3,208 research outputs found

    Efficient tilings of de Bruijn and Kautz graphs

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    Kautz and de Bruijn graphs have a high degree of connectivity which makes them ideal candidates for massively parallel computer network topologies. In order to realize a practical computer architecture based on these graphs, it is useful to have a means of constructing a large-scale system from smaller, simpler modules. In this paper we consider the mathematical problem of uniformly tiling a de Bruijn or Kautz graph. This can be viewed as a generalization of the graph bisection problem. We focus on the problem of graph tilings by a set of identical subgraphs. Tiles should contain a maximal number of internal edges so as to minimize the number of edges connecting distinct tiles. We find necessary and sufficient conditions for the construction of tilings. We derive a simple lower bound on the number of edges which must leave each tile, and construct a class of tilings whose number of edges leaving each tile agrees asymptotically in form with the lower bound to within a constant factor. These tilings make possible the construction of large-scale computing systems based on de Bruijn and Kautz graph topologies.Comment: 29 pages, 11 figure

    Cospectral digraphs from locally line digraphs

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    A digraph \G=(V,E) is a line digraph when every pair of vertices u,v∈Vu,v\in V have either equal or disjoint in-neighborhoods. When this condition only applies for vertices in a given subset (with at least two elements), we say that \G is a locally line digraph. In this paper we give a new method to obtain a digraph \G' cospectral with a given locally line digraph \G with diameter DD, where the diameter D′D' of \G' is in the interval [D−1,D+1][D-1,D+1]. In particular, when the method is applied to De Bruijn or Kautz digraphs, we obtain cospectral digraphs with the same algebraic properties that characterize the formers

    On hardware for generating routes in Kautz digraphs

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    In this paper we present a hardware implementation of an algorithm for generating node disjoint routes in a Kautz network. Kautz networks are based on a family of digraphs described by W.H. Kautz[Kautz 68]. A Kautz network with in-degree and out-degree d has N = dk + dk¿1 nodes (for any cardinals d, k>0). The diameter is at most k, the degree is fixed and independent of the network size. Moreover, it is fault-tolerant, the connectivity is d and the mapping of standard computation graphs such as a linear array, a ring and a tree on a Kautz network is straightforward.\ud The network has a simple routing mechanism, even when nodes or links are faulty. Imase et al. [Imase 86] showed the existence of d node disjoint paths between any pair of vertices. In Smit et al. [Smit 91] an algorithm is described that generates d node disjoint routes between two arbitrary nodes in the network. In this paper we present a simple and fast hardware implementation of this algorithm. It can be realized with standard components (Field Programmable Gate Arrays)

    The degree-diameter problem for sparse graph classes

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    The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree Δ\Delta and diameter kk. For fixed kk, the answer is Θ(Δk)\Theta(\Delta^k). We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is Θ(Δk−1)\Theta(\Delta^{k-1}), and for graphs of bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases for fixed kk. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given treewidth, graphs embeddable on a given surface, and apex-minor-free graphs

    Scaling metagenome sequence assembly with probabilistic de Bruijn graphs

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    Deep sequencing has enabled the investigation of a wide range of environmental microbial ecosystems, but the high memory requirements for {\em de novo} assembly of short-read shotgun sequencing data from these complex populations are an increasingly large practical barrier. Here we introduce a memory-efficient graph representation with which we can analyze the k-mer connectivity of metagenomic samples. The graph representation is based on a probabilistic data structure, a Bloom filter, that allows us to efficiently store assembly graphs in as little as 4 bits per k-mer, albeit inexactly. We show that this data structure accurately represents DNA assembly graphs in low memory. We apply this data structure to the problem of partitioning assembly graphs into components as a prelude to assembly, and show that this reduces the overall memory requirements for {\em de novo} assembly of metagenomes. On one soil metagenome assembly, this approach achieves a nearly 40-fold decrease in the maximum memory requirements for assembly. This probabilistic graph representation is a significant theoretical advance in storing assembly graphs and also yields immediate leverage on metagenomic assembly
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