DOCSLIB.ORG

Suffix and Factor Automata and Combinatorics on Words

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Suffix and Factor Automata and Combinatorics on Words Suffix and Factor Automata and Combinatorics on Words Gabriele Fici Workshop PRIN 2007–2009 Varese – 5 September 2011 Gabriele Fici Suffix and Factor Automata and Combinatorics on Words The Suffix Automaton Definition (A. Blumer et al. 85 — M. Crochemore 86) The Suffix Automaton of the word w is the minimal deterministic automaton recognizing the suffixes of w. Example The SA of w = aabbabb a a b b a b b 0 1 2 3 4 5 6 7 a b b 3′′ 4′′ b a b 3′ Gabriele Fici Suffix and Factor Automata and Combinatorics on Words The SA has several applications, for example in pattern matching music retrieval spam detection search of characteristic expressions in literary works speech recordings alignment ... Algorithmic Construction The SA allows the search of a pattern v in a text w in time and space O(jvj). Moreover: Theorem (A. Blumer et al. 85 — M. Crochemore 86) The SA of a word w over a fixed alphabet Σ can be built in time and space O(jwj). Gabriele Fici Suffix and Factor Automata and Combinatorics on Words Algorithmic Construction The SA allows the search of a pattern v in a text w in time and space O(jvj). Moreover: Theorem (A. Blumer et al. 85 — M. Crochemore 86) The SA of a word w over a fixed alphabet Σ can be built in time and space O(jwj). The SA has several applications, for example in pattern matching music retrieval spam detection search of characteristic expressions in literary works speech recordings alignment ... Gabriele Fici Suffix and Factor Automata and Combinatorics on Words Determinize by subset construction: a a b b a b b {0, 1, 2,..., 7} {1, 2, 5} {2} {3} {4} {5} {6} {7} a b b {3, 6} {4, 7} b b a {3, 4, 6, 7} One Way to Build the SA Build a naif non-deterministic automaton: w = aabbabb a a b b a b b 0 1 2 3 4 5 6 7 Gabriele Fici Suffix and Factor Automata and Combinatorics on Words a a b b a b b {0, 1, 2,..., 7} {1, 2, 5} {2} {3} {4} {5} {6} {7} a b b {3, 6} {4, 7} b b a {3, 4, 6, 7} One Way to Build the SA Build a naif non-deterministic automaton: w = aabbabb a a b b a b b 0 1 2 3 4 5 6 7 Determinize by subset construction: Gabriele Fici Suffix and Factor Automata and Combinatorics on Words Define on Fact(w) the equivalence: u ∼SA v () Endsetw (u) = Endsetw (v) Ending Positions We associate to each factor v of w the set of ending positions of v in w. We note this set Endsetw (v). Example w = a a b b a b b 1 2 3 4 5 6 7 Endsetw (ba) = f5g, Endsetw (abb) = Endsetw (bb) = f4; 7g. Gabriele Fici Suffix and Factor Automata and Combinatorics on Words Ending Positions We associate to each factor v of w the set of ending positions of v in w. We note this set Endsetw (v). Example w = a a b b a b b 1 2 3 4 5 6 7 Endsetw (ba) = f5g, Endsetw (abb) = Endsetw (bb) = f4; 7g. Define on Fact(w) the equivalence: u ∼SA v () Endsetw (u) = Endsetw (v) Gabriele Fici Suffix and Factor Automata and Combinatorics on Words Ending Positions u ∼SA v () Endsetw (u) = Endsetw (v) Remark ∗ u ∼SA v if and only if for any z 2 Σ one has uz 2 Suff(w) () vz 2 Suff(w) Remark Fact(w)= ∼SA is in bijection with the set of states of the SA of w. Gabriele Fici Suffix and Factor Automata and Combinatorics on Words The bounds are well known: jwj + 1 ≤ jSA(w)j ≤ 2jwj − 1 The upper bound is reached for w = abjw|−1, with a 6= b. And for the lower bound? Problem (J. Berstel and M. Crochemore) Characterize the languageL SA of words such that jSA(w)j = jwj + 1. The Number of States of the SA The number of states (classes) of the SA of w is therefore jSA(w)j = j Fact(w)= ∼SA j Gabriele Fici Suffix and Factor Automata and Combinatorics on Words The upper bound is reached for w = abjw|−1, with a 6= b. And for the lower bound? Problem (J. Berstel and M. Crochemore) Characterize the languageL SA of words such that jSA(w)j = jwj + 1. The Number of States of the SA The number of states (classes) of the SA of w is therefore jSA(w)j = j Fact(w)= ∼SA j The bounds are well known: jwj + 1 ≤ jSA(w)j ≤ 2jwj − 1 Gabriele Fici Suffix and Factor Automata and Combinatorics on Words And for the lower bound? Problem (J. Berstel and M. Crochemore) Characterize the languageL SA of words such that jSA(w)j = jwj + 1. The Number of States of the SA The number of states (classes) of the SA of w is therefore jSA(w)j = j Fact(w)= ∼SA j The bounds are well known: jwj + 1 ≤ jSA(w)j ≤ 2jwj − 1 The upper bound is reached for w = abjw|−1, with a 6= b. Gabriele Fici Suffix and Factor Automata and Combinatorics on Words Problem (J. Berstel and M. Crochemore) Characterize the languageL SA of words such that jSA(w)j = jwj + 1. The Number of States of the SA The number of states (classes) of the SA of w is therefore jSA(w)j = j Fact(w)= ∼SA j The bounds are well known: jwj + 1 ≤ jSA(w)j ≤ 2jwj − 1 The upper bound is reached for w = abjw|−1, with a 6= b. And for the lower bound? Gabriele Fici Suffix and Factor Automata and Combinatorics on Words The Number of States of the SA The number of states (classes) of the SA of w is therefore jSA(w)j = j Fact(w)= ∼SA j The bounds are well known: jwj + 1 ≤ jSA(w)j ≤ 2jwj − 1 The upper bound is reached for w = abjw|−1, with a 6= b. And for the lower bound? Problem (J. Berstel and M. Crochemore) Characterize the languageL SA of words such that jSA(w)j = jwj + 1. Gabriele Fici Suffix and Factor Automata and Combinatorics on Words b is right special a and b are bispecial Example w = aabbabb ab is left special Special Factors Definition v is a left special factor of w if there exist a 6= b such that av and bv are factors of w v is a right special factor of w if there exist a 6= b such that va and vb are factors of w v is a bispecial factor of w if it is both left and right special Gabriele Fici Suffix and Factor Automata and Combinatorics on Words b is right special a and b are bispecial Special Factors Definition v is a left special factor of w if there exist a 6= b such that av and bv are factors of w v is a right special factor of w if there exist a 6= b such that va and vb are factors of w v is a bispecial factor of w if it is both left and right special Example w = aabbabb ab is left special Gabriele Fici Suffix and Factor Automata and Combinatorics on Words a and b are bispecial Special Factors Definition v is a left special factor of w if there exist a 6= b such that av and bv are factors of w v is a right special factor of w if there exist a 6= b such that va and vb are factors of w v is a bispecial factor of w if it is both left and right special Example w = aabbabb ab is left special b is right special Gabriele Fici Suffix and Factor Automata and Combinatorics on Words Special Factors Definition v is a left special factor of w if there exist a 6= b such that av and bv are factors of w v is a right special factor of w if there exist a 6= b such that va and vb are factors of w v is a bispecial factor of w if it is both left and right special Example w = aabbabb ab is left special b is right special a and b are bispecial Gabriele Fici Suffix and Factor Automata and Combinatorics on Words Example (w = aabbabb) L Sw = 5 since the left special factors of w are , a; b; ab; abb Pw = 2 since a is left special in w L jSA(w)j = jwj + 1 + Sw − Pw = 7 + 1 + 5 − 2 = 11 The Number of States of the SA Theorem (G. Fici 09) L jSA(w)j = jwj + 1 + Sw − Pw L Sw = number of left special factors of w Pw = length of the shortest prefix of w which is not left special a a b b a b b 0 1 2 3 4 5 6 7 a b b 3′′ 4′′ b a b 3′ Gabriele Fici Suffix and Factor Automata and Combinatorics on Words The Number of States of the SA Theorem (G. Fici 09) L jSA(w)j = jwj + 1 + Sw − Pw L Sw = number of left special factors of w Pw = length of the shortest prefix of w which is not left special Example (w = aabbabb) a a b b a b b 0 1 2 3 4 5 6 7 a b b 3′′ 4′′ b a b L Sw = 5 since the left special factors3′ of w are , a; b; ab; abb Pw = 2 since a is left special in w L jSA(w)j = jwj + 1 + Sw − Pw = 7 + 1 + 5 − 2 = 11 Gabriele Fici Suffix and Factor Automata and Combinatorics on Words If jΣj = 2, LSA is the set of finite prefixes of standard Sturmian words, i.e., the set of left special factors of Sturmian words.
Load more

Recommended publications
  • Suffix Structure Lecture Moscow International Workshop ACM ICPC
    Suffix structure lecture Moscow International Workshop ACM ICPC 2015 19 November, 2015 1 Introduction Consider problem of pattern matching. One of possible statement is as follows: you are given text T and n patterns Si. For each of patterns you are to say whether it has occurrence in text. There are two ways in solving this problem. The first one (considered easier) is Aho-Corasick algorithm which constructs automaton that recognizes strings that contain some of patterns as substring. The second one (considered harder) is usage of suffix structures which are mainly suffix array, suffix automaton or suffix tree. This lecture will be dedicated to the last two, their relations and applications. 2 Naive solution Consider arbitrary substring s[pos; pos + len − 1]. It is the prefix of length len of suffix of string which starts in position pos. Given this, we can use following naive solution for the problem: let’s add all the suffixes of the string in trie. Then for each prefix of every string in trie will correspond exactly one vertex hence we can check whether Si is substring of T in O(Si). Disadvantages are obvious - such solution needs O(jT j2) time and memory. There are two ways of solving this problem which lead to suffix tree and suffix automaton. 1 3 Suffix tree We can see that every top-down way in trie is substring of T . Hence we can remove from trie all nodes which are, neither root nor vertex corresponding to some suffix and its degree equals 2 (i.e., nodes which are not crossroads - they have exactly one ingoing and exactly one outgoing edge).
    [Show full text]
  • Algorithms on Strings
    Algorithms on Strings Maxime Crochemore Christophe Hancart Thierry Lecroq Algorithms on Strings Cambridge University Press Algorithms on Strings – Maxime Crochemore, Christophe Han- cart et Thierry Lecroq Table of contents Preface VII 1 Tools 1 1.1 Strings and automata 2 1.2 Some combinatorics 8 1.3 Algorithms and complexity 17 1.4 Implementation of automata 21 1.5 Basic pattern matching techniques 26 1.6 Borders and prefixes tables 36 2 Pattern matching automata 51 2.1 Trie of a dictionary 52 2.2 Searching for several strings 53 2.3 Implementation with failure function 61 2.4 Implementation with successor by default 67 2.5 Searching for one string 76 2.6 Searching for one string and failure function 79 2.7 Searching for one string and successor by default 86 3 String searching with a sliding window 95 3.1 Searching without memory 95 3.2 Searching time 101 3.3 Computing the good suffix table 105 3.4 Automaton of the best factor 109 3.5 Searching with one memory 113 3.6 Searching with several memories 119 3.7 Dictionary searching 128 4 Suffix arrays 137 4.1 Searching a list of strings 138 4.2 Searching with common prefixes 141 4.3 Preprocessing the list 146 VI Table of contents 4.4 Sorting suffixes 147 4.5 Sorting suffixes on bounded integer alphabets 153 4.6 Common prefixes of the suffixes 158 5 Structures for index 165 5.1 Suffix trie 165 5.2 Suffix tree 171 5.3 Contexts of factors 180 5.4 Suffix automaton 185 5.5 Compact suffix automaton 197 6 Indexes 205 6.1 Implementing an index 205 6.2 Basic operations 208 6.3 Transducer of positions 213 6.4 Repetitions
    [Show full text]
  • Finite Automata Implementations Considering CPU Cache J
    Acta Polytechnica Vol. 47 No. 6/2007 Finite Automata Implementations Considering CPU Cache J. Holub The finite automata are mathematical models for finite state systems. More general finite automaton is the nondeterministic finite automaton (NFA) that cannot be directly used. It is usually transformed to the deterministic finite automaton (DFA) that then runs in time O(n), where n is the size of the input text. We present two main approaches to practical implementation of DFA considering CPU cache. The first approach (represented by Table Driven and Hard Coded implementations) is suitable forautomata being run very frequently, typically having cycles. The other approach is suitable for a collection of automata from which various automata are retrieved and then run. This second kind of automata are expected to be cycle-free. Keywords: deterministic finite automaton, CPU cache, implementation. 1 Introduction usage. Section 3 then describes general techniques for DFA implementation. It is mostly suitable for DFA that is run most The original formal study of finite state systems (neural of the time. Since DFA has a finite set of states, this kind of nets) is from 1943 by McCulloch and Pitts [14]. In 1956 DFA has to have cycles. Recent results in the implementation Kleene [13] modeled the neural nets of McCulloch and Pitts using CPU cache are discussed in Section 4. On the other by finite automata. In that time similar models were pre- hand we have a collection of DFAs each representing some sented by Huffman [12], Moore [17], and Mealy [15]. In 1959, document (e.g., in the form of complete index in case of Rabin and Scott introduced nondeterministic finite automata factor or suffix automata).
    [Show full text]
  • Starting the Matching from the End Enables Long Shifts. • the Horspool
    BNDM Starting the matching from the end enables long shifts. The Horspool algorithm bases the shift on a single character. • The Boyer–Moore algorithm uses the matching suffix and the • mismatching character. Factor based algorithms continue matching until no pattern factor • matches. This may require more comparisons but it enables longer shifts. Example 2.14: Horspool shift varmasti-aikai/sen-ainainen ainaisen-ainainen ainaisen-ainainen Boyer–Moore shift Factor shift varmasti-aikai/sen-ainainen varmasti-ai/kaisen-ainainen ainaisen-ainainen ainaisen-ainainen ainaisen-ainainen ainaisen-ainainen 86 Factor based algorithms use an automaton that accepts suffixes of the reverse pattern P R (or equivalently reverse prefixes of the pattern P ). BDM (Backward DAWG Matching) uses a deterministic automaton • that accepts exactly the suffixes of P R. DAWG (Directed Acyclic Word Graph) is also known as suffix automaton. BNDM (Backward Nondeterministic DAWG Matching) simulates a • nondeterministic automaton. Example 2.15: P = assi. ε ai ss -10123 BOM (Backward Oracle Matching) uses a much simpler deterministic • automaton that accepts all suffixes of P R but may also accept some other strings. This can cause shorter shifts but not incorrect behaviour. 87 Suppose we are currently comparing P against T [j..j + m). We use the automaton to scan the text backwards from T [j + m 1]. When the automaton has scanned T [j + i..j + m): − If the automaton is in an accept state, then T [j + i..j + m) is a prefix • of P . If i = 0, we found an occurrence. ⇒ Otherwise, mark the prefix match by setting shift = i. This is the ⇒ length of the shift that would achieve a matching alignment.
    [Show full text]
  • Text Searching Algorithms
    Department of Computer Science and Engineering Faculty of Electrical Engineering Czech Technical University in Prague TEXT SEARCHING ALGORITHMS SEMINARS Bo·rivoj Melichar, Jan Anto·s, Jan Holub, Tom¶a·s Polcar, and Michal Vor¶a·cek December 21, 2005 Preface \Practice makes perfect." The aim of this tutorial text is to facilitate deeper understanding of principles and ap- plications of text searching algorithms. It provides many exercises from di®erent areas of this topic. The front part of this text is devoted to review of basic notions, principles and algorithms from the theory of ¯nite automata. The reason for it is the reality that the next chapters devoted to di®erent aspects of the area of text searching algorithms are based on the use of ¯nite automata. Chapter 1 contains collection of de¯nitions of notions used in this tutorial. Chapter 2 is devoted to the basic algorithms from the area of ¯nite automata and has been written by Jan Anto·s. Chapter 3 contains basic algorithms and operations with ¯nite automata and it has been partly written by Jan Anto·s. Chapters 4 and 5 show construction of string matching automata for exact and approximate string matching and has been partly written by Tom¶a·s Polcar. Chapter 6, devoted to the simulation of string matching automata has been written by Jan Holub. Chapter 7 and 8 concerning construction of ¯nite automata accepting parts of strings has been partly written by Tom¶a·s Polcar. Chapter 9 describing computation of border arrays has been written by Michal Vor¶a·cek.
    [Show full text]
  • Text Searching: Theory and Practice 1 Introduction
    Text Searching: Theory and Practice Ricardo A. Baeza-Yates and Gonzalo Navarro Depto. de Ciencias de la Computaci´on, Universidad de Chile, Casilla 2777, Santiago, Chile. E-mail: rbaeza,gnavarro @dcc.uchile.cl. { } Abstract We present the state of the art of the main component of text retrieval systems: the search engine. We outline the main lines of research and issues involved. We survey the relevant techniques in use today for text searching and explore the gap between theoretical and practical algorithms. The main observation is that simpler ideas are better in practice. In theory, The best theory is inspired by practice. there is no difference between theory and practice. The best practice is inspired by theory. In practice, there is. Donald E. Knuth Jan L.A. van de Snepscheut 1 Introduction Full text retrieval systems have become a popular way of providing support for text databases. The full text model permits locating the occurrences of any word, sentence, or simply substring in any document of the collection. Its main advantages, as compared to alternative text retrieval models, are simplicity and efficiency. From the end-user point of view, full text searching of on- line documents is appealing because valid query patterns are just any word or sentence of the documents. In the past, alternatives to the full-text model, such as semantic indexing, had received some attention. Nowadays, those alternatives are confined to very specific domains while most text retrieval systems used in practice rely, in one way or another, on the full text model. The main component of a full text retrieval system is the text search engine.
    [Show full text]
  • The Finite Automata Approaches in Stringology
    Kybernetika Jan Holub The finite automata approaches in stringology Kybernetika, Vol. 48 (2012), No. 3, 386--401 Persistent URL: http://dml.cz/dmlcz/142945 Terms of use: © Institute of Information Theory and Automation AS CR, 2012 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz KYBERNETIKA—VOLUME 4 8 (2012), NUMBER 3, PAGES 386–401 THE FINITE AUTOMATA APPROACHES IN STRINGOLOGY Jan Holub We present an overview of four approaches of the finite automata use in stringology: deter- ministic finite automaton, deterministic simulation of nondeterministic finite automaton, finite automaton as a model of computation, and compositions of finite automata solutions. We also show how the finite automata can process strings build over more complex alphabet than just single symbols (degenerate symbols, strings, variables). Keywords: exact pattern matching, approximate pattern matching, finite automata, dy- namic programming, bitwise parallelism, suffix automaton, border array, de- generate symbol Classification: 93E12, 62A10 1. INTRODUCTION Many tasks in Computer Science can be solved using finite automata. The finite au- tomata theory is well developed formal system that is used in various areas. The original formal study of finite state systems (neural nets) is from 1943 by McCulloch and Pitts [38]. In 1956 Kleene [32] modeled the neural nets of McCulloch and Pitts by finite automata.
    [Show full text]
  • Suffix Arrays
    Suffix Arrays Outline for Today ● Review from Last Time ● Quick review of suffix trees. ● Suffix Arrays ● A space-efficient data structure for substring searching. ● LCP Arrays ● A surprisingly helpful auxiliary structure. ● Constructing Suffix Trees ● Converting from suffix arrays to suffix trees. ● Constructing Suffix Arrays ● An extremely clever algorithm for building suffix arrays. Review from Last Time Suffix Trees ● A suffix tree for $ e s a string T is an n o e Patricia trie of T$ n 8 $ n o s s $ where each leaf is e n s n e s labeled with the 7 e s n s 6 e index where the $ e $ n $ n e corresponding s $ 4 s 5 e suffix starts in T$. e $ 3 $ 1 2 0 nonsense$ 012345678 Suffix Trees ● If |T| = m, the $ e s suffix tree has n o e exactly m + 1 n 8 $ n o s s $ leaf nodes. e n s n e s 7 e s n ● For any T ≠ ε, all s 6 e $ e $ n $ internal nodes in n e s $ the suffix tree 4 s 5 e e $ 3 have at least two $ 1 children. 2 ● Number of nodes 0 in a suffix tree is nonsense$ 012345678 Θ( m). Space Usage ● Suffix trees are memory hogs. ● Suppose Σ = {A, C, G, T, $}. ● Each internal node needs 15 machine words: for each character, we need three words for the start/end index of the label and for a child pointer. ● This is still O(m), but it's a huge hidden constant. Suffix Arrays Suffix Arrays ● A suffix array for 8 $ a string T is an 7 e$ array of the 4 ense$ suffixes of T$, 0 nonsense$ stored in sorted 5 nse$ order.
    [Show full text]
  • Sliding Suffix Tree
    algorithms Article Sliding Suffix Tree Andrej Brodnik 1,2 and Matevž Jekovec 1,* 1 Faculty of Computer and Information Science, University of Ljubljana, 1000 Ljubljana, Slovenia; [email protected] 2 Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska, 6000 Koper-Capodistria, Slovenia * Correspondence: [email protected]; Tel.: +386-40-566543 Received: 20 June 2018; Accepted: 25 July 2018; Published: 3 August 2018 Abstract: We consider a sliding window W over a stream of characters from some alphabet of constant size. We want to look up a pattern in the current sliding window content and obtain all positions of the matches. We present an indexed version of the sliding window, based on a suffix tree. The data structure of size Q(jWj) has optimal time queries Q(m + occ) and amortized constant time updates, where m is the length of the query string and occ is its number of occurrences. Keywords: suffix tree; online pattern matching; sliding window 1. Introduction and Related Work Text indexing, pattern matching, and big data in general is a well studied field of computer science and engineering. An especially intriguing area is (infinite) streams of data, which are too big to fit onto disk and cannot be indexed in the traditional way, regardless of the data compression or succinct representation used (e.g., [1–3]). In practice, stream processing is usually done so that queries are defined in advance, and data streams are processed by efficient, carefully engineered filters. One way of implementing string matching including regular expressions is by using finite automata [4,5].
    [Show full text]
  • Suffix Arrays 3
    451: Suffix Arrays G. Miller, K. Sutner Carnegie Mellon University 2020/11/24 1 Suffix Arrays 2 Applications Recall: Suffix Trees/Automata 2 Suppose W is a string of length n. We can store the suffixes of W in two structures: Suffix Tree Path compacted, deterministic trie of size O(n). Can be constructed in linear time. Suffix Automaton Minimal PDFA for suff(W ) of size O(n). Can be constructed in linear time. Both structures are essentially finite state machines, so we implicitly also store all the factors. Linear is asymptotically optimal, but there is still room to improve the constants, and perhaps the difficulty of the algorithms. Suffix Arrays 3 Here is a computationally attractive alternative to suffix trees: so-called suffix arrays. To construct a suffix array for W , sort all the suffixes of W , and return a list of the corresponding indices in [n]. The result is the suffix arraySA (W ) of W . Technically, SA :[n] ! [n] but we will sometimes conflate the index SA(i) and the actual suffix w[SA(i):]. The brute-force approach to computing the suffix array is quadratic since the total length of all suffixes is quadratic. Longest Common Prefix 4 As it turns out, it is convenient to augment a suffix array with another array: the longest common prefix (LCP) array. LCP(i) = max jzj j z 2 pref(SA(i)) \ pref(SA(i+1)) In other words, if u and v are the two consecutive suffixes in lexicographic order, we need to compute the position k such that u[1:k] = v[1:k] whereas uk+1 6= vk+1.
    [Show full text]
  • Prefix and Right-Partial Derivative Automata ⋆
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Open Repository of the University of Porto Prefix and Right-Partial Derivative Automata ? Eva Maia, Nelma Moreira, Rog´erioReis CMUP & DCC, Faculdade de Ci^enciasda Universidade do Porto Rua do Campo Alegre, 4169-007 Porto, Portugal e-mail:femaia,nam,[email protected] Abstract. Recently, Yamamoto presented a new method for the conver- sion from regular expressions (REs) to non-deterministic finite automata (NFA) based on the Thompson "-NFA (AT). The AT automaton has two quotients discussed: the suffix automaton Asuf and the prefix automaton, Apre. Eliminating "-transitions in AT, the Glushkov automaton (Apos) is obtained. Thus, it is easy to see that Asuf and the partial derivative automaton (Apd) are the same. In this paper, we characterise the Apre automaton as a solution of a system of left RE equations and express it as a quotient of Apos by a specific left-invariant equivalence relation. − We define and characterise the right-partial derivative automaton (A pd). Finally, we study the average size of all these constructions both exper- imentally and from an analytic combinatorics point of view. 1 Introduction Conversion methods from regular expressions to equivalent nondeterministic fi- nite automata have been widely studied. Resulting NFAs can have "-transitions or not. The standard conversion with "-transitions is the Thompson automaton (AT) [15] and the standard conversion without "-transitions is the Glushkov (or position) automaton (Apos) [9]. Other conversions such as partial derivative au- tomaton (Apd) [1, 13] or follow automaton (Af ) [10] were proved to be quotients of the Apos, by specific right-invariant equivalence relations [6, 10].
    [Show full text]
  • 40 Years of Suffix Trees
    review articles DOI:10.1145/2810036 He first looked for the most frequent Tracing the first four decades in the life symbol and changed it into the most frequent letter of English, then simi- of suffix trees, their many incarnations, larly inferred the most frequent word, and their applications. then punctuation marks, and so on. Both before and after 1843, the BY ALBERTO APOSTOLICO, MAXIME CROCHEMORE, natural impulse when faced with MARTIN FARACH-COLTON, ZVI GALIL, AND S. MUTHUKRISHNAN some mysterious message has been to count frequencies of individual to- kens or subassemblies in search of a clue. Perhaps one of the most intense and fascinating subjects for this kind 40 Years of scrutiny have been biosequences. As soon as some such sequences be- came available, statistical analysts tried to link characters or blocks of of Suffix Trees characters to relevant biological func- tions. With the early examples of whole genomes emerging in the mid- 1990s, it seemed natural to count the occurrences of all blocks of size 1, 2, and so on, up to any desired length, looking for statistical characteriza- tions of coding regions, promoter re- gions, among others. This article is not about cryptogra- WHEN WILLIAM LEGRAND finally decrypted the string, phy. It is about a data structure and it did not seem to make much more sense than it its variants, and the many surprising and useful features it carries. Among did before. these is the fact that, to set up a sta- tistical table of occurrences for all substrings (also called factors), of any 53‡‡‡305))6*,48264‡.)4z);806”,48†8P60))85;1‡ length, of a text string of n characters, (;:‡*8†83(88)5*†,46(;88*96*?;8)* ‡ (;485);5*†2:* ‡ it only takes time and space linear in (;4956*2(5*Ñ4)8P8*;4069285);)6‡8)4‡‡;1(‡9;48081;8: the length of the text string.
    [Show full text]