Mobius Cubes Research Articles

Fault-Tolerant Hamiltonian Connectivity of Twisted Hypercube-Like Networks <italic>THLNs</italic>

The twisted hypercube-like networks (THLNs) include some well-known hypercube variants. A graph $G$ is $k$ -fault-tolerant Hamiltonian connected if $G-F$ remains Hamiltonian connected for every $F\subset V(G)\cup E(G)$ with $|F|\leq k$ . This paper is concerned with the fault-tolerant Hamiltonian connectivity of an $n$ -dimensional ( $n$ -D) THLN. Let $G_{n}$ be an $n$ -D THLN ( $n\geq 5$ ) and $F$ be a subset of $V(G_{n})\cup E(G_{n})$ with $|F|\leq n-2$ . We show that for arbitrary vertex-pair $(u,v)$ in $G_{n}-F$ , there exists a $(n-2)$ -fault-tolerant Hamiltonian path joining vertices $u$ and $v$ except $(u,v)$ being a weak vertex-pair in $G_{n}-F$ . The technical theorem proposed in this paper can be applied to several multiprocessor systems, including $n$ -D crossed cubes $CQ_{n}$ , $n$ -D twisted cubes $TQ_{n}$ for odd $n$ , $n$ -D locally twisted cubes $LTQ_{n}$ , and $n$ -D Mobius cubes $MQ_{n}$ .

MO<sup style=" margin-left:-10px;">¨</sup> bius立方体的h-好邻条件诊断度

条件诊断度是衡量互连网络可靠性能优劣的重要参数，“条件”意味着任何处理器的所有相邻处理器不能同时出现潜在故障。作为条件诊断度的一个改进参数，Peng等人通过限制要求每个无故障节点至少包含有h个无故障的邻点提出了h-好邻条件诊断度，是一种在节点失效的情况下衡量互连网络可靠性的重要指标。本文研究并证明了Mobius立方体(MNQ)在PMC模型和MM*模型下的h-好邻条件诊断度是(n-h+1)2h-1(0≤h≤n-3)，相比于传统诊断度成几何倍数的提高。 The conditional diagnosability is an important measure of the reliability of interconnection network. The “condition” means that all adjacent processors of any processor cannot be potentially faulty at the same time. As an improvement of conditional diagnosability, h-good-neighbor conditional diagnosability has been proposed by Peng et al. under the assumption that every fault-free node has at least h fault-free neighbors, which is an important indicator of reliability of interconnection network in the case of vertices failure. In this paper, we first introduced and investigated the h-good-neighbor conditional diagnosability of Mobius Cubes (MNQ). And we showed that the h-good- neighbor conditional diagnosability of Mo bius Cubes is (n-h+1)2h-1(0≤h≤n-3) under the PMC and MM* model, which can be several times higher than the classical diagnosability of MNQ.

Dimensional-Permutation-Based Independent Spanning Trees in Bijective Connection Networks

In recent years, there are many new findings on independent spanning trees (ISTs for short) in hypercubes, crossed cubes, locally twisted cubes, and Mobius cubes, which all belong to a more general network category called bijective connection networks (BC networks). However, little progress has been made for ISTs in general BC networks. In this paper, we first propose the definitions of conditional BC networks and V -dimensional-permutation. We then give a linear parallel algorithm of ISTs rooted at an arbitrary vertex in conditional BC networks, which include hypercubes, crossed cubes, locally twisted cubes, and Mobius cubes, based on the ascending circular dimensional-permutation, where the ISTs are all isomorphic to the binomial-like tree. In addition, we show that there exists an efficient algorithm to construct a spanning tree rooted at an arbitrary vertex in any BC network X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , and all V-dimensional-permutations can be used to construct spanning trees isomorphic to the n-level binomial tree and rooted at an arbitrary vertex in X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> .

A fully parallelized scheme of constructing independent spanning trees on Möbius cubes

A set of spanning trees in a graph is said to be independent (ISTs for short) if all the trees are rooted at the same node $$r$$r and for any other node $$v(\ne r)$$v(?r), the paths from $$v$$v to $$r$$r in any two trees are node-disjoint except the two end nodes $$v$$v and $$r$$r. It was conjectured that for any $$n$$n-connected graph there exist $$n$$n ISTs rooted at an arbitrary node. Let $$N=2^n$$N=2n be the number of nodes in the $$n$$n-dimensional Mobius cube $$MQ_n$$MQn. Recently, for constructing $$n$$n ISTs rooted at an arbitrary node of $$MQ_n$$MQn, Cheng et al. (Comput J 56(11):1347---1362, 2013) and (J Supercomput 65(3):1279---1301, 2013), respectively, proposed a sequential algorithm to run in $${\mathcal O}(N\log N)$$O(NlogN) time and a parallel algorithm that takes $${\mathcal O}(N)$$O(N) time using $$\log N$$logN processors. However, the former algorithm is executed in a recursive fashion and thus is hard to be parallelized. Although the latter algorithm can simultaneously construct $$n$$n ISTs, it is not fully parallelized for the construction of each spanning tree. In this paper, we present a non-recursive and fully parallelized approach to construct $$n$$n ISTs rooted at an arbitrary node of $$MQ_n$$MQn in $${\mathcal O}(\log N)$$O(logN) time using $$N$$N nodes of $$MQ_n$$MQn as processors. In particular, we derive useful properties from the description of paths in ISTs, which make the proof of independency to become easier than ever before.

Reliability Evaluation of BC Networks in Terms of the Extra Vertex- and Edge-Connectivity

Reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems. The extra connectivity and the extra edge-connectivity are two important parameters for the reliability evaluation of interconnection networks. The <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {n}}$</tex> </formula> -dimensional bijective connection network (in brief, BC network) includes several well known network models, such as, hypercubes, Möbius cubes, crossed cubes, and twisted cubes. In this paper, we explore the extra connectivity and the extra edge-connectivity of BC networks, and discuss the structure of BC networks with many faults. We obtain a sharp lower bound of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{g}}$</tex> </formula> -extra edge-connectivity of an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {n}}$</tex> </formula> -dimensional BC network for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{n}} \geq 4$</tex> </formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1 \leq { {g}} \leq {2^{[{{{n}} \over 2}]}}$</tex> </formula> . We also obtain a sharp lower bound of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${ {g}}$</tex> </formula> -extra connectivity of an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{n}}$</tex> </formula> -dimensional BC network for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{n}} \geq 4$</tex> </formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1 \leq { {g}} \leq 2{ {n}}$</tex> </formula> which improves the result in [“Reliability evaluation of BC networks,” IEEE Trans. Computers, DOI: 10.1109/tc.2012.106.] for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1 \leq { {g}} \leq { {n}} - 3$</tex> </formula> . Furthermore, we give a remark about exploring the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${ {g}}$</tex> </formula> -extra edge-connectivity of BC networks for the more general <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {g}}$</tex> </formula> , and we also characterize the structure of BC networks with many faulty nodes or links. As an application, we obtain several results on the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {g}}$</tex> </formula> -extra (edge-) connectivity and the structure of faulty networks on hypercubes, Möbius cubes, crossed cubes, and twisted cubes.

Reliability Evaluation of BC Networks

Reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems. Extra connectivity determination and faulty networks' structure analysis are two important aspects for the reliability evaluation of interconnection networks. An n-dimensional bijective connection network (in brief, BC network), denoted by Xn, is an n-regular graph with 2n vertices and n2n-1 edges. The hypercubes, Mobius cubes, crossed cubes, and twisted cubes are some examples of the BC networks. By exploring the boundary problem of the BC networks, we prove that when n ≥ 4 and 0 ≤ h ≤ n - 4 the h-extra connectivity of an n-dimensional BC network Xn is kh(Xn = n(h + 1)- 1/2h (h + 3). Furthermore, there exists a large connected component and the remaining small components have at most h vertices in total if the total number of faulty vertices is strictly less its h-extra connectivity. As an application, the results on the h-extra connectivity and structure of faulty networks on hypercubes, Mobius cubes, crossed cubes, and twisted cubes are obtained.

KMcube: the compound of Kautz digraph and Möbius cube

This paper introduces a novel interconnection network called KMcube (Kautz-Mobius cube). KMcube is a compound graph of a Kautz digraph and Mobius cubes. That is, it uses the Mobius cubes as the unit cluster and connects many such clusters by means of a Kautz digraph at the cost of only one additional arc being added to any node in each Mobius cubes. The topological benefits of both basic graphs are preserved in the compound network. It utilizes the topological properties of Mobius cubes to conveniently embed parallel algorithms into each cluster and the short diameter of a Kautz digraph to support efficient inter-cluster communication. Additionally, KMcube provides other attractive properties, such as the regularity, symmetry, and expandability. The proposed methodology for KMcube is further applied to the compound graphs of Kautz digraph and other Mobius-like graphs with the similar diameter to a Mobius cube. Moreover, other hybrid graphs of Kautz digraph and Mobius cubes are proposed and compared.

Parallel construction of independent spanning trees and an application in diagnosis on Möbius cubes

Independent spanning trees (ISTs) on networks have applications to increase fault-tolerance, bandwidth, and security. Mobius cubes are a class of the important variants of hypercubes. A recursive algorithm to construct n ISTs on n-dimensional Mobius cube M n was proposed in the literature. However, there exists dependency relationship during the construction of ISTs and the time complexity of the algorithm is as high as O(NlogN), where N=2 n is the number of vertices in M n and n?2. In this paper, we study the parallel construction and a diagnostic application of ISTs on Mobius cubes. First, based on a circular permutation n?1,n?2,?,0 and the definitions of dimension-backbone walk and dimension-backbone tree, we propose an O(N) parallel algorithm, called PMCIST, to construct n ISTs rooted at an arbitrary vertex on M n . Based on algorithm PMCIST, we further present an O(n) parallel algorithm. Then we provide a parallel diagnostic algorithm with high efficiency to diagnose all the vertices in M n by at most n+1 steps, provided the number of faulty vertices does not exceed n. Finally, we present simulation experiments of ISTs and an application of ISTs in diagnosis on 0-M 4.

Constructive Algorithm of Independent Spanning Trees on Mobius Cubes

Constructive Algorithm of Independent Spanning Trees on Mobius Cubes

A Systematic Algorithm for Identifying Faults on Hypercube-Like Networks Under the Comparison Model

There is a growing demand for fault diagnosis to increase the reliability of systems. Diagnosis by comparison is a realistic approach to the fault diagnosis of multiprocessor systems. In this paper, we consider <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional hypercube-like networks for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ≥ 5. We propose an efficient fault diagnosis algorithm for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional hypercube-like networks under the MM comparison model by exploiting the Hamiltonian and extended-star properties. Applying our algorithm, the faulty processors in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional hypercubes, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional crossed cubes, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional twisted cubes, and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional Möbius cubes can all be diagnosed in linear time provided the number of faulty processors is not more than the dimension <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> .

에지 고장이 있는 Restricted Hypercube-Like 그래프의 해밀톤 경로

Restricted Hypercube-Like(RHL) 그래프는 교차큐브, 뫼비우스큐브, 엠큐브, 꼬인큐브, 지역꼬인큐브, 다중꼬인큐브, 일반꼬인큐브와 같이 유용한 상호연결망들을 광범위하게 포함하는 그래프군이다. 본 논문에서는 <TEX>$m{\geq}4$</TEX> 인 m-차원 RHL 그래프 G에 대해서 임의의 에지 집합 <TEX>$F{\subset}E(G)$</TEX>, <TEX>${\mid}F{\mid}{\leq}m-2$</TEX>, 가 고장일 때, 고장 에지들을 제거한 그래프 <TEX>$G{\setminus}F$</TEX>는 임의의 서로 다른 두 정점 s와 t에 대해서 dist(s, V(F))<TEX>${\neq}1$</TEX> 이거나 dist(t, V(F))<TEX>${\neq}1$</TEX>이면 해밀톤 경로가 있음을 보인다. V(F)는 F에 속하는 에지들의 양 끝점들의 집합이고 dist(v, V(F))는 정점 v와 집합 V(F)의 정점들 간의 최소 거리이다. Restricted Hypercube-Like (RHL) graphs are a graph class that widely includes useful interconnection networks such as crossed cube, Mobius cube, Mcube, twisted cube, locally twisted cube, multiply twisted cube, and generalized twisted cube. In this paper, we show that for an m-dimensional RHL graph G, <TEX>$m{\geq}4$</TEX>, with an arbitrary faulty edge set <TEX>$F{\subset}E(G)$</TEX>, <TEX>${\mid}F{\mid}{\leq}m-2$</TEX>, graph <TEX>$G{\setminus}F$</TEX> has a hamiltonian path between any distinct two nodes s and t if dist(s, V(F))<TEX>${\neq}1$</TEX> or dist(t, V(F))<TEX>${\neq}1$</TEX>. Graph <TEX>$G{\setminus}F$</TEX> is the graph G whose faulty edges are removed. Set V(F) is the end vertex set of the edges in F and dist(v, V(F)) is the minimum distance between vertex v and the vertices in V(F).

(t,k)-Diagnosis for Component-Composition Graphs under the MM* Model

(t, k)-Diagnosis, which is a generalization of sequential diagnosis, requires that at least t faulty processors be identified and replaced in each iteration provided there are at most t faulty processors, where t ≥ k. Let κ(G) and n(G) be, respectively, the node connectivity and the number of nodes in a graph G. In this paper, we compute the (t, k)-diagnosability for a class of component composition graphs under the comparison diagnosis model. We show that the m-dimensional component-composition graph G (m ≥ 4) is (Ω(h),κ(G))-diagnosable, where h= {2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sup> × (m-3) × lg(m-1) (m-1)/(m-1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> if 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-2</sup> ≤ n(G)<; m! 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sup> × (m-3)/m-1 if n(G) ≥ m!. Based on this result, the (t, k)-diagnosability of several multiprocessor systems, including hypercubes, crossed cubes, twisted cubes, locally twisted cubes, multiply twisted cubes, generalized twisted cubes, recursive circulants, Mobius cubes, Mcubes, star graphs, bubble-sort graphs, pancake graphs, and burnt pancake graphs, can be computed efficiently.

Determining the Diagnosability of (1,2)-Matching Composition Networks and Its Applications

The classic problem of determining the diagnosability of a given network has been studied extensively. Under the PMC model, this paper addresses the problem of determining the diagnosability of a class of networks called (1,2)-Matching Composition Networks, each of which is constructed by connecting two graphs via one or two perfect matchings. By applying our results to multiprocessor systems, we can determine the diagnosability of hypercubes, twisted cubes, locally twisted cubes, generalized twisted cubes, recursive circulants G(2^{n},4) for odd n, folded hypercubes, augmented cubes, crossed cubes, Mobius cubes, and hyper-Petersen networks, all of which belong to the class of (1,2)-matching composition networks.

An efficient fault-tolerant routing algorithm in bijective connection networks with restricted faulty edges

In this paper, we study fault-tolerant routing in bijective connection networks with restricted faulty edges. First, we prove that the probability that all the incident edges of an arbitrary node become faulty in an n -dimensional bijective connection network, denoted by X n , is extremely low when n becomes sufficient large. Then, we give an O ( n ) algorithm to find a fault-free path of length at most n + 3 ⌈ log 2 ∣ F ∣ ⌉ + 1 between any two different nodes in X n if each node of X n has at least one fault-free incident edge and the number of faulty edges is not more than 2 n − 3 . In fact, we also for the first time provide an upper bound of the fault diameter of all the bijective connection networks with the restricted faulty edges. Since the family of BC networks contains hypercubes, crossed cubes, Möbius cubes, etc., all the results are appropriate for these cubes.

A family of Hamiltonian and Hamiltonian connected graphs with fault tolerance

Processor (vertex) faults and link (edge) faults may happen when a network is used, and it is meaningful to consider networks (graphs) with faulty processors and/or links. A k-regular Hamiltonian and Hamiltonian connected graph G is optimal fault-tolerant Hamiltonian and Hamiltonian connected if G remains Hamiltonian after removing at most k?2 vertices and/or edges and remains Hamiltonian connected after removing at most k?3 vertices and/or edges. In this paper, we investigate in constructing optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected graphs. Therefore, some of the generalized hypercubes, twisted-cubes, crossed-cubes, and Mobius cubes are optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected.

On conditional diagnosability and reliability of the BC networks

An n-dimensional bijective connection network (in brief, BC network), denoted by X n , is an n-regular graph with 2 n nodes and n2 n?1 edges. Hypercubes, crossed cubes, twisted cubes, and Mobius cubes all belong to the class of BC networks (Fan and He in Chin. J. Comput. 26(1):84---90, [2003]). We prove that the super connectivity of X n is 2n?2 for n?3 and the conditional diagnosability of X n is 4n?7 for n?5. As a corollary of this result, we obtain the super connectivity and conditional diagnosability of the hypercubes, twisted cubes, crossed cubes, and Mobius cubes.

(t, k) - Diagnosis for Matching Composition Networks under the MM* Model

(t, k)-diagnosis, which is a generalization of sequential diagnosis, requires at least k faulty processors identified and repaired in each iteration provided there are at most t faulty processors, where tgesk. In this paper, a (t, k)-diagnosis algorithm under the MM* model is proposed for matching composition networks, which include many well-known interconnection networks, such as hypercubes, crossed cubes, twisted cubes, and Mobius cubes. It is shown that a matching composition network of n dimensions is (Omega((2n*log n)/n), n)-diagnosable

Hamiltonian path embedding and pancyclicity on the Mobius cube with faulty nodes and faulty edges

A graph G=(V, E) is said to be pancyclic if it contains fault-free cycles of all lengths from 4 to |V| in G. Let F/sub v/ and F/sub e/ be the sets of faulty nodes and faulty edges of an n-dimensional Mobius cube MQ/sub n/, respectively, and let F=F/sub v//spl cup/F/sub e/. A faulty graph is pancyclic if it contains fault-free cycles of all lengths from 4 to |V-F/sub v/|. In this paper, we show that MQ/sub n/-F contains a fault-free Hamiltonian path when |F|/spl les/n-1 and n/spl ges/1. We also show that MQ/sub n/-F is pancyclic when |F|/spl les/n-2 and n/spl ges/2. Since MQ/sub n/ is regular of degree n, both results are optimal in the worst case.

Many-to-many disjoint path covers in hypercube-like interconnection networks with faulty elements

A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. We deal with the graph G/sub 0/ /spl oplus/ G/sub 1/ obtained from connecting two graphs G/sub 0/ and G/sub 1/ with n vertices each by n pairwise nonadjacent edges joining vertices in G/sub 0/ and vertices in G/sub 1/. Many interconnection networks such as hypercube-like interconnection networks can be represented in the form of G/sub 0/ /spl oplus/ G/sub 1/ connecting two lower dimensional networks G/sub 0/ and G/sub 1/. In the presence of faulty vertices and/or edges, we investigate many-to-many disjoint path coverability of G/sub 0/ /spl oplus/ G/sub 1/ and (G/sub 0/ /spl oplus/ G/sub 1/) /spl oplus/ (G/sub 2/ /spl oplus/ G/sub 3/ ), provided some conditions on the Hamiltonicity and disjoint path coverability of each graph G/sub i/ are satisfied, 0 /spl les/ i /spl les/ 3. We apply our main results to recursive circulant G(2/sup m/, 4) and a subclass of hypercube-like interconnection networks, called restricted HL-graphs. The subclasses includes twisted cubes, crossed cubes, multiply twisted cubes, Mobius cubes, Mcubes, and generalized twisted cubes. We show that all these networks of degree m with f or less faulty elements have a many-to-many k-DPC joining any k distinct source-sink pairs for any k /spl ges/ 1 and f /spl ges/ 0 such that f+2k /spl les/ m - 1.

(t; k)-Diagnosis for matching composition networks

(t, k)-diagnosis, which is a generalization of sequential diagnosis, requires at least k faulty processors identified and replaced in each iteration provided there are at most t faulty processors, where t /spl ges/k This paper proposes a (t,k) diagnosis algorithm for matching composition networks, which include many well-known interconnection networks such as hypercubes, crossed cubes, twisted cubes, and Mobius cubes. It is shown that matching composition networks of n dimensions are (/spl Omega/(2/sup n//spl middot/logn/n),n)-diagnosable, where n > 5.