Optimal Embeddings of Paths with Various Lengths in Twisted Cubes

• Authors: Jianxi Fan;Xiaohua Jia;Xiaola Lin
• Venue: IEEE Transactions on Parallel and Distributed Systems
• Year: 2007

Twisted cubes are variants of hypercubes. In this paper, we study the optimal embeddings of paths of all possible lengths between two arbitrary distinct nodes in twisted cubes. We use TQ_{n} to denote the n{\hbox{-}}{\rm dimensional} twisted cube and use {\rm dist}(TQ_{n},u,v) to denote the distance between two nodes u and v in TQ_{n}, where n \geq 1 is an odd integer. The original contributions of this paper are as follows: 1) We prove that a path of length l can be embedded between u and v with dilation 1 for any two distinct nodes u and v and any integer l with {\rm dist}(TQ_{n},u,v) + 2 \le l\le 2^{n} - 1 (n \geq 3) and 2) we find that there exist two nodes u and v such that no path of length {\rm dist}(TQ_{n},u,v) + 1 can be embedded between u and v with dilation 1 (n \geq 3). The special cases for the nonexistence and existence of embeddings of paths between nodes u and v and with length {\rm dist}(TQ_{n},u,v) + 1 are also discussed. The embeddings discussed in this paper are optimal in the sense that they have dilation 1.