Higher dimensional hexagonal networks

We define the higher dimensional hexagonal graphs as the generalization of a triangular plane tessellation, and consider it as a multiprocessor interconnection network. Nodes in a k-dimensional (k-D) hexagonal network are placed at the vertices of a k-D triangular tessellation, so that each node has up to 2k þ 2 neighbors. In this paper, we propose a simple addressing scheme for the nodes, which leads to a straightforward formula for computing the distance between nodes and a very simple and elegant routing algorithm. The number of shortest paths between any two nodes and their description are also provided in this paper. We then derive closed formulas for the surface area (volume) of these networks, which are defined as the number of nodes located at a given distance (up to a given distance, respectively) from the origin node. The number of nodes and the network diameter under a more symmetrical border conditions are also derived. We show that a k-D hexagonal network of size t has the same degree, the same or lower diameter, and fewer nodes than a ðk þ 1Þ-D mesh of size t. Simple embeddings between two networks are also described. That is, we show how to reduce the dimension of a mesh by removing some nodes, and converting it into a hexagonal network, while preserving the simplicity of basic data communication schemes such as routing and broadcasting. r 2003 Elsevier Inc. All rights reserved.