• 沒有找到結果。

多處理器系統診斷能力之量測

N/A
N/A
Protected

Academic year: 2021

Share "多處理器系統診斷能力之量測"

Copied!
78
0
0

加載中.... (立即查看全文)

全文

(1)

國 立 交 通 大 學

資訊工程學系

博 士 論 文

多處理器系統診斷能力之量測

Diagnosability Measures for Multiprocessor

Systems: A New Local Strategy

研 究 生:徐國晃

(2)

多處理器系統診斷能力之量測

Diagnosability Measures for Multiprocessor Systems:

A New Local Strategy

研 究 生:徐國晃 Student:Guo-Huang Hsu

指導教授:譚建民 博士 Advisor:Dr. Jimmy J. M. Tan

國 立 交 通 大 學

資 訊 工 程 系

博 士 論 文

A Dissertation Submitted to Department of Computer Science

College of Computer Science National Chiao Tung University in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Computer Science February 2009

(3)

誌 謝

本篇論文能夠順利的完成,要感謝很多人的付出及幫忙。首先要感謝

我的指導老師譚建民教授這幾年來的照顧和教導,讓我學習到很多做研究

的態度和方法。還有實驗室的伙伴們:皮卡丘、寬哥、峰哥、嚕嚕米、阿

鋼、韓吉與自良,感謝有你們的一路陪伴共同營造良好的學習研究環境,

有了你們的幫助和經驗分享,讓我在學習的過程能更加順利。由衷的感謝

大家的幫忙。在這裡,特別要感謝峰哥,每當我有研究上的困難時,你都

能不辭勞苦的協助我,使得我的研究瓶頸都能一一突破,再次的感謝你的

幫助。

此外,也要感謝我的論文計畫指導委員:蔡錫鈞老師與陳榮傑老師;校

內口試委員梁婷老師;與校外口試委員:陳健輝老師、張貿翔老師、許健

平老師、以及徐力行老師,非常感謝各位老師在口試時的指導與幫忙,讓

本論文能更加完美。

最後,最感謝我的家人。感謝哥哥在我念研究所初期時給予我經濟上

最大的支援。感謝老婆長久以來的支持與無怨無悔的付出。感謝父母為我

付出了那麼多的辛勞,教育我、養育我。有了你們大家做我的後盾,才讓

我能順利的完成學位。對於周遭曾經幫助過我的朋友與師長們,在此也要

致上最深的謝意。

(4)

多處理機系統診斷能力之量測

研究生:徐國晃 指導教授:譚建民 博士

國立交通大學資訊工程系

摘要

有關多處理機系統錯誤診斷問題已經在相當多的文獻被廣泛的討論,並且很多著名連結 網路的診斷能力也已經被提出來了。在這篇論文當中,我們針對多處理機系統研究了一 些不同的診斷問題。首先,我們介紹了一種新的診斷能力量測方法稱為局部診斷能力量

測,並且提出了一些架構用來決定系統中一個處理機在PMC診斷模式下是否為局部t-可

診斷的。針對超立方體網路(hypercube)和星狀網路(star graph),我們證明了網路中每一 個點的局部診斷能力等於它們自己的分支度。接著,我們針對系統診斷問題提出了一個 新的觀念稱為強局部可診斷特性。一個系統我們說他具有強局部可診斷特性即表示此系 統中每一個處理機的局部診斷能力等於它們自己的分支度。所以我們可以推得n維度超 立方體網路Qn和星狀網路Sn都有此很強的特性,當n ≥ 3。下一步我們接著研究當多處理 機系統具有一些壞掉的邊時,每一個點它們的局部診斷能力。對於具有一些壞掉的邊的 n維度超立方體網路Qn和星狀網Sn,我們證明了Qn在壞n - 2條邊以內其仍然保有此很強的 特性,而Sn在壞n - 3條邊以內也仍然保有此特性。假設網路在壞掉邊時每一個點具有至 少兩條好的邊時,在這樣的條件下,我們證明了Qn壞掉的邊數可以增加到3(n – 2) – 1條 仍然保有這種強特性,而Sn在此條件下無論壞多少條邊仍然可保有此特性。更進一步 地,我們考慮網路在壞掉邊時每一個點具有至少三條好的邊時,在這樣的條件下,我們 證明了Qn無論壞多少條邊仍可保有此很強的特性,並且我們所提出的這些壞邊數都是最 佳值。除此之外,我們針對一般系統也提出了一個新的診斷演算法。此演算法的時間複 雜度為O(N log N),此處N代表系統中處理機的總數。 條件式診斷能力量測是由賴等人所提出的,此量測方法在多處理機系統是另外一個有趣

(5)

任一個錯誤點集合不能包含任一個點的所有鄰居。本篇論文當中,我們根據這個條件去

計算一個n維度超立方體網路Qn在比較式診斷模式下它的條件式診斷能力,並且得到的

答案為3(n - 2) + 1,當n ≥ 5。此條件式診斷能力約是傳統診斷能力的三倍之多。最後, 我們延伸這個結果到BC(bijective connection)網路上,一個n維度BC網路記作Xn,此網路 是一個n-正規圖具有2n個點和n2n-1條邊。一般常見的超立方體網路(hypercube)、交錯超 立方體網路(crossed cube)、雙扭超立方體網路(twisted cube)和梅式超立方體網路(Mőbius cube)都是BC網路的一種。在這篇論文當中,我們也證明了一個n維度BC網路Xn在比較 式診斷模式下它的條件式診斷能力為3(n - 2) + 1,當n ≥ 5。根據這個結果,我們可以推 得所有立方體網路的條件式診斷能力。 關鍵字:PMC 診斷模式、比較式診斷模式、t-可診斷的、診斷能力、局部診斷能力、強 局部可診斷特性、條件式錯誤集合、條件式診斷能力、超立方體網路、星狀網路、BC 網路、診斷演算法。

(6)

Diagnosability Measures for Multiprocessor

Systems: A New Local Strategy

Student: Guo-Huang Hsu Advisor: Dr. Jimmy J. M. Tan

Department of Computer Science

College of Computer Science

National Chiao Tung University

Abstract

The problem of fault diagnosis has been discussed widely and the diagnosability of many well-known networks has been explored. In this thesis, we study some variants of diagnosis problems on multiprocessor systems. First of all, we introduce a new measure of diagnosability, called local diagnosability, and derive some structures for determining whether a vertex of a system is locally t-diagnosable under the PMC model. For hypercube network and star graph, we prove that the local diagnosability of each vertex is equal to its degree. Then, we propose a concept for system diagnosis, called strongly local-diagnosable property. A system G(V,E) is said to have a strongly local-diagnosable property, if the local diagnosability of each vertex is equal to its degree. We show that both Qn and Sn have this strong property for n ≥ 3, where the two notations Qn and Sn represent an n-dimensional hypercube and an n-dimensional star graph, respectively. Next, we study the local diagnosability of a faulty multiprocessor system. For a faulty hypercube Qn and a faulty star graph Sn, we prove that both Qn and Sn keep this strong property even if they have up to n – 2 faulty edges and n – 3 faulty edges, respectively. Assume that each vertex of a faulty hypercube Qn and a faulty star graph Sn is incident with at least two fault-free edges, we prove that Qn keeps this strong property even if it has up to 3(n – 2) – 1 faulty edges and Sn will also keep this strong property no matter how many edges are faulty. Furthermore, we prove Qn keeps this strong property no matter how many edges are faulty, provided that each vertex of

(7)

number of faulty edges are all tight. Besides, we propose a new diagnosis algorithm for general systems. The time complexity of our algorithm to diagnose all the faulty processors is bounded by O(N log N), where N is the total number of processors.

The conditional diagnosability measure, introduced by Lai et al., is another interesting issue for multiprocessor systems. They proposed this novel measure of diagnosability by adding an additional condition that any faulty set cannot contain all the neighbors of any vertex in a system. In this thesis, We make a contribution to the evaluation of diagnosability for hypercube networks under the comparison model and prove that the conditional diagnosability of n-dimensional hypercube Qn is 3(n – 2) + 1 for n ≥ 5. The conditional diagnosability of Qn is about three times larger than the classical diagnosability of Qn. Furthermore, we extend the result to bijective connection network (in brief, BC network). An

n-dimensional BC network, denoted by Xn, is an n-regular graph with 2n vertices and n2n-1

edges. The n-dimensional hypercube, crossed cube, twisted cube, and Mőbius cube are some examples of the n-dimensional BC networks. In this thesis, we also prove that the conditional diagnosability of Xn is 3(n – 2) + 1 under the comparison model, n ≥ 5. As a corollary of this result, we obtain the conditional diagnosability of the cube family.

Keywords: PMC model, comparison model, t-diagnosable, diagnosability, local

diagnosability, strongly local-diagnosable property, conditional faulty set, conditional diagnosability, hypercube network, star graph, BC network, diagnosis algorithm.

(8)

Contents

1 Introduction 1

1.1 Basic Terms and Notations . . . 4

1.2 Organization of the Thesis . . . 6

2 Diagnosis Model 7 2.1 The PMC Model and Some Previous Results . . . 7

2.2 The Comparison Model and Some Previous Results . . . 10

3 Local Diagnosability 14 3.1 The Local Diagnosability of Hypercube under the PMC Model . . . 21

3.2 The Local Diagnosability of Star Graph under the PMC Model . . . 23

3.3 Strongly Local-diagnosable Property . . . 25

3.4 Conditional Fault Local Diagnosability . . . 29

3.5 A Diagnosis Algorithm . . . 39

4 Conditionally Diagnosable Systems 42 4.1 Conditional Diagnosability of Hypercube under the Comparison Model . . 44 4.2 Conditional Diagnosability of BC Networks under the Comparison Model . 53

(9)

List of Figures

2.1 Illustration for a distinguishable pair (F1, F2) . . . 9

2.2 Description of distinguishability for Theorem 3. . . 12

3.1 A Type I structure T1(v; k) consists of 2k + 1 vertices and 2k edges. . . 18

3.2 A Type II structure T2(v; k, 2) consists of 2k + 5 vertices and 2k + 5 edges. 19 3.3 A cycle of length four and a Type I structure T1(v; 1) of order 1 at v. . . . 20

3.4 Some examples of local diagnosability. . . 21

3.5 A Q3 and a Type I structure T1(v; 3) of order 3 at vertex v. . . 23

3.6 The star graph S2, S3 and S4. . . 24

3.7 An indistinguishable pair (F1, F2) in Q2. . . 25

3.8 An indistinguishable pair (F1, F2), where |F1| = |F2| = n. . . 26

3.9 Q3 with one missing edge. The number labeled on each vertex represents its local diagnosability. . . 27

3.10 A 3-regular graph without the strong local diagnosability property after removing one edge. . . 29

3.11 An indistinguishable pair (A1, A2), where |A1| = |A2| = n. . . 30

3.12 The bipartite graph BG(v). . . 31

3.13 Illustration for the case 2 of Theorem 16 and Theorem 18. . . 33

3.14 A Type II structure T2(v; k − 2, 2) of order k at vertex v. . . 33

(10)

3.16 four different output states. . . 39

4.1 An indistinguishable pair (F1, F2). . . 42

4.2 An indistinguishable conditional-pair (F1, F2). . . 44

4.3 An indistinguishable conditional-pair (F1, F2), where |F1| = |F2| = 3(n −

2) + 2. . . 45 4.4 Two indistinguishable conditional-pairs for Q3 and Q4. . . 52

(11)

Chapter 1

Introduction

With the continuous increase in the size of a multiprocessor system, the complexity of the system can adversely affect its reliability. In order to maintain reliability, the system should be able to identify faulty processors and replace them with fault-free ones. The process of identifying faulty processors is called the diagnosis of the system, and the diagnosability of the system refers to the maximum number of faulty processors that can be identified by the system. The problem of identifying faulty processors in a multipro-cessor system has been widely studied in literatures [1, 3, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 25, 28, 29, 30, 31, 32, 33, 38, 45, 48, 49, 53, 54, 55, 58]. There are two funda-mental approaches to system-level diagnosis: tested-based diagnosis (PMC model) and comparison-based diagnosis (comparison model). In 1967, the Preparata, Metze, and Chien (PMC) model was proposed for system-level diagnosis in multiprocessor systems [45]. The PMC model uses tested-based diagnosis approach, under which a processor performs the diagnosis by testing on neighboring processors via the communication links between them. By analyzing the collection of all testing results, all of the faulty proces-sors are identified. The PMC model was also used [4, 5, 6, 8, 26, 27, 34, 35, 36, 45]. In [26], Hakimi and Amin proved that a system is t-diagnosable if it is t-connected with at least 2t + 1 vertices. They also gave a necessary and sufficient condition for verifying if a system is t-diagnosable under the PMC model.

(12)

multi-processor systems. An n-dimensional hypercube is denoted by Qn, and the diagnosability

of Qn is shown to be n [35] under the PMC model, n ≥ 3. An n-dimensional star graph

is denoted by Sn, and the diagnosability of Sn is shown to be n − 1 under the PMC

model, n ≥ 3 [37]. In [40], Lai et al. introduced a novel measure of diagnosability called conditional diagnosability by restricting that a faulty set cannot contain all the neighbors of any vertex. Based on this restriction, the conditional diagnosability of the n-dimensional hypercube is shown to be 4(n − 2) + 1. Besides, Lai et al. introduced a concept called a strongly t-diagnosable systems and proved that the n-dimensional hy-percube is strongly n-diagnosable. Essentially, it means that an n-dimensional hyhy-percube is almost (n + 1)-diagnosable except for the case where all the neighbors of some vertex are faulty simultaneously. In [50], Wang proved that the diagnosability of an incomplete hypercube under some conditions can be determined by simply checking the degree of each vertex under the PMC model. An incomplete hypercube is a hypercube with some missing edges. It is also called a faulty hypercube. There are some results concerning the diagnosability of several variations of the hypercube [4, 10, 21, 22, 26, 35, 50]. In clas-sical measures of system-level diagnosability for multiprocessor systems, it has generally been assumed that any subset of processors can potentially fail at the same time. As a consequence, the diagnosability of a system is upper bounded by its minimum degree.

We observe that the diagnosability of a system discussed in previous literatures are all in a global sense, but ignored some local information. A system is t-diagnosable if, all the faulty processors can be uniquely identified, provided that the number of faulty processors does not exceed t. However, it is possible to correctly indicate all the faulty processors in a t-diagnosable system when the number of faulty processors is greater than t. For example, consider a multiprocessor system generated by integrating two arbitrary subsystems with a few communication links in some way, where the two subsystems are m-diagnosable and n-diagnosable, respectively, and m >> n. The diagnosability of this system is limited by n, but it is possible to correctly point out all the faulty processors even if the number of the faulty ones is between m and n. Therefore, if only considering the global faulty/fault-free status, we lose some local systematic details.

(13)

and study the local diagnosability of each processor of a system. We can identify the diagnosability of a system by computing the local diagnosability of each processor. This measure of the local diagnosability leads us to study the local diagnosability of each processor instead of the whole system. We propose a necessary and sufficient condition, Theorem 6, to determine the local diagnosability of a processor. We also provide two useful structures, called T ype I structure and T ype II structure, to determine the local diagnosability of a processor under the PMC model. Based on these structures, the local diagnosability of each vertex of hypercube and star graph is shown to be equal to its own degree. Then, we propose a concept for system diagnosis, called strongly local-diagnosable property. A system G(V, E) is said to have a strongly local-local-diagnosable property, if the local diagnosability of each vertex is equal to its degree. We show that an n-dimensional hypercube Qn and an n-dimensional star graph Sn all have this strong

property. Then, we study the local diagnosability of an incomplete hypercube and an incomplete star graph. Firstly, we show that both Qn and Sn keep this strong property

even if it has up to n − 2 faulty edges and n − 3 faulty edges, respectively. Secondly, assume that each vertex of an incomplete hypercube Qnand an incomplete star graph Sn

is incident with at least two fault-free edges, we show Qn keeps this strong property even

if it has up to 3(n−2)−1 faulty edges and Snwill also keep this strong property no matter

how many edges are faulty. Furthermore, we show that Qn keeps this strong property no

matter how many edges are faulty, provided that each vertex of an incomplete hypercube Qn is incident with at least three fault-free edges. Our bounds on the number of faulty

edges are all tight. Besides, we propose a new diagnosis algorithm for general systems. The time complexity of our algorithm to diagnose all the faulty processors is bounded by O(N log N), where N is the total number of processors.

In 1980, Malek and Maeng introduced the comparison model using Comparison-based diagnosis approach, also known as the MM model [42, 43]. In this model, the number of faulty processors is limited and all faults are permanent. The MM model deals with the faulty diagnosis by sending the same input (or task) from a processor w to each pair of distinct neighbors, u and v, and then comparing their responses. The processor w is called the comparator of processors u and v. Different comparators may examine the same pair of processors. The result of the comparison is either the two responses agreed

(14)

or two responses disagreed. Based on the results of all the comparisons, one need to decide the faulty or fault-free status of the processors in the system. Using a comparison diagnosis model, Sengupta and Dahbura described a diagnosable system and presented a polynomial algorithm to determine the set of all faulty processors [47].

Reviewing some previous literatures [4, 10, 21, 22, 23, 26, 35, 39, 41, 46, 51], Qn, CQn,

T Qnand MQn, all have diagnosability n under the comparison model or the PMC model.

The diagnosability of the Star Sn is shown to be n − 1 under the comparison model [56].

In classical measures of system-level diagnosability for multiprocessor systems, if all the neighbors of some processor v are faulty simultaneously, it is not possible to determine whether processor v is fault-free or faulty. As a consequence, the diagnosability of a system is limited by its minimum degree. Hence, Lai et al. introduced a restricted diagnosability of multiprocessor systems called conditional diagnosability in [40]. Lai et al. considered a measure by restricting that, for each processor v in a system, all the processors which are directly connected to v do not fail at the same time. In this thesis, We make a contribution to the evaluation of diagnosability for hypercube networks under the comparison model and prove that the conditional diagnosability of n-dimensional hypercube Qn is 3(n − 2) + 1 for n ≥ 5. The conditional diagnosability of Qn is about

three times larger than the classical diagnosability of Qn. Furthermore, we extend the

result to bijective connection network (in brief, BC network). An n-dimensional BC network, denoted by Xn, is an n-regular graph with 2n vertices and n2n−1 edges. The

n-dimensional hypercube, crossed cube, twisted cube, and M¨obius cube are some examples of the n-dimensional BC networks. In this thesis, we also prove that the conditional diagnosability of Xn is 3(n − 2) + 1 under the comparison model, n ≥ 5. As a corollary

of this result, we obtain the conditional diagnosability of the cube family.

1.1

Basic Terms and Notations

A multiprocessor system can be represented by a graph G(V, E), where the set of vertices V (G) represents processors and the set of edges E(G) represents communication links between processors. Throughout this thesis, we focus on undirected graph without loops

(15)

and follow [52] for graph theoretical definitions and notations.

Let G(V, E) be a graph and v ∈ V (G) be a vertex. We use the notation EG(v) to

denote the set of edges incident with v. The cardinality |EG(v)| is called the degree of

v, denoted by degG(v) or simply deg(v). The maximum degree is denoted by ∆(G), the

minimum degree is δ(G), and G is regular if ∆(G) = δ(G). G is d-regular if deg(v) = d for every v ∈ V (G). The neighborhood N(v) of a vertex v in G is the set of all vertices that are adjacent to v in G. For a subset of vertices V′ ⊂ V (G), the neighborhood set of

the vertex set V′ is defined as N(V) = S

v∈V′

N(v) − V′. For a set of edges(respectively,

vertices) F , we use the notation G − F to denote the graph obtained from G by removing all the edges(respectively, vertices) in F . The components of a graph G are its maximal connected subgraphs. A component is trivial if it has no edges; otherwise, it is nontrivial. The connectivity κ(G) of a graph G(V, E) is the minimum number of vertices whose removal results in a disconnected or a trivial graph. Let G1 be a subgraph of G, we

shall write the vertex set of G1 as V (G1). The neighborhood set of V (G1) is defined as

N(V (G1)) = {u ∈ V (G) − V (G1) | there exists a vertex v ∈ V (G1) such that (u, v) ∈

E(G)}. The following is an useful characterization for the distinguishability of two sets of vertices under the PMC model and the comparison model. Let F1, F2 ⊆ V (G) be two

distinct sets. The symmetric difference of the two sets F1 and F2 is defined as the set

F1∆F2 = (F1− F2)S(F2− F1).

For studying the conditional diagnosability of a system, we also need some definitions for further discussion. Let G(V, E) be a graph. For any set of vertices U ⊆ V (G), G[U] denotes the subgraph of G induced by the vertex subset U. Let H be a subgraph of G and v be a vertex in H. We use V (H; 3) = {v ∈ V (H) | degH(v) ≥ 3} to represent the

set of vertices which has degree 3 or more in H. Let F1, F2 ⊆ V (G) be two distinct sets

and S = F1T F2. We use CF1∆F2,S to denote the subgraph induced by the vertex subset

(16)

1.2

Organization of the Thesis

The rest of this thesis is organized as follows. The details for the PMC model and the comparison model are described in Chapter 2, and the previous results for diagnosing a system are also provided in this chapter as well.

In Chapter 3, we introduce the concept of local diagnosability and propose a necessary and sufficient condition for verifying if it is locally t-diagnosable at a given processor in a system. Then, we define a strongly local-diagnosable property for a system and study the strong property in a faulty hypercube and a faulty star graph respectively. Next, we study the strong property in a conditional faulty hypercube and star graph. A diagnosis algorithm is proposed at the end of this chapter.

In Chapter 4, we focus on the measure of conditional diagnosability we study the conditional diagnosability of the hypercube Qn under the comparison model. Finally, our

(17)

Chapter 2

Diagnosis Model

The process of identifying faulty processors in a system is known as the system-level diagnosis. Several different approaches have been developed to diagnose faulty processors, among which there are two fundamental approaches on system-level diagnosis. One major approach is called PMC model established by Preparata, Metze and Chien [45]. Another major approach is the comparison model, proposed by Malek and Maeng [42, 43]. In the following, we describe the details of the two major models and give some previous results for diagnosing a system.

2.1

The PMC Model and Some Previous Results

The PMC diagnosis model is presented by Preparata, Metze and Chien [45]. In this model, a self-diagnosable system is often represented by a directed graph T (V, E) in which an edge directed from vertex u to vertex v means that u can test v. In this situation, u is called the tester and v is called the tested vertex. The outcome of a test (u, v) is 1(respectively, 0) if u evaluates v as faulty (respectively, fault-free). We assume that the testing results of fault-free vertices are always reliable and the testing results of faulty vertices are unreliable. The collection of all testing results is called a syndrome. Formally, a syndrome is a function σ : E → {0, 1}. The set of all faulty processors in the system is called a f aulty set. This can be any subset of V (T ). For a given syndrome σ, a subset

(18)

of vertices F ⊂ V (T ) is compatible with σ if the syndrome σ can be produced from the situation that all vertices in F are faulty and all vertices in V − F are fault-free. Since faulty testers can give arbitrary testing results, any syndrome compatible with a faulty set F can occur when faulty processors in the system are exactly those in F . Let σF be the set

of all syndromes which could be produced if F is the set of faulty vertices. Two distinct sets F1, F2 ⊆ V (G) are said to be distinguishable if σF1T σF2 = φ; otherwise, F1, F2 are

said to be indistinguishable. We say (F1, F2) is a distinguishable pair if σF1T σF2 = φ;

otherwise, (F1, F2) is an indistinguishable pair. For PMC model, some known results

about the definition of t-diagnosable system and related concepts are listed as follows. Some of these previous results are on directed graphs and others are on undirected.

Definition 1 [45] A system G is called t-diagnosable if, given the test outcomes obtained by the testing link, all the faulty vertices can be uniquely identified without replacement, provided that the number of faulty vertices does not exceed t.

Definition 2 [45] The maximum number of faulty vertices that a system G can guarantee to identify is called the diagnosability of G, written as t(G).

Dahbura and Masson [19] proposed a polynomial time algorithm to check whether a system is t-diagnosable.

Lemma 1 [19] A system G(V, E) is t-diagnosable under the PMC model if and only if for each pair F1, F2 ⊂ V with |F1|, |F2| ≤ t and F1 6= F2, there is at least one test from

V − (F1S F2) to F1∆F2.

The following two lemmas related to t-diagnosable systems are proposed by Preparata et al. [45] and Hakimi et al. [26], respectively.

Lemma 2 [45] Let G(V, E) be a graph and |V | = N. The following two conditions are necessary for G to be t-diagnosable;

(19)

2. each processor in G is tested by at least t other processors.

Lemma 3 [26] Let G(V, E) be a graph and |V | = N. G is t-diagnosable if 1. N ≥ 2t + 1, and

2. κ(G) ≥ t.

For a directed graph G(V, E) and vertex v ∈ V , let Γ(v) = {vi|(v, vi) ∈ E} and

Γ(X) = S

v∈XΓ(v) − X, X ⊂ V. Hakimi and Amin presented a necessary and sufficient

condition for a system G to be t-diagnosable as follows:

Theorem 1 [26] Let G(V, E) be the directed graph of a system G and |V | = N. Then

G is t-diagnosable under the PMC model if and only if: (i) N ≥ 2t + 1, (ii) din(v) ≥ t

for all v ∈ V , and (iii) for each integer p with 0 ≤ p ≤ t − 1, and each X ⊂ V with

|X| = N − 2t + p, |Γ(X)| > p.

In this thesis, we propose some new concepts on diagnosis, and we focus on undirected graph. The following lemma follows directly from Lemma 1.

Lemma 4 [19] Let G(V, E) be a graph. For any two distinct sets F1, F2 ⊂ V , (F1, F2)

is a distinguishable pair under the PMC model if and only if there exists a vertex u ∈

V − (F1S F2) and a vertex v ∈ F1∆F2 such that (u, v) ∈ E (see Figure 2.1).

b b b b

u

v

u

v

F1 F2 F1 F2 F1 (i) (ii)

Figure 2.1: Illustration for a distinguishable pair (F1, F2)

(20)

Lemma 5 [19] A system G(V, E) is t-diagnosable under the PMC model if and only if, for any two distinct sets F1, F2 ⊂ V with |F1| ≤ t and |F2| ≤ t, (F1, F2) is a distinguishable

pair.

The following Lemma 6 is equivalent to Lemma 5.

Lemma 6 [19] A system G(V, E) is t-diagnosable if and only if, for each indistinguishable pair F1, F2 ⊂ V , it implies that |F1| > t or |F2| > t.

By Lemma 2, a similar result for undirected graph is stated as follows.

Corollary 1 [45] Let G(V, E) be an undirected graph and |V | = N. The following two conditions are necessary for G to be t-diagnosable under the PMC model:

1. N ≥ 2t + 1, and 2. δ(G) ≥ t.

For our discussion later, a useful result presented by Lai [40] is stated below.

Theorem 2 [40] Let G(V, E) be a graph. G is t-diagnosable if and only if, for each set of vertices F ⊂ V with |F | = p, 0 ≤ p ≤ t − 1, each connected component of G − F has at least 2(t − p) + 1 vertices.

2.2

The Comparison Model and Some Previous

Re-sults

The comparison diagnosis model is proposed by Malek and Maeng [42, 43]. In this model, a self-diagnosable system is often represented by a multigraph M(V, C), where V is the same vertex set defined in G and C is the labeled edge set. Let (u, v)w be a labeled

(21)

edge. If (u, v) is an edge labeled by w, then (u, v)w is said to belong to C, which implies

that the vertex u and v are being compared by vertex w. The same pair of vertices may be compared by different comparators, so M is a multigraph. For (u, v)w ∈ C, we use

r((u, v)w) to denote the result of comparing vertices u and v by w such that r((u, v)w) = 0

if the outputs of u and v agree, and r((u, v)w) = 1 if the outputs disagree. In this model,

if r((u, v)w) = 0 and w is fault-free, then both u and v are fault-free. If r((u, v)w) = 1,

then at least one of the three vertices u, v, w must be faulty. If the comparator w is faulty, then the result of the comparison is unreliable that means both r((u, v)w) = 0 and

r((u, v)w) = 1 are possible outputs, and it outputs only one of these two possibilities. In

this thesis, we consider a complete diagnosis that means each vertex diagnoses all pairs of distinct neighbors. For an n-dimensional hypercube Qn, each vertex has degree n, and

therefore, there are n

2 comparisons for each vertex acting as a comparator. Furthermore,

there are 2n vertices in Q

n so the total number of comparisons is n22n= O(n22n).

As the description for the PMC model, the collection of all comparison results defined as a function σ: C → {0, 1}, is called the syndrome of the diagnosis. A subset F ⊂ V is said to be compatible with a syndrome σ if σ can arise from the circumstance that all vertices in F are faulty and all vertices in V − F are fault-free. A system is said to be diagnosable if, for every syndrome σ, there is a unique F ⊂ V that is compatible with σ. In [47], a system is called a t-diagnosable system if the system is diagnosable as long as the number of faulty vertices does not exceed t. The maximum number of faulty vertices that the system G can guarantee to identify is called the diagnosability of G, written as t(G). A faulty comparator can lead to unreliable results. So, a set of faulty vertices may produce different syndromes. Let σF = {σ | σ is compatible with F }. Two distinct sets

F1, F2 ⊂ V are said to be indistinguishable if and only if σF1T σF2 6= ∅; otherwise, F1, F2

are said to be distinguishable. There are several different ways to verify a system to be t-diagnosable under the comparison approach. The following theorem given by Sengupta and Dahbura [47] is a necessary and sufficient condition for ensuring distinguishability.

Theorem 3 [47] Let G(V, E) be a graph. For any two distinct sets F1, F2 ⊂ V , (F1, F2)

is a distinguishable pair under the comparison model if and only if at least one of the following conditions is satisfied (see Figure 2.2):

(22)

1. ∃u, w ∈ V − {F1S F2} and ∃v ∈ F1△F2 such that (u, v)w ∈ C,

2. ∃u, v ∈ F1 − F2 and ∃w ∈ V − {F1S F2} such that (u, v)w ∈ C, or

3. ∃u, v ∈ F2 − F1 and ∃w ∈ V − {F1S F2} such that (u, v)w ∈ C.

b b b b b b b b b b b b

F1

F2

V

(2) (3) (1) (1)

Figure 2.2: Description of distinguishability for Theorem 3.

The following result is a useful sufficient condition for checking whether (F1, F2) is a

distinguishable pair.

Theorem 4 Let G(V, E) be a graph. For any two distinct sets F1, F2 ⊂ V with |Fi| ≤ t,

i = 1, 2, and S = F1T F2. (F1, F2) is distinguishable under the comparison model if, the

subgraph CF1∆F2,S of G − S contains at least 2(t − |S|) + 1 vertices having degree 3 or

more.

Proof.

Given any pair of distinct sets of vertices F1, F2 ⊂ V with |Fi| ≤ t, i = 1, 2. Let

S = F1T F2, then 0 ≤ |S| ≤ t − 1, and |F1∆F2| ≤ 2(t − |S|). Consider the subgraph

CF1∆F2,S, the number of vertices having degree 3 or more is at least 2(t − |S|) + 1 in

CF1∆F2,S, the subgraph CF1∆F2,S contains at least 2(t − |S|) + 1 vertices. There is at least

one vertex with degree 3 or more lying in CF1∆F2,S− F1∆F2. Let u be one of such vertices

with degree 3 or more. Let i, j, and k be three distinct vertices linked to u. If one of i, j, and k lies in CF1∆F2,S− F1∆F2, condition 1 of Theorem 3 holds obviously. Suppose all

these three vertices belong to F1∆F2. Without loss of generality, assume i lies in F1− F2,

(23)

or, 2) if j lies in F2− F1, wherever k lies in F1− F2 or F2− F1, condition 2 or 3 of Theorem

3 holds. So (F1, F2) is a distinguishable pair and the proof is complete. 2

By Theorem 4, we now propose a sufficient condition to verify whether a system is t-diagnosable under the comparison diagnosis model.

Corollary 2 Let G(V, E) be a graph. G is t-diagnosable under the comparison model if, for each set of vertices S ⊂ V with |S| = p, 0 ≤ p ≤ t − 1, every connected component C of G − S contains at least 2(t − p) + 1 vertices having degree at least three. More precisely,

(24)

Chapter 3

Local Diagnosability

We first review some related results on system diagnosability of some well-known networks under the PMC model. In[35], Kavianpour et al. proved that the diagnosability of an n-dimensional hypercube Qnis n. In [21] and [22], Fan proved that an n-dimensional Crossed

cube and an n-dimensional M¨obius cube have diagnosability n under the PMC model. In [50], Wang proved that the diagnosability of a faulty hypercube can be determined by checking the degree of each vertex under the PMC model, provided that the minimum degree of the faulty hypercube is at least three.

We observe that the traditional diagnosability discussed in most literatures describes the global status of a system. In this thesis, we study the local status of each processor instead of the global status of a system. For example, for any two positive integers m and n with m >> n ≥ 3, the diagnosability of two hypercube systems Qm and Qn is m

and n, respectively. Combining Qm and Qn with a few edges in some way may cause the

diagnosability of the new system to become n. In this situation, the strong diagnosability of Qm is disregarded. For this reason, we are motivated to study the local status of each

processor. Given a single vertex, we require only identifying the status of this particular processor correctly. We now propose the following concept.

Definition 3 Let G(V, E) be a graph and v ∈ V be a vertex. G is locally t-diagnosable at vertex v if, given a syndrome σF produced by a set of faulty vertices F ⊆ V containing

vertex v with |F | ≤ t, every set of faulty vertices Fcompatible with σ

(25)

also contain vertex v.

Definition 4 Let G(V, E) be a graph and v ∈ V be a vertex. The local diagnosability of vertex v, written as tl(v), is defined to be the maximum value of t such that G is locally

t-diagnosable at vertex v.

The following result is another point of view for checking whether a vertex is locally t-diagnosable.

Lemma 7 Let G(V, E) be a graph and v ∈ V be a vertex. G is locally t-diagnosable at vertex v if and only if, for any two distinct sets of vertices F1, F2 ⊂ V , |F1| ≤ t, |F2| ≤ t

and v ∈ F1∆F2, (F1, F2) is a distinguishable pair.

In the following, we study some properties of a system being locally t-diagnosable at a given vertex, and its relationship between a system being t-diagnosable.

Proposition 1 Let G(V, E) be a graph and v ∈ V (G) be a vertex. G is locally t-diagnosable at vertex v under the PMC model, then |V (G)| ≥ 2t + 1.

Proof.

We show this by contradiction. Assume that |V (G)| ≤ 2t. We partition V (G) into two disjoint subsets F1, F2 with |F1| ≤ t, |F2| ≤ t. The vertex v is either in F1 or in F2.

Since V − (F1S F2) = ∅, there is no edge between V − (F1S F2) and F1∆F2. By Lemma

4, (F1, F2) is an indistinguishable pair, this contradicts the assumption that G is locally

t-diagnosable at vertex v. So the result follows. 2

Proposition 2 Let G(V, E) be a graph and v ∈ V be a vertex with deg(v) = n. The local diagnosability of vertex v is at most n under the PMC model.

(26)

Let F1 be the set of vertices adjacent to vertex v, F1 = NG(v) and |F1| = n. Let

F2 = F1S{v} with |F2| = n + 1. It is a simple matter to check that there is no edge

between V − (F1S F2) and F1∆F2. By Lemma 4, (F1, F2) is an indistinguishable pair.

Thus, G is not locally (n + 1)-diagnosable at vertex v, so tl(v) ≤ n = deg(v). We have

the stated result. 2

Proposition 3 Let G(V, E) be a graph. Under the PMC model, G is t-diagnosable if and only if G is locally t-diagnosable at every vertex.

Proof.

To prove the necessity, we assume that G is t-diagnosable. If the result is not true, there exists a vertex v ∈ V such that G is not locally t-diagnosable at vertex v. By Lemma 7, there exists a distinct pair of sets F1, F2 ⊂ V with |F1| ≤ t, |F2| ≤ t and

v ∈ F1∆F2, (F1, F2) is an indistinguishable pair. By Lemma 5, G is not t-diagnosable.

This contradicts the assumption, hence the necessary condition follows.

To prove the sufficiency, suppose on the contrary that G is not t-diagnosable, there exists a distinct pair of sets F1, F2 ⊂ V with |F1| ≤ t, |F2| ≤ t, (F1, F2) is an

indistinguish-able pair. Being distinct, the set F1∆F2 6= ∅, we can find a vertex v ∈ F1∆F2. By Lemma

7, G is not locally t-diagnosable at vertex v, which is a contradiction. This completes the

proof. 2

By Definition 4 and Proposition 3, we know that the diagnosability of a multiprocessor system is equal to the minimum local diagnosability of all vertices of the system. Thus, we have the following theorem.

Theorem 5 Let G(V, E) be a multiprocessor system. Under the PMC model, the diag-nosability of G is t if and only if

min{tl(v) | for every v ∈ V } = t.

(27)

diagnosability of each vertex. Because many well-known systems are vertex-symmetric, the diagnosability of these system can be easily identified by this effective method.

Before studying the local diagnosability of a vertex, we need some definitions for further discussion. Let F be a set of vertices and v be a vertex not in F . After deleting the vertices in F from G, we use Cv to denote the connected component which vertex v

belongs to. Now, we propose a necessary and sufficient condition for verifying if a system is locally t-diagnosable at a given vertex v.

Theorem 6 Let G(V, E) be a graph and v ∈ V be a vertex. G is locally t-diagnosable at vertex v under the PMC model if and only if, for each set of vertices F ⊂ V with |F | = p,

0 ≤ p ≤ t − 1 and v /∈ F , the connected component, which v belongs to in G − F , has at

least 2(t − p) + 1 vertices.

Proof.

To prove the necessity, we assume that G is locally t-diagnosable at vertex v. If the result does not hold, there exists a set of vertices F ⊂ V with |F | = p, 0 ≤ p ≤ t − 1, v /∈ F such that the connected component Cv has strictly less than 2(t − p) + 1 vertices,

|V (Cv)| ≤ 2(t−p). We then arbitrarily partition V (Cv) into two disjoint subsets, V (Cv) =

F1S F2 with |F1| ≤ t − p, |F2| ≤ t − p. Let A1 = F1S F and A2 = F2S F . It is clear

that |A1| ≤ (t − p) + p = t, |A2| ≤ (t − p) + p = t, the vertex v ∈ A1∆A2 and there is no

edge between V − (A1S A2) and A1∆A2. By Lemma 7, (A1, A2) is an indistinguishable

pair. This contradicts the assumption that G is locally t-diagnosable at vertex v.

We now prove the sufficiency by contradiction. Suppose G is not locally t-diagnosable at vertex v, then, there exists an indistinguishable pair (F1, F2) with |F1| ≤ t, |F2| ≤ t

and v ∈ F1∆F2. By Lemma 4, there is no edge between V − (F1S F2) and F1∆F2. Let

F = F1T F2 with |F | = p, 0 ≤ p ≤ t − 1 and v /∈ F . F1∆F2 is disconnected from other

parts after removing all the vertices in F from G. We observe that |F1∆F2| ≤ 2(t − p).

Thus, the connected component Cv has at most 2(t − p) vertices and |V (Cv)| ≤ 2(t − p).

This contradicts the assumption that the connected component Cvhas to satisfy |V (Cv)| ≥

(28)

We now propose two special subgraphs called Type I structure and Type II structure. They provide us with an efficient and simple method to identify the local diagnosability of each vertex of a system under the PMC diagnosis model.

Definition 5 Let G(V, E) be a graph, v ∈ V be a vertex and k be an integer, k ≥ 1, a Type I structure T1(v; k) of order k at vertex v is defined to be the following graph,

T1(v; k) = [V (v; k), E(v; k)]

which is composed of 2k + 1 vertices and of 2k edges as illustrated in Figure 3.1, where

• V (v; k) = {v} ∪ {xi, yi | 1 ≤ i ≤ k}, • E(v; k) = {(v, xi), (xi, yi)|1 ≤ i ≤ k}. b b b b b b b

...

v

x1 y1 x2 y2 xk yk

Figure 3.1: A Type I structure T1(v; k) consists of 2k + 1 vertices and 2k edges.

Following Theorem 6 and Definition 5, we propose a sufficient condition for verifying if it is locally t-diagnosable at a given processor in a system.

Theorem 7 Let G(V, E) be a graph and v ∈ V be a vertex. G is locally t-diagnosable at vertex v under the PMC model if G contains a Type I structure T1(v; t) of order t at

vertex v as a subgraph.

Proof.

We use Theorem 6 to prove this result. Assume that G contains a subgraph T1(v; t) at

(29)

number of vertices of the connected component including vertex v is at least 2t + 1. Let F ⊂ V (G) be a set of vertices with |F | = p, 0 ≤ p ≤ t − 1 and v /∈ F . After deleting F from V (G), there are at least (t − p) complete ei’s still remain in T1(v; t). Therefore, the

number of vertices of the connected component Cv is at least 2(t − p) + 1. By Theorem

6, G is locally t-diagnosable at vertex v. The proof is complete. 2 A Type II structure T2(v; k, 2) at a vertex v is defined as follows:

Definition 6 Let G(V, E) be a graph, v ∈ V be a vertex and k be an integer, k ≥ 1, a Type II structure T2(v; k, 2) of order k + 2 at vertex v is defined to be the following graph,

T2(v; k, 2) = [V (v; k, 2), E(v; k, 2)]

which is composed of 2k + 5 vertices and of 2k + 5 edges as illustrated in Figure 3.2, where • V (v; k, 2) = {v} ∪ {xi, yi | 1 ≤ i ≤ k} ∪ {z1, z2, z3, z4}, • E(v; k, 2) = {(v, xi), (xi, yi)|1 ≤ i ≤ k} ∪ {(v, z1), (v, z2), (z1, z3), (z2, z3), (z3, z4)}. b b b b b b b b b b b

...

v

x1 y1 x2 y2 xk yk z1 z2 z3 z4

Figure 3.2: A Type II structure T2(v; k, 2) consists of 2k + 5 vertices and 2k + 5 edges.

In the following, we propose another sufficient condition for verifying if it is locally t-diagnosable at a given processor in a system.

Theorem 8 Let G(V, E) be a graph and v ∈ V be a vertex. G is locally t-diagnosable at vertex v under the PMC model if G contains a Type II structure T2(v; k, 2) of order k + 2

(30)

Proof.

We use Theorem 6 to prove this result. Assume that G contains a subgraph T2(v; k, 2)

of order t = k + 2 at vertex v. The number of vertices of the connected component including vertex v is at least 2k + 5 = 2t + 1. Let F ⊂ V be a set of vertices with |F | = p, 0 ≤ p ≤ t − 1 and v /∈ F , the number of vertices of Cv is at least (2k + 5) − 2 ∗ 1

after removing one vertex in F , the number of vertices of Cv is at least (2k + 5) − 2 ∗ 2

after removing two vertices in F , and so on. Thus, the connected component Cv satisfies

|V (Cv)| ≥ (2k + 5) − 2p = 2(t − p) + 1. By Theorem 6, G is locally t-diagnosable at vertex

v. This proves the theorem. 2 In the following, we give some examples.

Example 1 Let us consider a cycle of length four as shown in Figure 3.3(a). We can find a Type I structure T1(v; 1) of order 1 at vertex v as shown in Figure 3.3(b), hence

vertex v is locally 1-diagnosable.

b b b b b b b

v

v

(a)

(b)

Figure 3.3: A cycle of length four and a Type I structure T1(v; 1) of order 1 at v.

Example 2 Consider examples as shown in Figure 3.4(a), 3.4(b) and 3.4(c). It is a rou-tine work to check that there is a subgraph T1(v1; 2), T1(v2; 2) and T2(v3; 1, 2) at vertex v1,

v2 and v3, respectively. Hence it is locally 2-diagnosable, 2-diagnosable and 3-diagnosable

at vertex v1, v2 and v3, respectively.

(31)

b b b b b

v

1 b b b b b b

v

2 b b b b b b b b

v3

(a)

(b)

(c)

Figure 3.4: Some examples of local diagnosability.

Theorem 9 Let G(V, E) be a graph and v ∈ V be a vertex with deg(v) = n. The local diagnosability of vertex v is n under the PMC model if G contains a subgraph, which is either a Type I structure T1(v; n) of order n or a Type II structure T2(v; n − 2, 2) of order

n, at vertex v.

3.1

The Local Diagnosability of Hypercube under

the PMC Model

In this section, we study the local diagnosability of hypercube under the PMC model. An n-dimensional hypercube can be modeled as a graph Qn, with the vertex set V (Qn) and

the edge set E(Qn). There are 2n vertices in Qn, and each vertex has degree n. Each

vertex v of Qn can be distinctly labeled by a binary n-bit string, v = vn−1vn−2...v1v0.

There is an edge between two vertices if and only if their binary labels differ in exactly one bit position. Let u and v be two adjacent vertices. If the binary labels of u and v differ in ith position, then the edge between them is said to be in ith dimension and the edge (u, v) is called an ith dimensional edge. Let i be a fixed position, we use Q0n−1 to denote

the subgraph of Qn induced by {v ∈ V (Qn) | vi = 0} and Q1n−1 to denote the subgraph

of Qn induced by {v ∈ V (Qn) | vi = 1}. Consequently, Qn is decomposed to Q0n−1 and

Q1

n−1 by dimension i, and Q0n−1 and Q1n−1 are (n − 1)-dimensional subcube of Qn induced

by the vertices with the ith bit position being 0 and 1 respectively. Q0

(32)

isomorphic to Qn−1. For each vertex v ∈ V (Q0n−1), there is exactly one vertex in Q1n−1,

denoted by v(1), such that (v, v(1)) ∈ E(Q

n). Conversely, for each vertex v ∈ V (Q1n−1),

there is exactly one vertex in Q0

n−1, denoted by v(0), such that (v, v(0)) ∈ E(Qn). Let Di

be the set of all edges with one end in Q0

n−1 and the other in Q1n−1. These edges are called

crossing edges in the ith dimension between Q0

n−1 and Q1n−1. We also call Di the set of

all ith dimensional edges.

Based on Theorem 9, we prove that the local diagnosability of each vertex in Qn is

equal to its degree.

Theorem 10 Let Qn be an n-dimensional hypercube. The local diagnosability of each

vertex in Qn is n under the PMC model, for n ≥ 3.

Proof.

We use Theorem 9 to prove this result, and we shall construct a Type I structure of order n at each vertex, for n ≥ 3. We prove this by induction on n. Since an n-dimensional hypercube Qn is vertex-symmetric, we can concentrate on the construction

of Type I structure at a given vertex v. For n = 3, deg(v) = 3 and it is clear that Q3 contains a Type I structure T1(v; 3) of order 3 at vertex v (see Figure 3.5). As the

inductive hypothesis, we assume that Qn−1 contains a Type I structure T1(v; n − 1) of

order n − 1 at each vertex, for some n ≥ 4. Now we consider Qn, Qn can be decomposed

into two subcubes Q0

n−1 and Q1n−1 by some dimension. Without loss of generality, we may

assume that the vertex v ∈ Q0

n−1. By the inductive hypothesis, Q0n−1 contains a Type I

structure T1(v; n − 1) of order n − 1 at vertex v. Consider the vertex v(1) in Q1n−1. Vertex

v(1) has an adjacent neighbor that is in Q1

n−1 due to deg(v(1)) = n, where n ≥ 3. Thus,

Qn contains a Type I structure T1(v; n) of order n at vertex v. By Theorem 9, the local

(33)

b b b b b b b b

b b b b b b b b

(a)

(b)

v

v

Figure 3.5: A Q3 and a Type I structure T1(v; 3) of order 3 at vertex v.

3.2

The Local Diagnosability of Star Graph under

the PMC Model

In this section, we study the local diagnosability of star graph under the PMC model. An n-dimensional star graph Sn is an (n − 1)-regular graph consisting of n! vertices and

(n − 1)n!/2 edges. The set of vertices V (Sn) = {u1u2...un|ui ∈ hni and ui 6= uj for i 6= j},

where hni is the set {1, 2, ..., n}. The adjacency is defined as follows: u1u2...ui...un is

adjacent to v1v2...vi...vn through an edge of dimension i, if v1 = ui, vi = u1, and vj = uj

for j /∈ {1, i}, where 2 ≤ i ≤ n. Let u = u1u2...ui...un be any vertex in Sn. We use (u)i

to denote the ith coordinate ui of u and Sn{i} to denote the ith subgraph of Sninduced by

those vertices u with (u)n = i. Obviously, Sn can be decomposed into n vertex disjoint

subgraphs Sn{i} for 1 ≤ i ≤ n, such that each Sn{i} is isomorphic to Sn−1. Thus, the

star graph can be constructed recursively. By the definition of Sn, there is exactly one

neighbor v of u such that u and v are adjacent through an edge of dimension i, for each 2 ≤ i ≤ n. For example, S4 contains 4! vertices in which two vertices u1u2u3u4 and

u4u2u3u1 are neighbors and joined through an edge of dimensional 4. Let (u)i denote the

unique i-neighbor of u. We have ((u)i)i = u and (u)n ∈ S{(u)1}

n . For 1 ≤ i, j ≤ n and

i 6= j, we use Ei,j to denote the set of edges between S{i}

n and Sn{j}. The star graph S2,

S3 and S4 are shown in Figure 3.6.

Based on Theorem 9, we prove that the local diagnosability of each vertex in Sn is

(34)

b c b c b c bc b c b c bc b c b c bc b c bc b c b c bc b c b b b b b b b b b b b b b b b b

S2

12 21

S3

123 213 312 132 231 321

S4

1234 4231 3214 2134 3241 2431 2314 3124 2341 3421 1324 4321 3412 2413 4312 1432 4213 1423 1342 4132 1243 4123 3142 2143 a f b e d b c f a c d e

Figure 3.6: The star graph S2, S3 and S4.

Theorem 11 Let Sn be an n-dimensional star graph. The local diagnosability of each

vertex in Sn is n − 1 under the PMC model, for n ≥ 3.

Proof.

We shall construct a Type I structure of order n − 1 at each vertex, for n ≥ 3. We prove this by induction on n. Since an n-dimensional star graph Sn is vertex-symmetric,

we can concentrate on an arbitrary vertex v = v1v2...vn. For n = 3, deg(v) = 2 and it is

clear that S3 contains a Type I structure T1(v; 2) of order 2 at vertex v. As the inductive

hypothesis, we assume that Sn−1 contains a Type I structure T1(v; n − 2) of order n − 2

at each vertex, for some n ≥ 4. Now we consider Sn. By the definition of star graphs,

Sn can be decomposed into n subgraphs S {v1} n , S{v 2} n , ..., and S{v n} n . So v ∈ S{v n} n . By

the inductive hypothesis, S{vn}

n contains a Type I structure T1(v; n − 2) of order n − 2

at vertex v. Consider the vertex (v)n in S{v1}

n . Vertex (v)n has at least one adjacent

neighbor in S{v1}

n due to deg((v)n) = n − 1, where n ≥ 3. Thus, Sn contains a Type I

structure T1(v; n − 1) of order n − 1 at vertex v. By Theorem 9, the local diagnosability

(35)

3.3

Strongly Local-diagnosable Property

In this section, we use hypercube as an example to introduce our concept of the strongly local-diagnosable property. In previous section, we presented two sufficient conditions, Theorem 7 and Theorem 8, for identifying the local diagnosability of a vertex. It seems that identifying the local diagnosability of a vertex is the same as counting its degree. We give an example to show that this is not true in general. As shown in Figure 3.7, we take a vertex v in two-dimensional hypercube Q2, let F1 = {v, 1} and F2 = {2, 3} with

|F1| = 2 and |F2| = 2. It is a simple matter to check that (F1, F2) is an indistinguishable

pair. Hence tl(v) 6= deg(v) = 2. We then propose the following two concepts.

b b b b

F1

F2

v

1 2 3

Figure 3.7: An indistinguishable pair (F1, F2) in Q2.

Definition 7 Let G(V, E) be a graph and v ∈ V be a vertex. Vertex v has the strongly local-diagnosable property if the local diagnosability of vertex v is equal to its degree.

Definition 8 Let G(V, E) be a graph. G has the strongly local-diagnosable property if, every vertex in the graph G has the strongly local- diagnosable property.

Following Definition 7, Definition 8, Theorem 10 and Theorem 11 imply the following two propositions.

Proposition 4 Let Qn be an n-dimensional hypercube, n ≥ 3. Qn has the strongly

local-diagnosable property under the PMC model.

Proposition 5 Let Sn be an n-dimensional star graph, n ≥ 3. Sn has the strongly

(36)

We now consider a system which is not vertex-symmetric. Let G(V, E) be a graph and F ⊂ E(G) be a set of edges. Removing the edges in F from G, the degree of each vertex in the resulting graph G − F is called the remaining degree of v, and is denoted by degG−F(v). We consider a faulty hypercube Qn with a faulty set F ⊂ E(Qn), n ≥ 3. We

shall prove that Qnhas the strongly local-diagnosable property even if it has up to (n − 2)

faulty edges. The number n − 2 is optimal in the sense that a faulty hypercube Qn cannot

be guaranteed to have this strong property if there are n − 1 faulty edges. As shown in Figure 3.8, we take a vertex v ∈ V (Qn) and a vertex x which is an adjacent neighbor of v.

Let F = {(y, x) ∈ E(Qn) | vertex y is directly adjacent to x}−{(v, x)}, then |F | = n − 1

and the remaining degree of v in Qn − F is n. Let F1 = (NQn−F(v) − {x})S{v} and

F2 = NQn−F(v), then |F1| = |F2| = n and v ∈ F1∆F2. It is clear that there is no edge

between V − (F1S F2) and F1∆F2. By Lemma 4, (F1, F2) is an indistinguishable pair,

hence tl(v) 6= degQn−F(v) = n. Therefore, Qn− F may not have this strong property, if

|F | ≥ n − 1. b b b b b b b b b b

(a)

(b)

: : :

x

v

v

x

|F |=n−1 F1 F2 faulty edges

Figure 3.8: An indistinguishable pair (F1, F2), where |F1| = |F2| = n.

Theorem 12 Let Qn be an n-dimensional hypercube with n ≥ 3, and F ⊂ E(Qn) be a set

of edges, 0 ≤ |F | ≤ n − 2. Removing all the edges in F from Qn, the local diagnosability

of each vertex is still equal to its remaining degree under the PMC model.

Proof.

We use Theorem 9 to prove this result, and we shall construct a Type I structure at each vertex. We prove this by induction on n. For n = 3, 0 ≤ |F | ≤ 1, if |F | = 0,

(37)

it is clear that Q3 contains a Type I structure T1(v; 3) of order 3 at every vertex. If

|F | = 1, a three-dimensional hypercube Q3 with one missing edge is shown in Figure 3.9.

It is a routine work to see that every vertex has a Type I structure T1(v; k) of order k

at it, where k is the remaining degree of the vertex. As the inductive hypothesis, we assume that the result is true for Qn−1, 0 ≤ |F | ≤ (n − 1) − 2, for some n ≥ 4. Now

we consider Qn, 0 ≤ |F | ≤ n − 2. If |F | = 0, refer to the proof of Theorem 10, Qn

contains a Type I structure T1(v; n) of order n at every vertex. If 1 ≤ |F | ≤ n − 2, we

choose an edge in F , the edge is in some dimension, decomposing Qn into two subcubes

Q0n−1 and Q1n−1 by this dimension, such that the edge is a crossing edge. Consider a

vertex v ∈ V (Qn). Let F0 = FT E(Q0n−1), 0 ≤ |F0| ≤ (n − 3) and F1 = F T E(Q1n−1),

0 ≤ |F1| ≤ (n − 3). Without loss of generality, we may assume that the vertex v is in

Q0

n−1 and degQ0

n−1−F0(v) = k. By the inductive hypothesis, Q

0

n−1 − F0 contains a Type

I structure T1(v; k) at v. Consider the crossing edge (v, v(1)). If (v, v(1)) ∈ F , Qn− F

contains a Type I structure T1(v; k) of order k at vertex v. If (v, v(1)) /∈ F , the remaining

degree of v in Qn− F is k + 1 and the vertex v(1) has at least an adjacent neighbor in Q1n−1

due to 0 ≤ |F1| ≤ (n − 1) − 2. Therefore, Qn− F contains a Type I structure T1(v; k + 1)

of order k + 1 at vertex v. By Theorem 9, removing all the edges in F from Qn, the local

diagnosability of each vertex is still equal to its remaining degree. 2

b

c

b

c

b

c

b

c

b

c

b

c

b

c

b

c

3 2 3 2 3 3 3 3

Figure 3.9: Q3 with one missing edge. The number labeled on each vertex represents its

local diagnosability.

We have the following corollary.

Corollary 3 Let Qn be an n-dimensional hypercube with n ≥ 3, and F ⊂ E(Qn) be a

set of edges, 0 ≤ |F | ≤ n − 2. Then, Qn− F has the strongly local-diagnosable property

(38)

We now consider a faulty star graph Snwith a faulty set F ⊂ E(Sn), n ≥ 3. Similarly,

we shall prove that Sn has the strongly local-diagnosable property even if it has up to

(n − 3) faulty edges and the number (n − 3) is also optimal.

Theorem 13 Let Sn be an n-dimensional star graph with n ≥ 3, and F ⊂ E(Sn) be a set

of edges, 0 ≤ |F | ≤ n − 3. Removing all the edges in F from Sn, the local diagnosability

of each vertex is still equal to its remaining degree under the PMC model.

Proof.

We prove this result by constructing a Type I structure T1 at each vertex. We prove

this by induction on n. For n = 3, |F | = 0, it is clear that S3 contains a Type I structure

T1(v; 2) of order 2 at every vertex. As the inductive hypothesis, we assume that the

result is true for Sn−1, 0 ≤ |F | ≤ (n − 1) − 3, for some n ≥ 4. Now we consider Sn,

0 ≤ |F | ≤ n − 3. If |F | = 0, refer to the proof of Theorem 11, Sn contains a Type I

structure T1(v; n − 1) of order n − 1 at every vertex. If 1 ≤ |F | ≤ n − 3, we choose an

edge e ∈ F in some dimension. The star graph can be decomposed into n subgraphs Sn{1}, Sn{2}, ..., and Sn{n}. By the symmetric property of Sn, we may assume that e is a

crossing edge between Sn{1}and Sn{2}. Consider a vertex v ∈ V (Sn). Let Fi = FT E(Sn{i}),

0 ≤ |Fi| ≤ (n − 4) for all 1 ≤ i ≤ n. Without loss of generality, we may assume that

vertex v is in Sn{1} and degS{1}

n −F1(v) = k. By the inductive hypothesis, S

{1}

n − F1 contains

a Type I structure T1(v; k) at v. Consider the crossing edge (v, (v)n). If (v, (v)n) ∈ F ,

Sn− F contains a Type I structure T1(v; k) of order k at vertex v. If (v, (v)n) /∈ F , the

remaining degree of v in Sn− F is k + 1 and the vertex (v)n has at least one adjacent

neighbor in S{(v)1}

n due to 0 ≤ |F{(v)1}| ≤ (n − 1) − 3. Therefore, Sn− F contains a Type

I structure T1(v; k + 1) of order k + 1 at vertex v. By Theorem 9, removing all the edges

in F from Sn, the local diagnosability of each vertex is still equal to its remaining degree.

2

With Theorem 13, we have the following corollary.

Corollary 4 Let Sn be an n-dimensional star graph with n ≥ 3, and F ⊂ E(Sn) be a set

(39)

the PMC model.

We now give an example to show that an n-regular graph G(V, E) has the strong local diagnosability property, but it may not keep this strong property after removing n − 2 edges from G. For example, a 3-regular graph is shown in Figure 3.10(a). The degree of each vertex is 3 and there exists a Type I structure T1(v; 3) of order 3 at each vertex. By

Theorem 9, Definition 7 and Definition 8, this graph has the strong local diagnosability property. Let F = {(2, 3)} be a set of one single edge, G − F is shown in Figure 3.10 (b). The vertex u does not have the strong local diagnosability property. The reason is as follows. Let F1 = {u, 1, 4} and F2 = {1, 2, 4} with |F1| ≤ 3, |F2| ≤ 3. Since there is no

edge between V (G) − (F1S F2) and F1∆F2, by Lemma 4, (F1, F2) is an indistinguishable

pair. Therefore, the local diagnosability of vertex u is at most 2 which is smaller than its degree. b b b b b b b b b b b b b b b b

(a)

(b)

1 2 3

u

4 1 2 3

u

4

Figure 3.10: A 3-regular graph without the strong local diagnosability property after removing one edge.

3.4

Conditional Fault Local Diagnosability

In previous section, we know that Qn does not have the strongly local-diagnosable

prop-erty, if there are n − 1 faulty edges, all these faulty edges are incident with a single vertex and this vertex is incident with only one fault-free edge. Therefore, we are led to the following question: How many edges can be removed from Qn such that Qn keeps the

(40)

hypercube Qn is incident with at least two fault-free edges? Firstly, we give an

exam-ple to show that a faulty hypercube Qn with 3(n − 2) faulty edges may not have the

strongly local-diagnosable property, even if each vertex of the faulty hypercube Qn is

incident with at least two fault-free edges. As shown in Figure 3.11(a), we take a cy-cle of length four in Qn, n ≥ 3. Let {v, a, b, c} be the four consecutive vertices on this

cycle, and F ⊂ E(Qn) be a set of edges, F = F1S F2S F3, where F1 is the set of all

edges incident with a except (v, a) and (b, a), F2 is the set of all edges incident with b

except (a, b) and (c, b), and F3 is the set of all edges incident with c except (v, c) and

(b, c), then |F1| = |F2| = |F3| = n − 2. The remaining degree of vertex v in Qn − F is

n, degQn−F(v) = n. As shown in Figure 3.11(b), let A1 = (NQn−F(v) − {c})S{v} and

A2 = (NQn−F(v) − {a})S{b}, then |A1| = |A2| = n and v ∈ A1∆A2. It is clear that

there is no edge between V (Qn) − (A1S A2) and A1∆A2. By Lemma 4, (A1, A2) is an

indistinguishable pair, hence tl(v) 6= degQn−F(v) = n. So some vertex of Qn− F may not

have this strong property, if |F | ≥ 3(n − 2). Then, we shall show that Qn− F has the

strongly local-diagnosable property, if each vertex of Qn− F is incident with at least two

fault-free edges and |F | ≤ 3(n − 2) − 1. We need the following results to construct a Type I structure or a Type II structure at a vertex of a faulty hypercube.

b b b b b b b b b b b b b b : : : : .. a b c v 1 2 n-2 v a 1 2 n-2 c b

A1

A2

(a)

(b)

|F1|=n-2 |F2|=n-2 |F3|=n-2

Figure 3.11: An indistinguishable pair (A1, A2), where |A1| = |A2| = n.

Theorem 14 [52] Let G(V, E) be a bipartite graph with bipartition (X, Y ). Then G has a matching that saturates every vertex in X if and only if

(41)

Theorem 15 [52] Let G(V, E) be a bipartite graph. The maximum size of a matching in

G equals the minimum size of a vertex cover of G.

Lemma 8 An n-dimensional hypercube Qn has no cycle of length three and any two

vertices have at most two common neighbors.

For our discussion later, we need some definitions. Let Qnbe an n-dimensional

hyper-cube and F ⊆ E(Qn) be a set of edges. Removing the edges in F from Qn, for a vertex v

in the resulting graph Qn− F , we define BG(v) = (L1(v)S L2(v), E) to be the bipartite

graph under v with bipartition (L1(v), L2(v)), where L1(v) = {x ∈ V (Qn) | vertex x is

adjacent to vertex v in Qn− F }, L2(v) = {y ∈ V (Qn) | there exists a vertex x ∈ L1(v)

such that (x, y) ∈ E(Qn) in Qn− F } − {v} and E(BG(v)) = {(x, y) ∈ E(Qn) | vertex

x ∈ L1(v) and vertex y ∈ L2(v)}. L1(v) (L2(v), respectively) is called the level one (level

two, respectively) vertex under v (see Figure 3.12).

b b b b b b b b

v

x1 x2 ... y1 y2 y3 ... .. .. .. .. .. .. .. L1(v) L2(v)

Figure 3.12: The bipartite graph BG(v).

Theorem 16 Let Qn be an n-dimensional hypercube with n ≥ 3, and F ⊂ E(Qn) be a

set of edges, 0 ≤ |F | ≤ 3(n − 2) − 1. Assume that each vertex of Qn− F is incident with

at least two fault-free edges. Removing all the edges in F from Qn, the local diagnosability

of each vertex is still equal to its remaining degree under the PMC model.

Proof.

According to Theorem 9, we can concentrate on the construction of Type I structure or Type II structure at each vertex. Consider a vertex v in Qn− F with degQn−F(v) = k.

(42)

As shown in Figure 3.12, let BG(v) = (L1(v)S L2(v), E) be the bipartite graph under v.

Then, |L1(v)| = k. Let M ⊂ E(BG(v)) be a maximum matching from L1(v) to L2(v). In

the following proof, we consider three cases by the size of M: 1) |M| = k, 2) |M| = k − 1 and 3) |M| ≤ k − 2.

Case 1: |M| = k

Since |M| = k and |L1(v)| = k, there exists a Type I structure T1(v; k) of order k at

vertex v. By Theorem 9, the local diagnosability of vertex v is equal to k. Case 2: |M| = k − 1

We shall show that there is a Type II structure of order k at vertex v. As shown in Figure 3.13, let L1(v) = {x1, x2, ..., xk} and let ML2(v) ⊂ L2(v) be the set of vertices

matched under M, ML2(v) = {y ∈ L2(v) | there exists a vertex x ∈ L1(v) such that

(x, y) ∈ M}. So |ML2(v)| = k − 1. Let ML2(v) = {y1, y2, ..., yk−1} and assume vertex

xi is matched with vertex yi for each i, 1 ≤ i ≤ k − 1. Then there exists a vertex

xk ∈ L1(v), xk is unmatched by M. Since each vertex of Qn− F is incident with at

least two fault-free edges, there exists a vertex yi ∈ ML2(v), i ∈ {1, 2, ..., k − 1}, such

that (xk, yi) ∈ E(BG(v)). Without loss of generality, let (xk, y1) ∈ E(BG(v)). If the

remaining degree of y1 is at least three, as shown in Figure 3.14, there exists a Type II

structure T2(v; k − 2, 2) of order k at vertex v. By Theorem 9, the local diagnosability

of vertex v is equal to k and the result follows. If the remaining degree of y1 is two,

the number of faulty edges incident with y1 is n − 2. Next, we divide the case into two

subcases: 2.1), both xk and x1 have remaining degree two and 2.2), one of xk and x1 has

remaining degree at least three and the other has at least two. Subcase 2.1: Both xk and x1 have remaining degree two.

This is an impossible case. Since the number of faulty edges incident with xk and x1

is 2(n − 2), the total number of faulty edges is at least 3(n − 2) which is greater than 3(n − 2) − 1, a contradiction.

Subcase 2.2: One of xk and x1 has remaining degree at least three and the

(43)

b b b b b b b b b b

v

x1 x2 xk−1 xk y1 y2 yk−1 ... ... ... .. .. L1(v) L2(v) M L2(v) BG(v)

Figure 3.13: Illustration for the case 2 of Theorem 16 and Theorem 18.

b b b b b b b b b b b

v

xk−1 x2 x1 xk yk−1 y2 y1 ... ... ... .. .. L1(v) L2(v) M L2(v) BG(v)

Figure 3.14: A Type II structure T2(v; k − 2, 2) of order k at vertex v.

Without loss of generality, assume xk has remaining degree at least three and x1 has

remaining degree at least two. Since degQn−F(xk) ≥ 3, there exist at least two vertices in

ML2(v) that are the neighbors of vertex xk. Then, we can find a vertex yi ∈ ML2(v) and

yi 6= y1, i ∈ {2, 3, ..., k − 1}, such that (xk, yi) ∈ E(BG(v)). Without loss of generality, let

(xk, y2) ∈ E(BG(v)). If the remaining degree of y2 is at least three, there exists a Type

II structure T2(v; k − 2, 2) of order k at vertex v. By Theorem 9, the local diagnosability

of vertex v is equal to k and the result follows. If the remaining degree of y2 is two, the

number of faulty edges incident with y2 is n − 2. We then consider two further cases:

Subcase 2.2.1: Vertex x1 has remaining degree two.

This is an impossible case. Since the number of faulty edges incident with x1 is n − 2,

數據

Figure 2.1: Illustration for a distinguishable pair (F 1 , F 2 ) It follows from Definition 1 that the following lemma holds.
Figure 3.1: A Type I structure T 1 (v; k) consists of 2k + 1 vertices and 2k edges. Following Theorem 6 and Definition 5, we propose a sufficient condition for verifying if it is locally t-diagnosable at a given processor in a system.
Figure 3.2: A Type II structure T 2 (v; k, 2) consists of 2k + 5 vertices and 2k + 5 edges
Figure 3.3: A cycle of length four and a Type I structure T 1 (v; 1) of order 1 at v.
+7

參考文獻

相關文件

Wang (2006), Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network, IEEE Trans- actions on Neural Networks,

‡網路作業系統( network operating system). ‡網路作業系統( network

1.大專以上學歷(不限特定科系) 2.行政文書處理與文字表達能力 3.外語能力(國際書信往來與客戶接待) 4.資訊應用能力(excel、ppt 等軟體操作)

„ An adaptation layer is used to support specific primitives as required by a particular signaling application. „ The standard SS7 applications (e.g., ISUP) do not realize that

„ Signaling over standard IP uses a common transport protocol that ensures reliable signaling delivery. „ Error-free

Each unit in hidden layer receives only a portion of total errors and these errors then feedback to the input layer.. Go to step 4 until the error is

Compensates for deep fades via diversity techniques over time, frequency and space. (Glass is

AWS中絕大多數的資料、備份、機器配 置,甚至是異地備份。Code Spaces表 示,這樣的情況已讓該公司無法再繼續 運作.