Three Round Adaptive Diagnosis in Hierarchical Multiprocessor Systems

System level diagnosis is an important technique for fault detection and location in multiprocessor computing systems. Adaptive diagnosis, proposed by Nakajima, is a practical system level diagnostic scheme; the main design objective of an adaptive diagnostic scheme is to reduce the number of test rounds, as well as the total number of tests. The hierarchical crossed cube draws upon constructions used within both the hypercube and the crossed cube, giving it many desirable features such as symmetry and logarithmic diameter, making it suitable for massively parallel systems with thousands of processors. In this paper, we first show that the diagnosability of a hierarchical crossed cube, HCC_k,n, is k+n, and then propose a scheme that completely diagnoses a HCCk,n within three test rounds, and at most N+⌈(k+n+2)/2⌉ × ⌊(k+n+2)/2⌋ tests, where N=2^k+2n is the number of vertices of HCC_k,n. Our diagnostic scheme has the optimal number of test rounds. Moreover, most of our proofs are applicable not just to hierarchical crossed cubes but also to hierarchical interconnection networks formed by replacing crossed cubes with other appropriate interconnection networks.