Two-dimensional prefix string matching and covering on square matrices

International audienceTwo linear time algorithms are presented. One for determining, for every position in a given square matrix, the longest prefix of a given pattern (also a square matrix) that occurs at that position and one for computing all square covers of a given two-dimensional square matrix

Algorithms for Approximate K-Covering of Strings

Computing approximate patterns in strings or sequences has important applications in DNA sequence analysis, data compression, musical text analysis, and so on. In this paper, we introduce approximate k-covers and study them under various commonly used distance measures. We propose the following problem: "Given a string x of length n, a set U of m strings of length k, and a distance measure, compute the minimum number t such that U is a set of approximate k-covers for x with distance t". To solve this problem, we present three algorithms with time complexity O(km(n - k)), O(mn2) and O(mn2) under Hamming, Levenshtein and edit distance, respectively. A World Wide Web server interface has been established at http://www.uncg.edu/mat/kcover/ for automated use of the programs

Quasi-Periodicity Under Mismatch Errors

Tracing regularities plays a key role in data analysis for various areas of science, including coding and automata theory, formal language theory, combinatorics, molecular biology and many others. Part of the scientific process is understanding and explaining these regularities. A common notion to describe regularity in a string T is a cover or quasi-period, which is a string C for which every letter of T lies within some occurrence of C. In many applications finding exact repetitions is not sufficient, due to the presence of errors. In this paper we initiate the study of quasi-periodicity persistence under mismatch errors, and our goal is to characterize situations where a given quasi-periodic string remains quasi-periodic even after substitution errors have been introduced to the string. Our study results in proving necessary conditions as well as a theorem stating sufficient conditions for quasi-periodicity persistence. As an application, we are able to close the gap in understanding the complexity of Approximate Cover Problem (ACP) relaxations studied by [Amir 2017a, Amir 2017b] and solve an open question

Approximate Cover of Strings

Regularities in strings arise in various areas of science, including coding and automata theory, formal language theory, combinatorics, molecular biology and many others. A common notion to describe regularity in a string T is a cover, which is a string C for which every letter of T lies within some occurrence of C. The alignment of the cover repetitions in the given text is called a tiling. In many applications finding exact repetitions is not sufficient, due to the presence of errors. In this paper, we use a new approach for handling errors in coverable phenomena and define the approximate cover problem (ACP), in which we are given a text that is a sequence of some cover repetitions with possible mismatch errors, and we seek a string that covers the text with the minimum number of errors. We first show that the ACP is NP-hard, by studying the cover-size relaxation of the ACP, in which the requested size of the approximate cover is also given with the input string. We show this relaxation is already NP-hard. We also study another two relaxations of the ACP, which we call the partial-tiling relaxation of the ACP and the full-tiling relaxation of the ACP, in which a tiling of the requested cover is also given with the input string. A given full tiling retains all the occurrences of the cover before the errors, while in a partial tiling there can be additional occurrences of the cover that are not marked by the tiling. We show that the partial-tiling relaxation has a polynomial time complexity and give experimental evidence that the full-tiling also has polynomial time complexity. The study of these relaxations, besides shedding another light on the complexity of the ACP, also involves a deep understanding of the properties of covers, yielding some key lemmas and observations that may be helpful for a future study of regularities in the presence of errors

Internal Quasiperiod Queries

Internal pattern matching requires one to answer queries about factors of a given string. Many results are known on answering internal period queries, asking for the periods of a given factor. In this paper we investigate (for the first time) internal queries asking for covers (also known as quasiperiods) of a given factor. We propose a data structure that answers such queries in $O(\log n \log \log n)$ time for the shortest cover and in $O(\log n (\log \log n)^2)$ time for a representation of all the covers, after $O(n \log n)$ time and space preprocessing.Comment: To appear in the SPIRE 2020 proceeding