A possible optimal design of one-way Hamming network H(n, 3) based on the minimum transmission latency

Abstract

One-way Hamming network H(n, 3), namely directed Hamming network, is the cartesian product of n complete graphs $K_{3}$ and has been widely used in hypercube parallel computer for its high communication rate and availability. As one of the critical parameters for evaluating the one-way Hamming network performance, the transmission latency which is the time for the information transmits from the source to the destination is proportional to the network diameter, and it can be reduced by optimizing the network diameter, especially, the minimum transmission latency corresponds to the oriented diameter which is the minimum diameter of one-way network. Currently, although the problems in the design and optimization of H(n, 2) with the oriented diameter and the minimum transmission latency have been solved, studies on the one-way Hamming network H(n, 3) are not found the best of our knowledge. This paper studies the one-way Hamming network H(n, 3) with the possible oriented diameter and the possible minimum transmission latency. Specifically, we first present a lemma and a mathematical model for the one-way Hamming network H(n, 3) with the possible oriented diameter and the possible minimum transmission latency, and then propose a recursive method to obtain  $n\le \overrightarrow{d}(H(n,3))\le n+1$, where  $\overrightarrow{d}(H(n,3))$ denotes the oriented diameter of H(n, 3). Finally, a practical example is utilized to intuitively describe such a method in this paper. Results show that the optimal design of the one-way Hamming network H(n, 3) helps reduce the information transmission latency by  $100\%$ as n tends to infinity when 2n is the baseline.