Optimal Path Length for Octagon-Cell Interconnection Networks: Fundamental Theorems with Proofs

Interconnection Networks are widely used in many applications such as parallel and massively parallel systems, industrial applications, VLSI circuits, electronic devices and telephone switches etc. The realization of data communication in parallel system is possible all the way through interconnection networks. The whole performance of the system depends on the type of interconnection networks used. Interconnection networks with embed ability by some other interconnection structure, high resilience, simple routing, constant node degree, and scalability are highly desirable. We had introduced Octagon-Cell Networks (OCN), in the past research, which is a new interconnection network. In this research, we introduce and prove the fundamental theorems for finding the optimal path length between any pair of source and destination nodes in Octagon-Cell Network of any depth. The basic objective of these theorems (excluding Horizontal and Vertical Moves) to calculate the optimal path length, is to check whether any intermediate node exists along the optimal path, for which its line number or serial number matches with the line number or the serial number of destination node respectively. We also introduce the theorems to find the total number of horizontal lines, which is the maximum line number of a node, as well the total number of vertical lines, which is the maximum serial number of a node in Octagon-Cell Network of any depth. The theorems are proved mathematically and are tested successfully on all possible cases.