The diagnosability of hypercubes with arbitrarily missing links Dajin Wang * Department of Computer Science, Montclair State University, Upper Montclair, NJ 07043, USA Received 16 September 1998; accepted 16 February 1999 Abstract We study the problem of determining diagnosability for incomplete hypercubes that have arbitrarily distributed missing links, under the classic PMC diagnostic model and its variant, the BGM model. Based on the result proved in this paper, for both models, in most cases the diagnosability of an incomplete hypercube can be determined by simply checking the link degree of each node. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Diagnosability; Diagnosis; Fault-tolerance; Graph theory; Hypercubes; Link failure; Multiprocessor networks 1. Introduction Many topologies have been proposed to inter- connect processors in a multiprocessor computer system. Among them, the hypercube has drawn the greatest attention. A hypercube possesses many attractive properties and has become the most popular architecture for multiprocessor systems. Commercial multiprocessor systems based on hypercube structure have already been available [3,5,6,13]. Because of its importance for achieving high performance, the fault-tolerant computing for hypercubes has been the interest of many re- searchers. It is well known that the diagnosability of an n-dimensional hypercube is n under the classic PMC model [11]. That is, an n-dimensional hyper- cube can correctly detect all faulty nodes provided that the number of faulty nodes does not exceed n. When the adopted fault bound is maximum, n, all the links will be involved in the diagnosis. Algo- rithms are available to determine the faulty pro- cessors provided that the diagnosability of the system is known [1,9]. We can apply diagnosis al- gorithms only when we know the diagnosability of the system. Having diagnosability n in an n-di- mensional hypercube implies that all links among nodes are functioning, i.e., there are no missing links. It is then a natural question to ask how the diagnosability decreases if some links are missing. The past literature in this ®eld has seen studies of diagnosability for hypercubes with regularly en- hanced links [15] or regularly decreased links [17]. In this paper, we present results that establish the diagnosability for hypercubes in the presence of arbitrarily distributed missing links. We will give a simple algorithm to decide the diagnosability for incomplete hypercubes with missing links (failing, or in use other than diagnosis). The diagnosability of incomplete hypercubes under a variant of PMC model, the BGM model [2], is also studied in this paper. A simple algo- rithm is presented that decides the diagnosability for incomplete hypercubes under the BGM model. As an immediate corollary of the main results in www.elsevier.com/locate/sysarc Journal of Systems Architecture 46 (2000) 519±527 * E-mail address: wang@pegasus.montclair.edu (D. Wang). 1383-7621/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 3 - 7 6 2 1 ( 9 9 ) 0 0 0 1 5 - 6