Pancyclicity of OTIS (swapped) networks based on properties of the factor graph

M. Malekimajd, M.R. Hoseiny-Farahabady, A. Movaghar, H. Sarbazi-azad
2011 Information Processing Letters  
The plausibility of embedding cycles of different lengths in the graphs of a network (known as the pancyclicity property) has important applications in interconnection networks, parallel processing systems, and the implementation of a number of either computational or graph problems such as those used for finding storage schemes of logical data structures, layout of circuits in VLSI, etc. In this paper, we present the sufficient condition of the pancyclicity property of OTIS networks. The OTIS
more » ... etwork (also referred to as two-level swapped network) is composed of n clones of an n-node original network constituting its clusters. It has received much attention due to its many favorable properties such as high degree of scalability, regularity, modularity, package-ability and high degree of algorithmic efficiency. Many properties of OTIS networks have been studied in the literature. In this work, we show that the OTIS networks have the pancyclicity property when the factor graph is Hamiltonian. More precisely, using a constructive method, we prove that if the factor graph G of an OTIS network contains cycles of length {3, 4, 5, l}, then all cycles of length {3, . . . ,l 2 }, can be embedded in the OTIS-G network. This result resolves the open question posed and tracked in Day and AlAyyoub (2002) [2], Hoseiny Farahabady and Sarbazi Azad (2007) [4] and Shafiei et al. (2011) [14]. processors of an N 2 processors within an OTIS system are partitioned into N groups of N processors [11] . It has been shown in [6] that when the number of processors in a group equals the number of groups, both the bandwidth and the power consumption in a group shaped network are optimized while both the system area and the volume of system are minimized. The OTIS-hypercube and OTISmesh are two of the most widely studied instances of the OTIS architecture [2, 12, 14] . A number of algorithms have been developed for OTIS networks, such as routing, selection, and data rearrangement and sorting [13, 16] , matrix multiplication [15] , and broadcasting [2]. Many of topological properties of these systems such as node degree, diameter, β-cut, and bisection width are addressed in previous studies [2] . Furthermore in [4] it was proved that if G is a Hamiltonian-connected graph, so is the OTIS-G. In [12] the performance merits of the OTIS-hypercube and the effect of different structural and workload parameters 0020-0190/$ -see front matter
doi:10.1016/j.ipl.2011.07.020 fatcat:yxl4em2zyjasfb4ppaiphx2wma