10,604 research outputs found

    A Unifying Theory for Graph Transformation

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    The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO

    Computational Analyses of Metagenomic Data

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    Metagenomics studies the collective microbial genomes extracted from a particular environment without requiring the culturing or isolation of individual genomes, addressing questions revolving around the composition, functionality, and dynamics of microbial communities. The intrinsic complexity of metagenomic data and the diversity of applications call for efficient and accurate computational methods in data handling. In this thesis, I present three primary projects that collectively focus on the computational analysis of metagenomic data, each addressing a distinct topic. In the first project, I designed and implemented an algorithm named Mapbin for reference-free genomic binning of metagenomic assemblies. Binning aims to group a mixture of genomic fragments based on their genome origin. Mapbin enhances binning results by building a multilayer network that combines the initial binning, assembly graph, and read-pairing information from paired-end sequencing data. The network is further partitioned by the community-detection algorithm, Infomap, to yield a new binning result. Mapbin was tested on multiple simulated and real datasets. The results indicated an overall improvement in the common binning quality metrics. The second and third projects are both derived from ImMiGeNe, a collaborative and multidisciplinary study investigating the interplay between gut microbiota, host genetics, and immunity in stem-cell transplantation (SCT) patients. In the second project, I conducted microbiome analyses for the metagenomic data. The workflow included the removal of contaminant reads and multiple taxonomic and functional profiling. The results revealed that the SCT recipients' samples yielded significantly fewer reads with heavy contamination of the host DNA, and their microbiomes displayed evident signs of dysbiosis. Finally, I discussed several inherent challenges posed by extremely low levels of target DNA and high levels of contamination in the recipient samples, which cannot be rectified solely through bioinformatics approaches. The primary goal of the third project is to design a set of primers that can be used to cover bacterial flagellin genes present in the human gut microbiota. Considering the notable diversity of flagellins, I incorporated a method to select representative bacterial flagellin gene sequences, a heuristic approach based on established primer design methods to generate a degenerate primer set, and a selection method to filter genes unlikely to occur in the human gut microbiome. As a result, I successfully curated a reduced yet representative set of primers that would be practical for experimental implementation

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Almost covering all the layers of hypercube with multiplicities

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    Given a hypercube Qn:={0,1}n\mathcal{Q}^{n} := \{0,1\}^{n} in Rn\mathbb{R}^{n} and k{0,,n}k \in \{0, \dots, n\}, the kk-th layer Qkn\mathcal{Q}^{n}_{k} of Qn\mathcal{Q}^{n} denotes the set of all points in Qn\mathcal{Q}^{n} whose coordinates contain exactly kk many ones. For a fixed tNt \in \mathbb{N} and k{0,,n}k \in \{0, \dots, n\}, let PR[x1,,xn]P \in \mathbb{R}\left[x_{1}, \dots, x_{n}\right] be a polynomial that has zeroes of multiplicity at least tt at all points of QnQkn\mathcal{Q}^{n} \setminus \mathcal{Q}^{n}_{k}, and PP has zeros of multiplicity exactly t1t-1 at all points of Qkn\mathcal{Q}^{n}_{k}. In this short note, we show that deg(P)max{k,nk}+2t2.deg(P) \geq \max\left\{ k, n-k\right\}+2t-2.Matching the above lower bound we give an explicit construction of a family of hyperplanes H1,,HmH_{1}, \dots, H_{m} in Rn\mathbb{R}^{n}, where m=max{k,nk}+2t2m = \max\left\{ k, n-k\right\}+2t-2, such that every point of Qkn\mathcal{Q}^{n}_{k} will be covered exactly t1t-1 times, and every other point of Qn\mathcal{Q}^{n} will be covered at least tt times. Note that putting k=0k = 0 and t=1t=1, we recover the much celebrated covering result of Alon and F\"uredi (European Journal of Combinatorics, 1993). Using the above family of hyperplanes we disprove a conjecture of Venkitesh (The Electronic Journal of Combinatorics, 2022) on exactly covering symmetric subsets of hypercube Qn\mathcal{Q}^{n} with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest. We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.Comment: 16 pages, substantial changes from previous version, title and abstract changed to better reflect the content of the pape

    A tamed family of triangle-free graphs with unbounded chromatic number

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    We construct a hereditary class of triangle-free graphs with unbounded chromatic number, in which every non-trivial graph either contains a pair of non-adjacent twins or has an edgeless vertex cutset of size at most two. This answers in the negative a question of Chudnovsky, Penev, Scott, and Trotignon. The class is the hereditary closure of a family of (triangle-free) twincut graphs G1,G2,G_1, G_2, \ldots such that GkG_k has chromatic number kk. We also show that every twincut graph is edge-critical

    Twin-Width V: Linear Minors, Modular Counting, and Matrix Multiplication

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    Reconfiguration of Digraph Homomorphisms

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    For a fixed graph H, the H-Recoloring problem asks whether, given two homomorphisms from a graph G to H, one homomorphism can be transformed into the other by changing the image of a single vertex in each step and maintaining a homomorphism to H throughout. The most general algorithmic result for H-Recoloring so far has been proposed by Wrochna in 2014, who introduced a topological approach to obtain a polynomial-time algorithm for any undirected loopless square-free graph H. We show that the topological approach can be used to recover essentially all previous algorithmic results for H-Recoloring and that it is applicable also in the more general setting of digraph homomorphisms. In particular, we show that H-Recoloring admits a polynomial-time algorithm i) if H is a loopless digraph that does not contain a 4-cycle of algebraic girth 0 and ii) if H is a reflexive digraph that contains no triangle of algebraic girth 1 and no 4-cycle of algebraic girth 0

    Fast Macroscopic Forcing Method

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    The macroscopic forcing method (MFM) of Mani and Park and similar methods for obtaining turbulence closure operators, such as the Green's function-based approach of Hamba, recover reduced solution operators from repeated direct numerical simulations (DNS). MFM has been used to quantify RANS-like operators for homogeneous isotropic turbulence and turbulent channel flows. Standard algorithms for MFM force each coarse-scale degree of freedom (i.e., degree of freedom in the RANS space) and conduct a corresponding fine-scale simulation (i.e., DNS), which is expensive. We combine this method with an approach recently proposed by Sch\"afer and Owhadi (2023) to recover elliptic integral operators from a polylogarithmic number of matrix-vector products. The resulting Fast MFM introduced in this work applies sparse reconstruction to expose local features in the closure operator and reconstructs this coarse-grained differential operator in only a few matrix-vector products and correspondingly, a few MFM simulations. For flows with significant nonlocality, the algorithm first "peels" long-range effects with dense matrix-vector products to expose a local operator. We demonstrate the algorithm's performance for scalar transport in a laminar channel flow and momentum transport in a turbulent one. For these, we recover eddy diffusivity operators at 1% of the cost of computing the exact operator via a brute-force approach for the laminar channel flow problem and 13% for the turbulent one. We observe that we can reconstruct these operators with an increase in accuracy by about a factor of 100 over randomized low-rank methods. We glean that for problems in which the RANS space is reducible to one dimension, eddy diffusivity and eddy viscosity operators can be reconstructed with reasonable accuracy using only a few simulations, regardless of simulation resolution or degrees of freedom.Comment: 16 pages, 10 figures. S. H. Bryngelson and F. Sch\"afer contributed equally to this wor

    Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods

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    We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix AA, defined as tr(f(A))\operatorname{tr}(f(A)) where f(x)=xlogxf(x)=-x\log x. After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.Comment: 32 pages, 10 figure
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