String Periods in the Order-Preserving Model

The order-preserving model (op-model, in short) was introduced quite recently but has already attracted significant attention because of its applications in data analysis. We introduce several types of periods in this setting (op-periods). Then we give algorithms to compute these periods in time O(n), O(n log log n), O(n log^2 log n/log log log n), O(n log n) depending on the type of periodicity. In the most general variant the number of different periods can be as big as Omega(n^2), and a compact representation is needed. Our algorithms require novel combinatorial insight into the properties of such periods

String periods in the order-preserving model

In the order-preserving model, two strings match if they share the same relative order between the characters at the corresponding positions. This model is quite recent, but it has already attracted significant attention because of its applications in data analysis. We introduce several types of periods in this setting (op-periods). Then we give algorithms to compute these periods in time O(n), O(nlog⁡log⁡n), O(nlog2⁡log⁡n/log⁡log⁡log⁡n), O(nlog⁡n) depending on the type of periodicity. In the most general variant, the number of different op-periods can be as big as Ω(n2), and a compact representation is needed. Our algorithms require novel combinatorial insight into the properties of op-periods. In particular, we characterize the Fine–Wilf property for coprime op-periods. © 2019 Elsevier Inc.Supported by ISF grants no. 824/17 and 1278/16 and by an ERC grant MPM under the EU's Horizon 2020 Research and Innovation Programme (grant no. 683064).Supported by the Ministry of Science and Higher Education of the Russian Federation, project 1.3253.2017.A part of this work was done during the workshop StringMasters in Warsaw 2017 that was sponsored by the Warsaw Center of Mathematics and Computer Science. The authors thank the participants of the workshop, especially Hideo Bannai and Shunsuke Inenaga, for helpful discussions

A Parameterized Study of Maximum Generalized Pattern Matching Problems

The generalized function matching (GFM) problem has been intensively studied starting with [Ehrenfeucht and Rozenberg, 1979]. Given a pattern p and a text t, the goal is to find a mapping from the letters of p to non-empty substrings of t, such that applying the mapping to p results in t. Very recently, the problem has been investigated within the framework of parameterized complexity [Fernau, Schmid, and Villanger, 2013]. In this paper we study the parameterized complexity of the optimization variant of GFM (called Max-GFM), which has been introduced in [Amir and Nor, 2007]. Here, one is allowed to replace some of the pattern letters with some special symbols "?", termed wildcards or don't cares, which can be mapped to an arbitrary substring of the text. The goal is to minimize the number of wildcards used. We give a complete classification of the parameterized complexity of Max-GFM and its variants under a wide range of parameterizations, such as, the number of occurrences of a letter in the text, the size of the text alphabet, the number of occurrences of a letter in the pattern, the size of the pattern alphabet, the maximum length of a string matched to any pattern letter, the number of wildcards and the maximum size of a string that a wildcard can be mapped to.Comment: to appear in Proc. IPEC'1

Weak factor automata : the failure of failure factor oracles?

In indexing of, and pattern matching on, DNA and text sequences, it is often important to represent all factors of a sequence. One e cient, compact representation is the factor oracle (FO). At the same time, any classical deterministic nite automaton (DFA) can be transformed to a so-called failure one (FDFA), which may use failure transitions to replace multiple symbol transitions, potentially yielding a more compact representation. We combine the two ideas and directly construct a failure factor oracle (FFO) from a given sequence, in contrast to ex post facto transformation to an FDFA. The algorithm is suitable for both short and long sequences. We empirically compared the resulting FFOs and FOs on number of transitions for many DNA sequences of lengths 4 - 512, showing gains of up to 10% in total number of transitions, with failure transitions also taking up less space than symbol transitions. The resulting FFOs can be used for indexing, as well as in a variant of the FO-using backward oracle matching algorithm. We discuss and classify this pattern matching algorithm in terms of the keyword pattern matching taxonomies of Watson, Cleophas and Zwaan. We also empirically compared the use of FOs and FFOs in such backward reading pattern matching algorithms, using both DNA and natural language (English) data sets. The results indicate that the decrease in pattern matching performance of an algorithm using an FFO instead of an FO may outweigh the gain in representation space by using an FFO instead of an FO.http://www.journals.co.za/ej/ejour_comp.htmlam201