7,923 research outputs found
Graph Embeddings and Simplicial Maps
An undirected graph is viewed as a simplicial complex. The notion of a graph embedding of a guest graph in a host graph is generalized to the realm of simplicial maps. Dilation is redefined in this more general setting. Lower bounds on dilation for various guest and host graphs are considered. Of particular interest are graphs that have been proposed as communication networks for parallel architectures. Bhatt et al. provide a lower bound on dilation for embedding a planar guest graph in a butterfly host graph. Here, this lower bound is extended in two directions. First, a lower bound that applies to arbitrary guest graphs is derived, using tools from algebraic topology. Second, this lower bound is shown to apply to arbitrary host graphs through a new graph-theoretic measure, called bidecomposability. Bounds on the bidecomposability of the butterfly graph and of the k-dimensional torus are determined. As corollaries to the main lower bound theorem, lower bounds are derived for embedding arbitrary planar graphs, genus g graphs, and k-dimensional meshes in a butterfly host graph
Embedding Schemes for Interconnection Networks.
Graph embeddings play an important role in interconnection network and VLSI design. Designing efficient embedding strategies for simulating one network by another and determining the number of layers required to build a VLSI chip are just two of the many areas in which graph embeddings are used. In the area of network simulation we develop efficient, small dilation embeddings of a butterfly network into a different size and/or type of butterfly network. The genus of a graph gives an indication of how many layers are required to build a circuit. We have determined the exact genus for the permutation network called the star graph, and have given a lower bound for the genus of the permutation network called the pancake graph. The star graph has been proposed as an alternative to the binary hypercube and, therefore, we compare the genus of the star graph with that of the binary hypercube. Another type of embedding that is helpful in determining the number of layers is a book embedding. We develop upper and lower bounds on the pagenumber of a book embedding of the k-ary hypercube along with an upper bound on the cumulative pagewidth
On the Effect of Quantum Interaction Distance on Quantum Addition Circuits
We investigate the theoretical limits of the effect of the quantum
interaction distance on the speed of exact quantum addition circuits. For this
study, we exploit graph embedding for quantum circuit analysis. We study a
logical mapping of qubits and gates of any -depth quantum adder
circuit for two -qubit registers onto a practical architecture, which limits
interaction distance to the nearest neighbors only and supports only one- and
two-qubit logical gates. Unfortunately, on the chosen -dimensional practical
architecture, we prove that the depth lower bound of any exact quantum addition
circuits is no longer , but . This
result, the first application of graph embedding to quantum circuits and
devices, provides a new tool for compiler development, emphasizes the impact of
quantum computer architecture on performance, and acts as a cautionary note
when evaluating the time performance of quantum algorithms.Comment: accepted for ACM Journal on Emerging Technologies in Computing
System
Embedding multidimensional grids into optimal hypercubes
Let and be graphs, with , and a one to one map of their vertices. Let , where is the distance
between vertices and of . Now let = , over all such maps .
The parameter is a generalization of the classic and well studied
"bandwidth" of , defined as , where is the path on
points and . Let
be the -dimensional grid graph with integer values through in
the 'th coordinate. In this paper, we study in the case when and is the hypercube
of dimension , the hypercube of
smallest dimension having at least as many points as . Our main result is
that
provided for each . For such , the bound
improves on the previous best upper bound . Our methods include
an application of Knuth's result on two-way rounding and of the existence of
spanning regular cyclic caterpillars in the hypercube.Comment: 47 pages, 8 figure
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