48,901 research outputs found
Efficient tilings of de Bruijn and Kautz graphs
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
Quasi-isometric classification of non-geometric 3-manifold groups
We describe the quasi-isometric classification of fundamental groups of
irreducible non-geometric 3-manifolds which do not have "too many" arithmetic
hyperbolic geometric components, thus completing the quasi-isometric
classification of 3--manifold groups in all but a few exceptional cases.Comment: Minor revision (added footnote in the Introduction
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