Embedding long paths in k-ary n-cubes with faulty nodes and links

Stewart, I.A.; Xiang, Y.

Abstract

Let $k \geq 4$ be even and let $n \geq 2$. Consider a faulty k-ary n-cube $Q_n^k$ in which the number of node faults $f_n$ and the number of link faults $f_e$ are such that $f_n + f_e \leq 2n-2$. We prove that given any two healthy nodes s and e of $Q_n^k$, there is a path from s to e of length at least $k^n - 2f_n - 1$ (resp. $k^n - 2f_n - 2$) if the nodes s and e have different (resp. the same) parities (the parity of a node in $Q_n^k$ is the sum modulo 2 of the elements in the n-tuple over {0, 1, ..., k-1} representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.

Citation

Stewart, I., & Xiang, Y. (2008). Embedding long paths in k-ary n-cubes with faulty nodes and links. IEEE Transactions on Parallel and Distributed Systems, 19(8), 1071-1085. https://doi.org/10.1109/tpds.2007.70787

• Journal Article Type: Article
• Publication Date: Aug 1, 2008
• Deposit Date: Jun 29, 2009
• Publicly Available Date: Jul 2, 2009
• Journal: IEEE Transactions on Parallel and Distributed Systems
• Print ISSN: 1045-9219
• Electronic ISSN: 1558-2183
• Publisher: Institute of Electrical and Electronics Engineers
• Peer Reviewed: Peer Reviewed
• Volume: 19
• Issue: 8
• Pages: 1071-1085
• DOI: https://doi.org/10.1109/tpds.2007.70787
• Public URL: https://durham-repository.worktribe.com/output/1587638
• Publisher URL: http://www.dur.ac.uk/i.a.stewart/Papers/embedding.pdf