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Issue No.08 - August (2008 vol.57)
pp: 1046-1056
Carmen Martínez , University of Cantabria, Santander
Ramón Beivide , University of Cantabria, Santander
Esteban Stafford , University of Cantabria, Santander
Miquel Moretó , UPC UPC , Barcelona Barcelona
Ernst M. Gabidulin , Moscow Institute of Physics and Technology, Moscow
ABSTRACT
In this paper we consider a broad family of toroidal networks, denoted as Gaussian networks, which include many previously proposed and used topologies. We will define such networks by means of the Gaussian Integers, the subset of the Complex numbers with integer real and imaginary parts. Nodes in Gaussian networks are labeled by Gaussian integers, which confer these topologies an algebraic structure based on quotient rings of the Gaussian integers. In this sense, Gaussian integers reveal themselves as the appropriate tool for analyzing and exploiting any type of toroidal network. Using this algebraic approach, we can characterize the main distance-related properties of Gaussian networks, providing closed expressions for their diameter and average distance. In addition, we solve some important applications, like unicast and broadcast packet routing or the perfect placement of resources over these networks.
INDEX TERMS
Packet-switching networks, Network topology, Graph theory, Graphs and networks, Packet routing
CITATION
Carmen Martínez, Ramón Beivide, Esteban Stafford, Miquel Moretó, Ernst M. Gabidulin, "Modeling Toroidal Networks with the Gaussian Integers", IEEE Transactions on Computers, vol.57, no. 8, pp. 1046-1056, August 2008, doi:10.1109/TC.2008.57
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