An $O(\log _3N)$ Algorithm for Reliability Assessment of 3-Ary $n$ -Cubes Based on $h$ -Extra Edge Connectivity.

Reliability evaluation of multiprocessor systems is of great significance to the design and maintenance of these systems. As two generalizations of traditional edge connectivity, extra edge connectivity and component edge connectivity are two important parameters to evaluate the fault-tolerant capability of multiprocessor systems. Fast identifying the extra edge connectivity and the component edge connectivity of high order remains a scientific problem for many useful multiprocessor systems. In this article, we determine the $h$ -extra edge connectivity of the 3-ary $n$ -cube $Q_n^3$ for $h\in [1, \frac{3^n-1}{2}]$. Specifically, we divide the interval $[1, \frac{3^n-1}{2}]$ into some subintervals and characterize the monotonicity of $\lambda _h(Q_n^3)$ in these subintervals and then deduce a recursive closed formula of $\lambda _h(Q_n^3)$. Based on this formula, an efficient algorithm with complexity $O(\log _3\,N)$ is designed to determine the exact values of $h$ -extra edge connectivity of the 3-ary $n$ -cube $Q_n^3$ for $h\in [1, \frac{3^n-1}{2}]$ completely. Moreover, we also determine the $g$ -component edge connectivity of the 3-ary $n$ -cube $Q_n^3(n\geq 6$) for $1\leq g\leq 3^{\lceil \frac{n}{2}\rceil }$. [ABSTRACT FROM AUTHOR]