IRJMST Vol 6 Issue 10 [Year 2015] ISSN 2250 – 1959 (0nline) 2348 – 9367 (Print) International Research Journal of Management Science & Technology http://www.irjmst.com Page 216 Critical Analysis on the Classical Topological Properties of the Star Graph, Hypercube and Perfect Difference Network for Distributed Systems Manish Bhardwaj * E-mail: manish@csirhrdg.res.in CSIR Human Resource Development Group, CSIR Complex, New Delhi-110012, India Prof. Rakesh Kumar Katare E-mail:katare1962@gmail.com Department of Computer Science, Awadhesh Pratap Singh University, Rewa- 486003, M .P. Abstract Design of interconnection networks for processor organization requires a comprehensive knowledge of implicit topological properties inherited by these networks. The architecture of novel interconnection networks is defined by its polylogrithmic time using polynomial number of processors. An undirected graph with processors as nodes and links between the processors as an edge represents a graphical model for interconnection networks with the assumption of the relation that processors are connected with a polynomial in polylogrithmic time. Star Graph, hypercube and a recently developed perfect difference network are popular interconnection networks of interest to researchers with principal to have efficient use of processors in minimal time with minimal connectivity. We restrict our attention to study and compare these interconnection networks topological properties. To measure the combinatorial properties of the compared interconnection networks, we had defined mapping function to map data according to the top ology of the interconnection network. I NTRODUCTION In order to increase the performance of computer systems, computer engineers try to harness the power latent in parallel architecture by designing the optimized interconnection networks and defining the new efficient algorithms for them. [1] The basic representation of processor organization is developed with a relation of nodes in the form of mapping function according to the topological properties of the interconnection network. A graph depicting the processing elements (PEs) and a line connecting a pair of node acting as communication medium between them represents an edge. Inter-processor communication system consists of intelligent exchanges of information between themselves via an interconnection formation. In parallel systems the topology of interconnection systems determines the performance of network. The feasibility cost of the network depends on the graphical position of PE’s and the possible links between them. Graph theory as a tool plays an important role in analyzing the distributed system. Performance of communication arrangement is done by mapping the quality parameters of information transfer to the classical combinatorial attributes importantly degree, diameter and connectivity of a graph depicting an interconnection network. The change of era from Von Neumann architecture and sequential algorithms to PRAM (parallel random access machine) fully exploits the processing power of PEs through efficient interconnection models like star graph, hypercube and perfect difference network. Binary tree represents the basic structure to achieve the parallelism. It incorporates the famous recursive divide and conquer algorithm to define the time complexity Θ (log p) for ہp/2 ۂprocessors. The product of time complexity and number of p processors give the cost as Θ (p log p). Hence, parallel algorithms are defined with polylogarithmic time complexity. [1]. This paper is intended to define different data structures according to their properties and introduce important interconnection models viz. Star graph, Hypercube (n-cube) and Perfect Difference Network (PDN). The authors have attempted to compare these three popular interconnection networks in order to critically study their inter-processor organization. In our study we assume that the data/ information traffic volume among all the three networks discussed is similar. Keywords—Star Graph, Hypercube, Perfect Difference Network, Interconnection Networks, Topology. PRELIMENARIES In order to make it easy and comprehensible, authors feel it appropriate to give a background of interconnection network topological properties classification. Interconnection networks (ICN) classification is based on the type of connectivity between the two communicating PEs. ICN are broadly classified as direct and indirect interconnection networks. While in direct networks, a group of PE communicates with another PE via point-to-point links and the PE is responsible for computing and routing of the message packets. They are represented using a graph model with nodes as vertices and links as edges. On the other hand, indirect network PEs is connected via switches and the switches denote the nodes and edges as links when modeled using a graph theory. Efficient parallel algorithms implemented on real hardware are characterized by best processor topologies which are based on the metrics as given in table 3 in the later part of this paper. For clarity, the terms used frequently in this paper are described hereunder: ____________ *Author for correspondence