A new quadratic algorithm to convert a regular expression into an automaton

Ponty, Jean-Luc; Ziadi, Djelloul; Champarnaud, Jean-Marc

doi:10.1007/3-540-63174-7_9

Cited by 25 publications

(36 citation statements)

References 6 publications

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“…It was also observed earlier, e.g., [9,25,14], that First and Last-sets (and nullability) can be defined in a syntaxdirected way over the parse tree of e. For instance, if lab(n) = and Lchild (n), Rchild (n) are non-nullable then First(n) = First(Lchild (n)) and Last(n) = Last(Rchild (n)). We define now the Boolean properties SupFirst and SupLast for every node n, where n denotes parent(n): SupFirst(n) iff lab(n ) = , n = Rchild (n ), and…”

Section: Structure Of Regular Expressionsmentioning

confidence: 64%

“…Note that First(n) and Last(n) are non-empty for every node n of e. For instance, for the expression e0 in Figure 1 First(n2) = {p1, p2} and Last(n2) = {p5}. Given two nodes u, v of e, let LCA(u, v) denote the lowest common ancestor of u and v in e. The next lemma was stated before, e.g., in [9,25], but not in terms of LCA.…”

Section: Structure Of Regular Expressionsmentioning

confidence: 90%

“…It says that testing if two positions follow each other in e (this means the Glushkov automaton has a transition between the positions) can be realized in constant time. This is achieved by preprocessing e's parse tree for LCA [1] and by using LCA queries to realize a structural relationship of follow positions known from [9,25]. Hence, we do not build the Glushkov automaton.…”

Section: Introductionmentioning

confidence: 99%

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Deterministic regular expressions in linear time

Groz

Maneth

Staworko

2012

Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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Deterministic regular expressions are widely used in XML processing. For instance, all regular expressions in DTDs and XML Schemas are required to be deterministic. In this paper we show that determinism of a regular expression e can be tested in linear time. The best known algorithms, based on the Glushkov automaton, require O(σ|e|) time, where σ is the number of distinct symbols in e. We further show that matching a word w against an expression e can be achieved in combined linear time O(|e| + |w|), for a wide range of deterministic regular expressions: (i) star-free (for multiple input words), (ii) bounded-occurrence, i.e., expressions in which each symbol appears a bounded number of times, and (iii) bounded plus-depth, i.e., expressions in which the nesting depth of alternating plus (union) and concatenation symbols is bounded. Our algorithms use a new structural decomposition of the parse tree of e. For matching arbitrary deterministic regular expressions we present an O(|e| + |w| log log |e|) time algorithm.

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Section: Structure Of Regular Expressionsmentioning

confidence: 64%

Section: Structure Of Regular Expressionsmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

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Deterministic regular expressions in linear time

Groz

Maneth

Staworko

2012

Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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“…The first one, of order O(m + n 2 ), where m = |α|, was proposed by Brüggemann-Klein in 1993 [BK93] and it is primarily based on the prior transformation of α into star-normal form. Other quadratic, however sophisticated, algorithms have been introduced in 1996 and 1997, respectively in [PZC97] and [CP97]. Our goal in the next section, is to present an alternative recursive definition of Follow(α), that only involves disjoint unions of sets, allowing for simple implementations of that construction in time O(n 2 ).…”

Section: The Glushkov Automatonmentioning

confidence: 99%

The Average Transition Complexity of Glushkov and Partial Derivative Automata

Broda

Machiavelo

Moreira

et al. 2011

Developments in Language Theory

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Abstract. In this paper, the relation between the Glushkov automaton (Apos) and the partial derivative automaton (A pd ) of a given regular expression, in terms of transition complexity, is studied. The average transition complexity of Apos was proved by Nicaud to be linear in the size of the corresponding expression. This result was obtained using an upper bound of the number of transitions of Apos. Here we present a new quadratic construction of Apos that leads to a more elegant and straightforward implementation, and that allows the exact counting of the number of transitions. Based on that, a better estimation of the average size is presented. Asymptotically, and as the alphabet size grows, the number of transitions per state is on average 2. Broda et al. computed an upper bound for the ratio of the number of states of A pd to the number of states of Apos, which is about 1 2 for large alphabet sizes. Here we show how to obtain an upper bound for the number of transitions in A pd , which we then use to get an average case approximation. Some experimental results are presented that illustrate the quality of our estimate.

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“…It requires the expression to be first rewritten in starnormal form, which can be done non-trivially in O(m). Several other quadratic algorithms have been given: that of [9] which is based on an optimization of the follow recursion, and that of [23], based on the ZPC structure, which consists of two mutually linked copies of the syntactic tree of the expression.…”

Section: Introductionmentioning

confidence: 99%

A Unified Construction of the Glushkov, Follow, and Antimirov Automata

Allauzen

Mohri

2006

Lecture Notes in Computer Science

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Abstract. Many techniques have been introduced in the last few decades to create -free automata representing regular expressions: Glushkov automata, the so-called follow automata, and Antimirov automata. This paper presents a simple and unified view of all these -free automata both in the case of unweighted and weighted regular expressions. It describes simple and general algorithms with running time complexities at least as good as that of the best previously known techniques, and provides concise proofs. The construction methods are all based on two standard automata algorithms: epsilon-removal and minimization. This contrasts with the multitude of complicated and special-purpose techniques and proofs put forward by others to construct these automata. Our analysis provides a better understanding of -free automata representing regular expressions: they are all the results of the application of some combinations of epsilon-removal and minimization to the classical Thompson automata. This makes it straightforward to generalize these algorithms to the weighted case, which also results in much simpler algorithms than existing ones. For weighted regular expressions over a closed semiring, we extend the notion of follow automata to the weighted case. We also present the first algorithm to compute the Antimirov automata in the weighted case. IntroductionThe construction of finite automata representing regular expressions has been widely studied due to its multiple applications to pattern-matching and many Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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A new quadratic algorithm to convert a regular expression into an automaton

Cited by 25 publications

References 6 publications

Deterministic regular expressions in linear time

Deterministic regular expressions in linear time

The Average Transition Complexity of Glushkov and Partial Derivative Automata

A Unified Construction of the Glushkov, Follow, and Antimirov Automata

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