A Fast $f(r, k+1)/k$f(r, k+1)/k-Diagnosis for Interconnection Networks Under MM* Model

Yanze Huang, Limei Lin, Sun-Yuan Hsieh

Abstract: Cyberspace is not a “vacuum space”, and it is normal that there are inevitable viruses and worms in cyberspace. Cyberspace security threats stem from the problem of endogenous security, which is caused by the incompleteness of theoretical system and technology of the information field itself. Thus it is impossible and unnecessary for us to build an “aseptic” cyberspace. On the contrast, we must focus on improving the “self-immunity” of network. Literally, endogenous security is an endogenous effect from its own structural factors rather than external ones. The <inline-formula><tex-math notation="LaTeX">$t/k$</tex-math></inline-formula> -diagnosis strategy plays a very important role in measuring endogenous network security without prior knowledge, which can significantly enhance the self-diagnosing capability of network. As far as we know, few research involves <inline-formula><tex-math notation="LaTeX">$t/k$</tex-math></inline-formula> -diagnosis algorithm and <inline-formula><tex-math notation="LaTeX">$t/k$</tex-math></inline-formula> -diagnosability of interconnection networks under MM* model. In this article, we propose a fast <inline-formula><tex-math notation="LaTeX">$f(r,k+1)/k$</tex-math></inline-formula> -diagnosis algorithm of complexity <inline-formula><tex-math notation="LaTeX">$O(Nr^2)$</tex-math></inline-formula> , say <inline-formula><tex-math notation="LaTeX">$G$</tex-math></inline-formula> MIS <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> DIAGMM*, for a general <inline-formula><tex-math notation="LaTeX">$r$</tex-math></inline-formula> -regular network <inline-formula><tex-math notation="LaTeX">$G$</tex-math></inline-formula> under MM* model by designing a 0-comparison subgraph <inline-formula><tex-math notation="LaTeX">$M_0(G)$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> is the size of <inline-formula><tex-math notation="LaTeX">$G$</tex-math></inline-formula> . We determine that the <inline-formula><tex-math notation="LaTeX">$t/k$</tex-math></inline-formula> -diagnosability <inline-formula><tex-math notation="LaTeX">$(t(G)/k)^M$</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">$G$</tex-math></inline-formula> under MM* model is <inline-formula><tex-math notation="LaTeX">$f(r,k+1)$</tex-math></inline-formula> by <inline-formula><tex-math notation="LaTeX">$G$</tex-math></inline-formula> MIS <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> DIAGMM* algorithm. Moreover, we establish the <inline-formula><tex-math notation="LaTeX">$(t(G)/k)^M$</tex-math></inline-formula> of some interconnection networks under MM* model, including BC networks, <inline-formula><tex-math notation="LaTeX">$(n,l)$</tex-math></inline-formula> -star graph networks, and data center network DCells. Finally, we compare <inline-formula><tex-math notation="LaTeX">$(t(G)/k)^M$</tex-math></inline-formula> with diagnosability, conditional diagnosability, pessimistic diagnosability, extra diagnosability, and good-neighbor diagnosability under MM* model. It can be seen that <inline-formula><tex-math notation="LaTeX">$(t(G)/k)^M$</tex-math></inline-formula> is greater than other fault diagnosabilities in most cases.