Recent Advances in Text-to-Pattern Distance Algorithms

We study the classic Text-to-Pattern Hamming Distances problem: given a pattern P of length m and a text T of length n, both over a polynomial-size alphabet, compute the Hamming distance between P and T [i . . i + m − 1] for every shift i, under the standard Word-RAM model with Θ(log n)-bit words. • We provide an O(n √ m) time Las Vegas randomized algorithm for this problem, beating the decades-old O(n √ m log m) running time [Abrahamson, SICOMP 1987]. We also obtain a deterministic algorithm, with a slightly higher O(n √ m(log m log log m) 1/4 ) running time. Our randomized algorithm extends to the k-bounded setting, with running time O n + nk √ m , removing all the extra logarithmic factors from earlier algorithms [Gawrychowski and Uznański, ICALP 2018; Chan, Golan, Kociumaka, Kopelowitz and Porat, STOC 2020].

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“…It has long been asked (see e.g. [Uzn20b]) whether one can reduce 3SUM to Text-to-Pattern Hamming Distances.…”

Section: Theorem 13 (Deterministic Exact Algorithm) the Text-to-patte...mentioning

confidence: 99%

“…5 Other variations of (1 + ε)-approximation text-to-pattern matching problem have also been studied in the literature, such as replacing Hamming distance by other ℓ p norms [LP11, GU18, SU19, Uzn20a], or restricting to algorithms that do not use FFT [CGK + 20, Uzn20a]. See also the survey by Uznański [Uzn20b]. k-mismatch.…”

Section: Introductionmentioning

confidence: 99%

Approximating text-to-pattern Hamming distances

Chan

Golan

Kociumaka

et al. 2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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“…As most real-world textual data (e.g., involving genetic data or text written by humans) contains errors, considering string-processing algorithms in an approximate setting is natural and has been heavily investigated. See, e.g., the recent papers [16,31,30,47], and the references therein, as well as classical results such as [1,40,34]. Closer to the topic of this paper, the problem of approximate pattern matching was also considered in the context of regular expression matching -see [6,40] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Following [22], it would be interesting to try to optimise the algorithms for all classes from the table (except Reg, where the upper and conditional lower bounds match). In the case of Reg, it would be interesting to consider the problem for regular patterns with a constant number of variables; already in the case of two variables (also known as approximate string matching under Hamming distance) the known complexity upper and lower bounds do not match anymore [30,47]. Another direction is to consider the two problems for other distance functions (e.g., edit distance) instead of the Hamming distance.…”

Section: Introductionmentioning

confidence: 99%

Matching Patterns with Variables under Hamming Distance

Gawrychowski¹,

Manea²,

Siemer³

2021

Preprint

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A pattern α is a string of variables and terminal letters. We say that α matches a word w, consisting only of terminal letters, if w can be obtained by replacing the variables of α by terminal words. The matching problem, i.e., deciding whether a given pattern matches a given word, was heavily investigated: it is NP-complete in general, but can be solved efficiently for classes of patterns with restricted structure. In this paper, we approach this problem in a generalized setting, by considering approximate pattern matching under Hamming distance. More precisely, we are interested in what is the minimum Hamming distance between w and any word u obtained by replacing the variables of α by terminal words. Firstly, we address the class of regular patterns (in which no variable occurs twice) and propose efficient algorithms for this problem, as well as matching conditional lower bounds. We show that the problem can still be solved efficiently if we allow repeated variables, but restrict the way the different variables can be interleaved according to a locality parameter. However, as soon as we allow a variable to occur more than once and its occurrences can be interleaved arbitrarily with those of other variables, even if none of them occurs more than once, the problem becomes intractable. ACM Subject ClassificationTheory of computation → Design and analysis of algorithms, Theory of computation → Formal languages and automata

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“…Considering string problems under string distances is natural, as most real-world string-based data is prone to errors due to, e.g., mutations in genetic data or typos in handcrafted texts. Even though many concepts are already from the 50's and 60's (Hamming distance [85] and edit distance [121,120]), it is a very active area of research with a long history of results as it can be seen in, e.g., the recent papers [42,81,80,161], and the references therein, as well as classical results…”

Section: Word Equationsmentioning

confidence: 99%

Matching Patterns with Variables in Approximate Settings

Siemer

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First and foremost, I hereby express my sincere gratitude and thank my supervisor, Florin Manea, for the excellent guidance and support throughout my studies. Florin is very knowledgeable in many areas of theoretical computer science, encouraging and helpful with new research ideas, and a highly cheerful person throughout my entire studies and everyday work. I am very grateful that he is swift in answering requests and always finds time to discuss research.Secondly, I would like to thank my second supervisor, Carsten Damm. I have been working as a Tutor for Carsten since my Bachelor studies and enjoyed taking his advanced courses in theoretical computer science. Carsten also convinced me to pursue doctoral studies with the back then "new" professor at our institute, Florin Manea. Looking back, I am very grateful for all of these things.Further, I express my gratitude towards Jens Grabowski, who advised me very well with all the formal requirements as a member of the thesis advisory committee.I also extend my many thanks to the entire working group. A special thank you goes to Maria Kosche, Tore Koß, and Paul Sarnighausen-Cahn as my future academic siblings, co-authors, and thesis proofreaders. I am thankful for the amicable and relaxed work environment I experienced with the previously mentioned group members and our master students Tina Ringleb, Timo Specht, and Maximilian Winkler, who work on the same floor. Speaking of people working on the same floor, I want to express my deepest gratitude to Henrik Brosenne, whom I closely collaborated with in teaching, as well as Heike Jachinke and Patricia Nitzke, who were always helping us regarding formal problems, such as traveling, room bookings, and much more, making it exceptionally convenient to deal with.

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Recent Advances in Text-to-Pattern Distance Algorithms

Cited by 4 publications

References 28 publications

Approximating text-to-pattern Hamming distances

Approximating text-to-pattern Hamming distances

Matching Patterns with Variables under Hamming Distance

Matching Patterns with Variables in Approximate Settings

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