Hawaii International Conference on System Sciences

Due to recent developments of parallel and distributed computing, the design and analysis of various interconnection networks has been a main topic of research for the past few years and is still stimulated by the new technologies of communication networks such as optic fibers. There are many advantages in using Cayley (di)graphs as models for interconnection networks. This work first surveys some classes of Cayley graphs which are well studied as models of interconnection networks. Results and problems related to routings in networks are then presented, with emphasis on loads of nodes and links in routings.

Cayley graphs and interconnection networks

A network implements orthogonal directional traffic classes including: interior traffic, interior to exterior traffic (source traffic), exterior to interior traffic (destination traffic), and transitory traffic. Classified traffic transits networks which are an 'ordered' (i.e. numbered) set of Network Entities ('NE' or elements) commonly referred to and including links, switches, and stations. Each NE is 'ordered' based on a network 'center' which is functionally determined by an NE's connectedness (i.e. the quality and quantity of connections), and by its centeredness (i.e. how close it is to the center of the network). An assigned numeric address ('host number') designated during ordering, specifies the 'relative' location of each element, and provides information both with respect to the nodes' 'centeredness' and 'connectedness' (i.e., expressed as 'relative' to the 'center' of an Ordered Network). To effect data flow (traffic), topologically static switching and mapping are used. Each multi-domain network, subdivided into subnets or 'control areas', uses a distributed map instead of a forwarding table to determine forwarding links. Only locally relevant mapping information is maintained for data forwarding. Network objects and support servers provide inter-networking communication between network entities both local and remote. Network objects, which are distributed at each node, include a SWITCH object and a LINK object. Network objects (SWITCH and LINK) use the same control mechanism regardless of an object's function, position, or particular data structure. Support servers include: and inter-Domain Map Server (DMS); a Domain Name Server (DNS); an Interior Map Server (IMS); and an Interior Name Server (INS). The support servers provide a communication support function for network operation. Ordered networking is implemented according to a methodology that initially determines link sets in a domain. From the link sets, a map establishing the Ordered Network topology is generated by the IMS. A path switch matrix for each node is then generated from the map, and is distributed among the nodes in the domain. The path switch matrix is generated as a function of the four traffic classes. The path matrix located in each node takes the source address, the destination address and the traffic class and uses them to determine which link to forward the traffic on. In addition, the path switch matrix handles temporary links without protocols, as provided by the orthogonality of classes.

System and method for network flow optimization using traffic classes

Arrangement graphs: a class of generalized star graphs

Cayley graphs as classifiers for data mining: The influence of asymmetries

Undertakes a comparative study of two important interconnection network topologies: the star graph and the hypercube, from the graph theory point of view. Topological properties are derived for the star graph and are compared with the corresponding properties of the hypercube. Among other results, the authors determine necessary and sufficient conditions for shortest path routing and characterize maximum-sized families of parallel paths between any two nodes of the star graph. These parallel paths are proven of minimum length within a small additive constant. They also define greedy and asymptotically balanced spanning trees to support broadcasting and personalized communication on the star graph. These results confirm the already claimed topological superiority of the star graph over the hypercube. >

A comparative study of topological properties of hypercubes and star graphs

We obtain the fault diameter of k-ary n-cube interconnection networks (also known as n-dimensional k-torus networks). We start by constructing a complete set of node-disjoint paths (i.e., as many paths as the degree) between any two nodes of a k-ary n-cube. Each of the obtained paths is of length zero, two, or four plus the minimum length except for one path in a special case (when the Hamming distance between the two nodes is one) where the increase over the minimum length may attain eight. These results improve those obtained by B. Bose et al. (1995) where the length of some of the paths has a variable increase (which can be arbitrarily large) over the minimum length. These results are then used to derive the fault diameter of the k-ary n-cube, which is shown to be /spl Delta/+1 where /spl Delta/ is the fault free diameter.

Fault diameter of k-ary n-cube networks

We study the cross product as a method for generating and analyzing interconnection network topologies for multiprocessor systems. Consider two interconnection graphs G/sub 1/ and G/sub 2/ each with some established properties such as symmetry, low degree and diameter, scalability, simple optimal routing, recursive structure (partitionability), fault tolerance, existence of node-disjoint paths, low cost embedding, and efficient broadcasting. We investigate and evaluate the corresponding properties for the cross product of G/sub 1/ and G/sub 2/ based on the properties of G/sub 1/ and those of G/sub 2/. We also give a mathematical characterization of product families of graphs which are closed under the cross product operation. This investigation is useful in two ways. On one hand, it gives a new tool for further studying some of the known interconnection topologies, such as the hypercube and the mesh, which can be defined using the cross product operation. On the other hand, it can be used in defining and evaluating new interconnection graphs using the cross product operation on known topologies.

The cross product of interconnection networks

We conduct a general study of the topological properties of optical transpose interconnection systems (OTIS). We first obtain their basic topological metrics of size, degree, shortest distance and diameter, and then we obtain results related to the recursive structure and efficient embedding of meshes, cubes, spanning trees and cycles. We also present minimal one-to-one routing and optimal broadcasting algorithms, and we show how to construct node-disjoint paths between any two nodes of an OTIS network. Recent studies have addressed only particular members of the general class of OTIS networks. In this paper, we present unified tools for obtaining the topological properties of an arbitrary OTIS network based on the properties of the corresponding factor network.

Khaled Day

Papers

A comparative study of topological properties of hypercubes and star graphs

Arrangement graphs: a class of generalized star graphs

Fault diameter of k-ary n-cube networks

The cross product of interconnection networks

Topological properties of OTIS-networks