A (4 n − 9)/3 diagnosis algorithm on n-dimensional cube network

Abstract

As a generalization of the precise and pessimistic diagnosis strategies of system-level diagnosis of multicomputers, the t/ k diagnosis strategy can significantly improve the self-diagnosing capability of a system at the expense of no more than k fault-free processors (nodes) being mistakenly diagnosed as faulty. In the case k ⩾ 2, to our knowledge, there is no known t/ k diagnosis algorithm for general diagnosable system or for any specific system. Hypercube is a popular topology for interconnecting processors of multicomputers. It is known that an n-dimensional cube is (4 n − 9)/3-diagnosable. This paper addresses the (4 n − 9)/3 diagnosis of n-dimensional cube. By exploring the relationship between a largest connected component of the 0-test subgraph of a faulty hypercube and the distribution of the faulty nodes over the network, the fault diagnosis of an n-dimensional cube can be reduced to those of two constituent ( n − 1)-dimensional cubes. On this basis, a diagnosis algorithm is presented. Given that there are no more than 4 n − 9 faulty nodes, this algorithm can isolate all faulty nodes to within a set in which at most three nodes are fault-free. The proposed algorithm can operate in O( N log 2 N) time, where N = 2 n is the total number of nodes of the hypercube. The work of this paper provides insight into developing efficient t/ k diagnosis algorithms for larger k value and for other types of interconnection networks.