The $t/s$ -Diagnosability of Hypercube Networks Under the PMC and Comparison Models

Abstract

A t/s-diagnosable system, a generalization of t/t-diagnosable system, refers to such a system that all the faulty nodes of the system can be isolated within a set of size at most s in the presence of at most t faulty nodes. In this paper, the t/s-diagnosability of the hypercubes under the PMC model (the comparison model) is evaluated. First, several novel properties of hypercube are proposed, which are previously unknown in the literatures. Second, based on the above properties of hypercubes, we show that an n-dimensional (n ≥ 5) hypercube is (kn - ((k(k + 1))/2) + 1)/(kn - ((k(k + 1))/2) + k - 1)-diagnosable in terms of both the PMC and the comparison models, where 2 ≤ k ≤ n - 2. Furthermore, we introduce a fast diagnosis algorithm to isolate the faulty nodes in a subset of the system under the PMC model (the comparison model). And the time complexity of the algorithm is O(n2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) for an n-dimensional hypercube.