1) inhibits implementation of a sliding cover, but [2] shows how to compute all the rooted covers of indeterminate x from its prefix array in O(n 2 ) worst case time, Θ(n) in the average case.…”

Section: Extensionsmentioning

confidence: 99%

“…Since the longest cover of x[1..j] is also a cover of x [1..i], γ implicitly specifies all the covers of every prefix of x. A recent paper [2] extends the computation of γ to "indeterminate strings" (see below for definition).…”

Section: Introductionmentioning

confidence: 99%

Enhanced Covers of Regular & Indeterminate Strings using Prefix Tables

Alatabbi,

Islam,

Rahman

et al. 2015

Preprint

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A cover of a string x = x[1..n] is a proper substring u of x such that x can be constructed from possibly overlapping instances of u. A recent paper [12] relaxes this definition -an enhanced cover u of x is a border of x (that is, a proper prefix that is also a suffix) that covers a maximum number of positions in x (not necessarily all) -and proposes efficient algorithms for the computation of enhanced covers. These algorithms depend on the prior computation of the border array β[1..n], where β[i] is the length of the longest border of x[1..i], 1 ≤ i ≤ n. In this paper, we first show how to compute enhanced covers using instead the prefix table: an array π[1..n] such that π[i] is the length of the longest substring of x beginning at position i that matches a prefix of x. Unlike the border array, the prefix table is robust: its properties hold also for indeterminate strings -that is, strings defined on subsets of the alphabet Σ rather than individual elements of Σ. Thus, our algorithms, in addition to being faster in practice and more space-efficient than those of [12], allow us to easily extend the computation of enhanced covers to indeterminate strings. Both for regular and indeterminate strings, our algorithms execute in expected linear time. Along the way we establish an important theoretical result: that the expected maximum length of any border of any prefix of a regular string x is approximately 1.64 for binary alphabets, less for larger ones.

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“…Combinatorial properties of covers were presented in [2,19,39]. Covers were also considered in indeterminate [1,6,24,36] and weighted strings [10,11,35]. A survey by Apostolico and Breslauer [7] describes in detail the algorithms [8,13,14,41], whereas a survey by Iliopoulos and Mouchard [39] describes the algorithms [8,9,38].…”

Section: Introductionmentioning

confidence: 99%

Experimental Evaluation of Algorithms for Computing Quasiperiods

Czajka¹,

Radoszewski²

2019

Preprint

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Quasiperiodicity is a generalization of periodicity that was introduced in the early 1990s. Since then, dozens of algorithms for computing various types of quasiperiodicity were proposed. Our work is a step towards answering the question: "Which algorithm for computing quasiperiods to choose in practice?". The central notions of quasiperiodicity are covers and seeds. We implement algorithms for computing covers and seeds in the original and in new simplified versions and compare their efficiency on various types of data. We also discuss other known types of quasiperiodicity, distinguish partial covers as currently the most promising for large real-world data, and check their effectiveness using real-world data.

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Computing covers using prefix tables

Cited by 8 publications

References 17 publications

Crochemore’s Partitioning on Weighted Strings and Applications

Crochemore’s Partitioning on Weighted Strings and Applications

Enhanced Covers of Regular & Indeterminate Strings using Prefix Tables

Experimental Evaluation of Algorithms for Computing Quasiperiods

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