A Note on the Nature Diagnosability of Alternating Group Graphs Under the PMC Model and MM^* Model

Abstract

Many multiprocessor systems have interconnection networks as underlying topologies and an interconnection network is usually represented by a graph where nodes represent processors and links represent communication links between processors. No faulty set can contain all the neighbors of any fault-free node in the system, which is called the nature diagnosability of the system. Diagnosability of a multiprocessor system is one important study topic. As a favorable topology structure of interconnection networks, the n-dimensional alternating group graph AGn has many good properties. In this paper, we prove the following. (1) The nature diagnosability of AGn is 4n − 10 for n ≥ 5 under the PMC model and MM * model. (2) The nature diagnosability of the 4-dimensional alternating group graph AG 4 under the PMC model is 5. (3) The nature diagnosability of AG 4 under the MM * model is 4. system G is said to be t-diagnosable if all faulty processors can be identified without replacement, provided that the number of presented faults does not exceed t. The diagnosability t(G) of G is the maximum value of t such that G is t-diagnosable. 4,5,13 For a t-diagnosable system, Dahbura and Masson 4 proposed an algorithm with time complex O(n 2.5 ), which can effectively identify the set of faulty processors.Several diagnosis models were proposed to identify the faulty processors. One major approach is the Preparata, Metze, and Chien's (PMC) diagnosis model introduced by Preparata et al. 17 The diagnosis of the system is achieved through two linked processors testing each other. Another major approach is the Maeng and Malek's (MM) diagnosis model, namely the comparison diagnosis model, proposed by Maeng and Malek. 15 In the MM model, to diagnose a system, a node sends the same task to two of its neighbors, and then compares their responses. In the event of a random node failure, it is very unlikely that all the neighbors of any node fail simultaneously in the system. This reason has motivated research on the restricted diagnosability of the system. In 2005, Lai et al. 13 introduced the restricted diagnosability of the system called the conditional diagnosability. They considered the situation that no faulty set can contain all the neighbors of any node in the system. Since the probability that all the neighbors of a fault node fail and create faults is more to the probability that all the neighbors of a fault-free node fail and create faults in the system, we consider the situation that no faulty set can contain all the neighbors of any fault-free node in the system, which is called the nature diagnosability of the system. In 2012, Peng et al. 16 proposed a measure for fault diagnosis of systems, namely, the g-good-neighbor diagnosability (which is also called the g-goodneighbor conditional diagnosability), which requires that every fault-free node has at least g fault-free neighbors. In Ref. 16, they studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the PMC model. In 2016, Wang and Han 20 studied the g-...

g-Good-Neighbor Diagnosability of Arrangement Graphs under the PMC Model and MM* Model

“…In [7], Wang and Han studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the MM* model. There is a significant amount of research on the g-good-neighbor diagnosability of graphs [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].…”