Wheeler languages

Alanko, Jarno; D’Agostino, Giovanna; Policriti, Alberto; Prezza, Nicola

doi:10.1016/j.ic.2021.104820

Cited by 18 publications

(27 citation statements)

References 15 publications

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“…The (classic) Myhill-Nerode Theorem, among many other things, establishes a bijection between equivalence classes of ≡ L and the states of the minimum DFA recognizing L. This minimum automaton is also unique up to isomorphism and a similar result, fully proved in [2], holds for Wheeler languages as well. In order to state such an analogous of Myhill-Nerode Theorem for Wheeler languages, the equivalence ≡ L is replaced by the equivalence ≡ c L defined below.…”

Section: Preliminariesmentioning

confidence: 72%

“…-The first one consists in identifying some local properties of ≤ D , used to define a general notion of a co-lex (possibly partial) order over the states of an NFA (see [2]). Turning back to DFA's, one can easily prove that ≤ D is the maximum co-lex (partial) order over D. In general, co-lex orders over NFA's can still be used for indexing, with index-construction complexity parametric on the width (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, co-lex orders are not as well behaving on NFA's as they are on DFA's: over an NFA we cannot guarantee the existence of a maximum co-lex order and also finding a maximal one turns out to be an NP-complete problem [6]. To overcome such difficulties, in [2] a new class of automata was introduced: the reduced NFA's. On reduced NFA's distinguished states have different incoming languages.…”

Section: Introductionmentioning

confidence: 99%

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Ordering Regular Languages and Automata: Complexity

D’Agostino¹,

Martincigh²,

Policriti³

2022

Preprint

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Given an order of the underlying alphabet we can lift it to the states of a finite deterministic automaton: to compare states we use the order of the strings reaching them. When the order on strings is the co-lexicographic one and this order turns out to be total, the DFA is called Wheeler. This recently introduced class of automata-the Wheeler automata-constitute an important data-structure for languages, since it allows the design and implementation of a very efficient tool-set of storage mechanisms for the transition function, supporting a large variety of substring queries. In this context it is natural to consider the class of regular languages accepted by Wheeler automata, i.e. the Wheeler languages. An inspiring result in this area is the following: it has been shown that, as opposed to the general case, the classic determinization by powerset construction is polynomial on Wheeler automata. As a consequence, most classical problems, when considered on this class of automata, turn out to be "easy"-that is, solvable in polynomial time. In this paper we consider computational problems related to Wheelerness, but starting from non-deterministic automata. We also consider the case of reduced non-deterministic ones-a class of NFA where recognizing Wheelerness is still polynomial, as for DFA's. Our collection of results shows that moving towards non-determinism is, in most cases, a dangerous path leading quickly to intractability. Moreover, we start a study of "state complexity" related to Wheeler DFA and languages, proving that the classic construction for the intersection of languages turns out to be computationally simpler on Wheeler DFA than in the general case. We also provide a construction for the minimum Wheeler DFA recognizing a given Wheeler language.

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Section: Preliminariesmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ordering Regular Languages and Automata: Complexity

D’Agostino¹,

Martincigh²,

Policriti³

2022

Preprint

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“…..f A (a i [d]), then any potential vertex u has an implicit label βS = βB i f A (a i [1])f A (a i [2]). ..f A (a i [d]) [1,3] where β must be 3, and the edge (u, v) already exists.…”

Section: Discussionmentioning

confidence: 99%

“…Many graphs allow for extending Burrows-Wheeler Transformation (BWT) based techniques for efficient pattern matching. Sufficient conditions for doing this are captured by the definition of Wheeler graphs, introduced in [16], and further studied in [3,4,12,15,20]. De Bruijn graphs are themselves Wheeler graphs, hence on a de Bruijn graph exact pattern matching is solvable in linear time.…”

Section: Exact Matching Approximate Matching Solvable In Linear Timementioning

confidence: 99%

The Complexity of Approximate Pattern Matching on De Bruijn Graphs

Gibney¹,

Thankachan²,

Aluru³

2022

Preprint

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Aligning a sequence to a walk in a labeled graph is a problem of fundamental importance to Computational Biology. For finding a walk in an arbitrary graph with |E| edges that exactly matches a pattern of length m, a lower bound based on the Strong Exponential Time Hypothesis (SETH) implies an algorithm significantly faster than O(|E|m) time is unlikely [Equi et al., ICALP 2019]. However, for many special graphs, such as de Bruijn graphs, the problem can be solved in linear time [Bowe et al., WABI 2012]. For approximate matching, the picture is more complex. When edits (substitutions, insertions, and deletions) are only allowed to the pattern, or when the graph is acyclic, the problem is again solvable in O(|E|m) time. When edits are allowed to arbitrary cyclic graphs, the problem becomes NP-complete, even on binary alphabets [Jain et al., RECOMB 2019]. These results hold even when edits are restricted to only substitutions. Despite the popularity of de Bruijn graphs in Computational Biology, the complexity of approximate pattern matching on de Bruijn graphs remained open. We investigate this problem and show that the properties that make de Bruijn graphs amenable to efficient exact pattern matching do not extend to approximate matching, even when restricted to the substitutions only case with alphabet size four. Specifically, we prove that determining the existence of a matching walk in a de Bruijn graph is NP-complete when substitutions are allowed to the graph. In addition, we demonstrate that an algorithm significantly faster than O(|E|m) is unlikely for de Bruijn graphs in the case where only substitutions are allowed to the pattern. This stands in contrast to pattern-to-text matching where exact matching is solvable in linear time, like on de Bruijn graphs, but approximate matching under substitutions is solvable in subquadratic Õ(n √ m) time, where n is the text's length [Abrahamson, SIAM J. Computing 1987].

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