A Linear Algorithm for Embedding of Cycles in Crossed Cubes with Edge-Pncyclic

A cycle structure is a fundamental network for multiprocessor systems and suitable for developing simple algorithms with low communication cost. Many efficient algorithms were designed with respect to cycles for solving a variety of algebraic problems, graph problems, and some parallel applications, such as those in image and signal processing [2, 16]. To carry out a cycle-structure algorithm on a multiprocessor computer or a distributed system, the processes of the parallel algorithm need to be mapped to the nodes of the interconnection network in the system such that any two adjacent processes in the cycle are mapped to two adjacent nodes of the network. Besides, to execute a parallel program efficiently, the size of the allocated cycle must accord to the problem size. Thus, many researchers focus their studies on how to embed cycles of different sizes into an interconnection network. In distributed systems, each node and each edge may be assigned with distinct resource and distinct bandwidth, respectively. For this purpose, it is meaningful to study the problem of how to embed cycles into a network such that these cycles pass through a special node/edge. The crossed cube proposed by Efe [6] is one of the most notable variations of hypercube, but some properties of the former are superior to those of the latter. For example, the diameter of the crossed cube is almost the half of that of the hypercube. With regard to cycles embedding of crossed cubes, many interesting results have received considerable attention [1, 3-5, 7-15, 18-20]. Finding cycles of arbitrary length passing through any given edge in crossed cubes has attracted special attention from researchers. In this paper, we consider the problem of embedding a cycle of arbitrary numbers of nodes that passes through a given edge on the crossed cube. A concept, cycle pattern, is in use to construct an efficient algorithm for embedding a desired cycle with arbitrary given edge into the crossed cube. Furthermore,