Longest common substring made fully dynamic

Given two strings S and T, each of length at most n, the longest common substring (LCS) problem is to find a longest substring common to S and T. This is a classical problem in computer science with an O(n)-time solution. In the fully dynamic setting, edit operations are allowed in either of the two strings, and the problem is to find an LCS after each edit. We present the first solution to this problem requiring sublinear time in n per edit operation. In particular, we show how to find an LCS after each edit operation in Õ(n2/3) time, after Õ(n)-time and space preprocessing. 1 This line of research has been recently initiated in a somewhat restricted dynamic variant by Amir et al. [SPIRE 2017]. More specifically, they presented an Õ(n)-sized data structure that returns an LCS of the two strings after a single edit operation (that is reverted afterwards) in Õ(1) time. At CPM 2018, three papers (Abedin et al., Funakoshi et al., and Urabe et al.) studied analogously restricted dynamic variants of problems on strings. We show that the techniques we develop can be applied to obtain fully dynamic algorithms for all of these variants. The only previously known sublinear-time dynamic algorithms for problems on strings were for maintaining a dynamic collection of strings for comparison queries and for pattern matching, with the most recent advances made by Gawrychowski et al. [SODA 2018] and by Clifford et al. [STACS 2018]. As an intermediate problem we consider computing the solution for a string with a given set of k edits, which leads us, in particular, to answering internal queries on a string. The input to such a query is specified by a substring (or substrings) of a given string. Data structures for answering internal string queries that were proposed by Kociumaka et al. [SODA 2015] and by Gagie et al. [CCCG 2013] are used, along with new ones, based on ingredients such as the suffix tree, heavy-path decomposition, orthogonal range queries, difference covers, and string periodicity

Chaining with Overlaps Revisited

Chaining algorithms aim to form a semi-global alignment of two sequences based on a set of anchoring local alignments as input. Depending on the optimization criteria and the exact definition of a chain, there are several O(n log n) time algorithms to solve this problem optimally, where n is the number of input anchors. In this paper, we focus on a formulation allowing the anchors to overlap in a chain. This formulation was studied by Shibuya and Kurochkin (WABI 2003), but their algorithm comes with no proof of correctness. We revisit and modify their algorithm to consider a strict definition of precedence relation on anchors, adding the required derivation to convince on the correctness of the resulting algorithm that runs in O(n log2 n) time on anchors formed by exact matches. With the more relaxed definition of precedence relation considered by Shibuya and Kurochkin or when anchors are non-nested such as matches of uniform length (k-mers), the algorithm takes O(n log n) time. We also establish a connection between chaining with overlaps and the widely studied longest common subsequence problem. 2012 ACM Subject Classification Theory of computation ! Pattern matching; Theory of computation ! Dynamic programming; Applied computing ! Genomics.Peer reviewe

Methods for the identification of common RNA motifs

Löwes B. Methods for the identification of common RNA motifs. Bielefeld: Universität Bielefeld; 2017.For a long time, non-coding RNAs were given less attention than messenger RNAs, even though their existence was proposed at a similar time in 1971, because the research focus was mostly on protein coding genes. With the discovery of catalytically active RNA molecules and micro RNAs, which are involved in the post-transcriptional regulation of gene expression, non-coding RNAs have gained widespread attention. It was revealed early on that non-coding RNAs are often more conserved in structure than in sequence. Since determining the function of non-coding RNAs includes costly and time consuming laboratory experiments, computational methods can help identifying further homologs of experimentally validated RNA families. But a question remains: can we identify potential RNAs with novel functions solely by using *in silico* methods? In this thesis, we perform an evaluation of 4,667 viral reference genomes in order to identify common RNA motifs shared by multiple taxonomically distant viruses. One potential mechanism that might explain similar motifs in taxonomically distant viruses that infect common hosts by interacting with their cellular components is convergent evolution. Convergent evolution is used to describe the phenomenon that two different species that are originated from two ancestors share related or similar traits. By looking for long stretches of exact RNA structure matches with low sequence conservation, we want to maximize the chance that the common motifs are the result of structural convergence due to similar selection criteria in common host organisms. Viruses are an excellent fit when it comes to the discovery of shared RNA motifs without the involvement of conserved sequence regions because of their high mutation rates. We were able to identify 69 RNA motifs, which could not be assigned to any of the existing RNA families, with a length of at least 50 nucleotides that are shared among at least three taxonomically distant viruses. The secondary structure of an RNA molecule can be represented as a string. Finding maximal repeats in strings can be done using well-known string matching techniques based on suffix trees and arrays. In contrast to normal RNA sequences, secondary structure strings represent base pairing interactions within a single molecule. Thus, not every substring of the secondary structure defines a well-formed RNA structure. Therefore, we describe a new data structure, the viable suffix tree, that takes the constraints on the RNA secondary structure into account and only returns maximal repeats that are well-formed structures. But this data structure is not limited to RNA structures, it can also be used for any other problem domain for which a set of allowed words can be defined, e.g. by using a grammar. However, the overall complexity of constructing the viable suffix tree cannot be lower than the complexity of the word problem for the language of such a grammar. A limitation of exact structure matching is the need for long common stretches of secondary structures that are not allowed to have a mismatch at any position. Therefore, we need to allow small mismatches to find more potential targets, but current state of the art techniques use computationally too expensive methods for sequence and structure comparisons and exhibit high false positive rates around 50%. We present a new approach that uses smaller RNA sequence and structure seed motifs that do not require long stretches of the secondary structure to be identical. The sequence and structure motifs can be hashed into integer values, which can be compared much faster. An evaluation using the three well understood hammerhead ribozyme families showed that our approach is able to detect 70% to 80% of the hammerhead motifs with a similar false positive rate as the other approaches. Whenever the performance of new and existing tools should be compared, there is a need for a benchmark data set with an underlying gold standard. BRaliBase is a widely used benchmark for assessing the accuracy of RNA secondary structure alignment methods. In most case studies based on the BRaliBase benchmark, one can observe a puzzling drop in accuracy in the 40% to 60% sequence identity range, the so-called “BRaliBase dent”. We show that this dent is due to a bias in the composition of the BRaliBase benchmark, namely the inclusion of a disproportionate number of tRNAs, which exhibit a very conserved secondary structure. Furthermore, we show that a simple sampling approach that restricts the presence of the most abundant RNA families can prevent such artifacts during the performance evaluation