Algebraic Approach to Transformations on Hypercube System

Abstract

In this study some algebraic transformations on hypercube system are proposed. The presented transformations of forbidden subcubes provide defining set of all maximal and nonfaulty subcubes, and subset of minimal and nonfaulty subcubes included certain given vertices. It provides formal and simple findings of all the current sources and targets of information, and also the pathways between them. The correctness of all these transformations is proved by the aid of three theorems. In the proposed transformations there were three special operations of cube algebra applied, as subtraction, product and intersection on coordinates.Hypercube is a distributed parallel system consisting of 2 n identicalprocessors, each provided with its own sizable memory and interconnected with n neighbors. It has a homogeneous symmetric structure and has necessarily rich connectivity. It also has useful topology in which many other topologies, such as meshes, rings, trees and etc. can be embedded [13,2,3,6, 11, 15,1]. Hypercube multiprocessor systems usually have a large number of processors (nodes), so the probabilitythat some processor fails can be high. In order to prevent and determine the faulty nodes and links in the data communication there are many different kind of methods to reach the shortest paths between the source and the target nodes. For solving this problem it is applied traditional mathematical theory or heuristics, which can not completely take into account the peculiarities of the hypercube structure. These methods depend on the number of faulty components of hypercube and can not achieve necessary reliabilitywhen the number of faulty elements exceeds of cube dimension [2,3, 11, 15 ]. The last years for this purpose it