Symmetric properties and two variants of shuffle-cubes

TL;DR

The diameter of the shuffle-cube is approximately a quarter of the diameter of the hypercube of the same dimension, making it a competitive candidate network topology for massive parallel and distributed systems.

Abstract

Li et al. in [Inf. Process. Lett. 77 (2001) 35--41] proposed the shuffle cube $SQ_{n}$ as an attractive interconnection network topology for massive parallel and distributed systems. By far, symmetric properties of the shuffle cube remains unknown. In this paper, we show that $SQ_{n}$ is not vertex-transitive for all $n>2$, which is not an appealing property in interconnection networks. To overcome this limitation, two novel vertex-transitive variants of the shuffle-cube, namely simplified shuffle-cube $SSQ_{n}$ and balanced shuffle cube $BSQ_{n}$ are introduced. Then, routing algorithms of $SSQ_{n}$ and $BSQ_{n}$ for all $n>2$ are given respectively. Furthermore, we show that both $SSQ_{n}$ and $BSQ_{n}$ possess Hamiltonian cycle embedding for all $n>2$. Finally, as a by-product, we mend a flaw in the Property 3 in [IEEE Trans. Comput. 46 (1997) 484--490].

Figures (4)

• Figure 1: $SQ_6$.
• Figure 2: $SSQ_6$.
• Figure 3: $BSQ_6$.
• Figure 4: Construction of Hamiltonian cycle of $BSQ_n$.

Theorems & Definitions (21)

• Definition 1
• Lemma 1
• Lemma 2
• Theorem 3
• Corollary 4
• Definition 2
• Theorem 5
• Definition 3
• Definition 4
• Theorem 6