We propose an alternative way to represent graphs via OBDDs based on the observation that a partition of the graph nodes allows sharing among the employed OBDDs. In the second part of the paper we present a method to compute at the same time the quotient w.r.t. the maximum bisimulation and the OBDD representation of a given graph. The proposed computation is based on an OBDD-rewriting of the notion of Ackermann encoding of hereditarily finite sets into natural numbers.
We collaborate in a research program aimed at creating a rigorous framework, experimental infrastructure, and computational environment for understanding, experimenting with, manipulating, and modifying a diverse set of fundamental biological processes at multiple scales and spatio-temporal modes. The novelty of our research is based on an approach that (i) requires coevolution of experimental science and theoretical techniques and (ii) exploits a certain universality in biology guided by a parsimonious model of evolutionary mechanisms operating at the genomic level and manifesting at the proteomic, transcriptomic, phylogenic, and other higher levels. Our current program in "systems biology" endeavors to marry largescale biological experiments with the tools to ponder and reason about large, complex, and subtle natural systems. To achieve this ambitious goal, ideas and concepts are combined from many different fields: biological experimentation, applied mathematical modeling, computational reasoning schemes, and large-scale numerical and symbolic simulations. From a biological viewpoint, the basic issues are many: (i) understanding common and shared structural motifs among biological processes; (ii) modeling biological noise due to interactions among a small number of key molecules or loss of synchrony; (iii) explaining the robustness of these systems in spite of such noise; and (iv) cataloging multistatic behavior and adaptation exhibited by many biological processes.
Suffix trees are an important data structure at the core of optimal solutions to many fundamental string problems, such as exact pattern matching, longest common substring, matching statistics, and longest repeated substring. Recent lines of research focused on extending some of these problems to vertex-labeled graphs, although using ad-hoc approaches which in some cases do not generalize to all input graphs. In the absence of a ubiquitous tool like the suffix tree for labeled graphs, we introduce the labeled direct product of two graphs as a general tool for obtaining optimal algorithms: we obtain conceptually simpler algorithms for the quadratic problems of string matching (SMLG) and longest common substring (LCSP) in labeled graphs. Our algorithms are also more efficient, since they run in time linear in the size of the labeled product graph, which may be smaller than quadratic for some inputs, and their run-time is predictable, because the size of the labeled direct product graph can be precomputed efficiently. We also solve LCSP on graphs containing cycles, which was left as an open problem by Shimohira et al. in 2011. To show the power of the labeled product graph, we also apply it to solve the matching statistics (MSP) and the longest repeated string (LRSP) problems in labeled graphs. Moreover, we show that our (worst-case quadratic) algorithms are also optimal, conditioned on the Orthogonal Vectors Hypothesis. Finally, we complete the complexity picture around LRSP by studying it on undirected graphs.
The Lempel-Ziv factorization (LZ77) and the Run-Length encoded Burrows-Wheeler Transform (RLBWT) are two important tools in text compression and indexing, being their sizes $z$ and $r$ closely related to the amount of text self-repetitiveness. In this paper we consider the problem of converting the two representations into each other within a working space proportional to the input and the output. Let $n$ be the text length. We show that $RLBWT$ can be converted to $LZ77$ in $\mathcal{O}(n\log r)$ time and $\mathcal{O}(r)$ words of working space. Conversely, we provide an algorithm to convert $LZ77$ to $RLBWT$ in $\mathcal{O}\big(n(\log r + \log z)\big)$ time and $\mathcal{O}(r+z)$ words of working space. Note that $r$ and $z$ can be \emph{constant} if the text is highly repetitive, and our algorithms can operate with (up to) \emph{exponentially} less space than naive solutions based on full decompression.
In the present work, we tackle the regular language indexing problem by first studying the hierarchy of $p$-sortable languages: regular languages accepted by automata of width $p$. We show that the hierarchy is strict and does not collapse, and provide (exponential in $p$) upper and lower bounds relating the minimum widths of equivalent NFAs and DFAs. Our bounds indicate the importance of being able to index NFAs, as they enable indexing regular languages with much faster and smaller indexes. Our second contribution solves precisely this problem, optimally: we devise a polynomial-time algorithm that indexes any NFA with the optimal value $p$ for its width, without explicitly computing $p$ (NP-hard to find). In particular, this implies that we can index in polynomial time the well-studied case $p=1$ (Wheeler NFAs). More in general, in polynomial time we can build an index breaking the worst-case conditional lower bound of $\Omega(|P| m)$, whenever the input NFA's width is $p \in o(\sqrt{m})$.
Abstract Summary: Chimera is a Bioconductor package that organizes, annotates, analyses and validates fusions reported by different fusion detection tools; current implementation can deal with output from bellerophontes, chimeraScan, deFuse, fusionCatcher, FusionFinder, FusionHunter, FusionMap, mapSplice, Rsubread, tophat-fusion and STAR. The core of Chimera is a fusion data structure that can store fusion events detected with any of the aforementioned tools. Fusions are then easily manipulated with standard R functions or through the set of functionalities specifically developed in Chimera with the aim of supporting the user in managing fusions and discriminating false-positive results. Availability and implementation: Chimera is implemented as a Bioconductor package in R. The package and the vignette can be downloaded at bioconductor.org. Contact: raffaele.calogero@unito.it Supplementary information: Supplementary data are available at Bioinformatics online.
