Automata Theory on Sliding Windows

Ganardi, Moses; Hucke, Danny; König, Daniel; Lohrey, Markus; Mamouras, Konstantinos

doi:10.4230/lipics.stacs.2018.31

Cited by 6 publications

(17 citation statements)

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“…For Boolean properties several papers [17][18][19] have considered streaming algorithms for testing membership in regular and context-free languages. Another variant of the problem was considered in [20][21][22], where the regular property is verified on the last N letters of the stream, instead of the entire prefix up to the current position.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Data Structures for Timed Automata Acceptance

et al. 2022

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We study a variant of the classical membership problem in automata theory, which consists of deciding whether a given input word is accepted by a given automaton. We do so through the lenses of parameterized dynamic data structures: we assume that the automaton is fixed and its size is the parameter, while the input word is revealed as in a stream, one symbol at a time following the natural order on positions. The goal is to design a dynamic data structure that can be efficiently updated upon revealing the next symbol, while maintaining the answer to the query on whether the word consisting of symbols revealed so far is accepted by the automaton. We provide complexity bounds for this dynamic acceptance problem for timed automata that process symbols interleaved with time spans. The main contribution is a dynamic data structure that maintains acceptance of a fixed one-clock timed automaton $${\mathcal {A}}$$ A with amortized update time $$2^{{\mathcal {O}}(|{\mathcal {A}}|)}$$ 2 O ( | A | ) per input symbol.

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Section: Introductionmentioning

confidence: 99%

Dynamic Data Structures for Timed Automata Acceptance

et al. 2022

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“…It started with a seminal paper of Porat and Porat in FOCS 2009 [52], who showed streaming algorithms for exact pattern matching and for the k-mismatches problem. The result of Porat and Porat was followed by a series of works on streaming pattern matching [7,13,34,14,57,33,30,16,32,53], search of repetitions in streams [20,18,19,28,48,47,29], and recognising formal languages in streams [45,24,25,22,23,27,4,21,26].…”

Section: Introductionmentioning

confidence: 99%

Small space and streaming pattern matching with k edits

Kociumaka,

Porat,

Starikovskaya

2021

Preprint

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In this work, we revisit the fundamental and well-studied problem of approximate pattern matching under edit distance. Given an integer k, a pattern P of length m, and a text T of length n ≥ m, the task is to find substrings of T that are within edit distance k from P . Our main result is a streaming algorithm that solves the problem in Õ(k 5 ) space and Õ(k 8 ) amortised time per character of the text, providing answers correct with high probability. (Hereafter, Õ(•) hides a poly(log n) factor.) This answers a decade-old question: since the discovery of a poly(k log n)-space streaming algorithm for pattern matching under Hamming distance by Porat and Porat [FOCS 2009], the existence of an analogous result for edit distance remained open. Up to this work, no poly(k log n)-space algorithm was known even in the simpler semi-streaming model, where T comes as a stream but P is available for readonly access. In this model, we give a deterministic algorithm that achieves slightly better complexity.Our central technical contribution is a new space-efficient deterministic encoding of two strings, called the greedy encoding, which encodes a set of all alignments of cost ≤ k with a certain property (we call such alignments greedy). On strings of length at most n, the encoding occupies Õ(k 2 ) space. We use the encoding to compress substrings of the text that are close to the pattern. In order to do so, we compute the encoding for substrings of the text and of the pattern, which requires read-only access to the latter.In order to develop the fully streaming algorithm, we further introduce a new edit distance sketch parametrised by integers n ≥ k. For any string of length at most n, the sketch is of size Õ(k 2 ) and it can be computed with an Õ(k 2 )-space streaming algorithm. Given the sketches of two strings, in Õ(k 3 ) time we can compute their edit distance or certify that it is larger than k. This result improves upon Õ(k 8 )-size sketches of Belazzougui and Zhu [FOCS 2016] and very recent Õ(k 3 )-size sketches of Jin, Nelson, and Wu [STACS 2021].

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“…Space complexity of sliding window algorithms. The space complexity of sliding window algorithms for formal languages in the one-way (fixed-size and variablesize) model has been studied in [15,16,17,19,20] and in [18,Section 9] for the two-way variable-size model. For regular languages, the main result of [17] is a space trichotomy for the one-way case: the space complexity of a regular language is either constant, logarithmic or linear.…”

Section: Introductionmentioning

confidence: 99%

“…For the two-way variable-size model a space trichotomy has been shown in [18]. Table 1 summarizes some of the main results of [16,17,18] in more detail (ignore the word size bound for the moment). The results on the two-way fixed-size model in Table 1 are shown in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Sliding Window Algorithms for Regular Languages

Ganardi

Hucke

Lohrey

2018

Language and Automata Theory and Applications

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We study the problem of recognizing regular languages in a variant of the streaming model of computation, called the sliding window model. In this model, we are given a size of the sliding window n and a stream of symbols. At each time instant, we must decide whether the suffix of length n of the current stream ("the active window") belongs to a given regular language.Recent works [14,15] showed that the space complexity of an optimal deterministic sliding window algorithm for this problem is either constant, logarithmic or linear in the window size n and provided natural language theoretic characterizations of the space complexity classes. Subsequently, [16] extended this result to randomized algorithms to show that any such algorithm admits either constant, double logarithmic, logarithmic or linear space complexity.In this work, we make an important step forward and combine the sliding window model with the property testing setting, which results in ultra-efficient algorithms for all regular languages. Informally, a sliding window property tester must accept the active window if it belongs to the language and reject it if it is far from the language. We consider deterministic and randomized sliding window property testers with onesided and two-sided errors. In particular, we show that for any regular language, there is a deterministic sliding window property tester that uses logarithmic space and a randomized sliding window property tester with two-sided error that uses constant space.

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Automata Theory on Sliding Windows

Cited by 6 publications

References 0 publications

Dynamic Data Structures for Timed Automata Acceptance

Dynamic Data Structures for Timed Automata Acceptance

Small space and streaming pattern matching with k edits

Sliding Window Algorithms for Regular Languages

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