Mobius Cubes Research Articles

Adaptive Diagnosis of Variants of the Hypercube

System-level fault diagnosis deals with the problem of identifying faulty nodes (processors) in a multiprocessor system. Each node is faulty or fault-free, and it can test other nodes in the system, and outputs the test results. The test result from a node is reliable if the node is fault-free, but the result is unreliable if it is faulty. In this paper, we prove that four variants of the hypercube: the crossed cube, the twisted cube, the Mobius cube, and the enhanced cube, are adaptively diagnosed using at most 4 parallel testing rounds, with at most n faulty nodes (for the enhanced cube, with at most n + 1 faulty nodes), where each processor participates in at most one test in each round. Furthermore, we propose another diagnosis algorithm for the n-dimensional enhanced cube with at most n + 1 faulty nodes, and show that it is adaptively diagnosed with at most 5 rounds in the worst case, but with at most 3 rounds if the number of existing faulty nodes is at most n - ⌊log (n + 1)⌋.

The t/k-diagnosability of the BC graphs

Processor fault diagnosis takes an important role in fault-tolerant computing on multiprocessor systems. There are two classical diagnosis strategies - the precise strategy and the pessimistic strategy, both of which are based on the well-known PMC diagnostic model. Nevertheless, the degree of diagnosability of the system is limited under these two strategies. A better method, called the t/k-diagnosis strategy, is proposed by Somani and Peleg, in which the identified fault-set is allowed to contain at most k fault-free processors. Using this diagnosis strategy, the degree of diagnosability of the hypercube increases greatly as the number of the fault-free processors in the fault-set increases. We study the t/k-diagnosability of so-called BC graphs that include hypercubes, crossed cubes, Mobius cubes, and twisted cubes, etc. We show that any n-dimensional BC graph is t(n, k)/k-diagnosable when n/spl ges/4 and 0/spl les/k/spl les/n, where t(n,k)=(k+1)n-1/2(k+1)(k+2)+1. Therefore, the crossed cube, the Mobius cube, and the twisted cube all have the same t/k-diagnosability as the hypercube. As a result, the algorithms developed for diagnosis on the hypercube may also be used to diagnose multiprocessor systems whose network topologies are based on BC graphs.

Fault-Tolerant Pancyclicity of the Mobius Cubes

The Mobius cube MQ n proposed by Cull et al. is an alternative to the popular hypercube network. Recently, MQ n was shown to be pancyclic, i.e., cycles of any lengths at least four can be embedded into it. Due to the importance of the fault tolerance in the parallel processing area, in this paper, we study an injured MQ n with mixed node and link faults. We show that it is (n - 2)-fault-tolerant pancyclic for n ≥ 3, that is, an injured n-dimensional MQ n is still pancyclic with up to (n-2) faults. Furthermore, our result is optimal.

The diagnosability of the matching composition network under the comparison diagnosis model

The classical problem of diagnosability is discussed widely and the diagnosability of many well-known networks has been explored. We consider the diagnosability of a family of networks, called the matching composition network (MCN); a perfect matching connects two components. The diagnosability of MCN under the comparison model is shown to be one larger than that of the component, provided some connectivity constraints are satisfied. Applying our result, the diagnosability of the hypercube Qn, the crossed cube CQ/sub n/, the twisted cube TQ/sub n/, and the Mobius cube MQ/sub n/ can all proven to be n, for n/spl ges/4. In particular, we show that the diagnosability of the four-dimensional hypercube Q/sub 4/ is 4, which is not previously known.

Diagnosability of the Mobius cubes

The recently introduced interconnection networks, the Mobius cubes, are hypercube variants that have some better properties than hypercubes. The n-dimensional Mobius cube M/sub n/ is a regular graph with 2/sup n/ nodes and n2/sup n-1/ edges. The diameter of M/sub n/ is about one half that of the n-dimensional hypercube Q/sub n/ and the average number of steps between nodes for M/sub n/ is about two-thirds of the average for Q/sub n/, and 1-M/sub n/ has dynamic performance superior to that of Q/sub n/. Of course, the symmetry of M/sub n/ is not superior to that of Q/sub n/, i.e., Q/sub n/ is both node symmetric and edge symmetric , whereas M/sub n/ is, in general, neither node symmetric (n/spl ges/4) nor edge symmetric (n/spl ges/3). In this paper, we study the diagnosability of M/sub n/. We use two diagnosis strategies, both based on the so-called PMC diagnostic model-the precise (one-step) diagnosis strategy proposed by Preparata et al. (1967) and the pessimistic diagnosis strategy proposed by Friedman (1975). We show that the diagnosability of M/sub n/ is the same as that of Q/sub n/, i.e., M/sub n/ is n-diagnosable under the precise diagnosis strategy and (2n-2)/(2n-2)-diagnosable under the pessimistic diagnosis strategy.

The Mobius cubes

The Mobius cubes are hypercube variants that give better performance with the same number of links and processors. We show that the diameter of the Mobius cubes is about one half the diameter of the equivalent hypercube, and that the average number of steps between processors for a Mobius cube is about two-thirds of the average for a hypercube. We give an efficient routing algorithm for the Mobius cubes. This routing algorithm finds a shortest path and operates in time proportional to the dimension of the cube. We also give efficient broadcast algorithms for the Mobius cubes. We show that the Mobius cubes contain ring networks and other networks. We report results of simulation studies on the dynamic message-passing performance of the hypercube, the Twisted Cube of P.A.J. Hilbers et al. (1987), and the Mobius cubes. Our results are in agreement with S. Abraham (1990), showing that the Twisted Cube has worse dynamic performance than the hypercube, but our results show that the 1-Mobius cube has dynamic performance superior to that of the hypercube. This contradicts current literature, which implies that twisted cube variants will have worse dynamic performance. >