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資訊科學與工程研究所

在 PMC 模式下超立方體之

g

-good-neighbor 條件式診斷能力

The g-good-neighbor Conditional Diagnosability

of Hypercube under PMC Model

研 究 生:彭紹綸

指導教授:譚建民 教授

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在 PMC 模式下超立方體之

g

-good-neighbor 條件式診斷能力

The g-good-neighbor Conditional Diagnosability

of Hypercube under PMC Model

研 究 生:彭紹綸 Student:Shao-Lun Peng

指導教授:譚建民 Advisor:Jimmy J. M. Tan

國 立 交 通 大 學

資 訊 科 學 與 工 程 研 究 所

碩 士 論 文

A Thesis

Submitted to Institute of Computer Science and Engineering College of Computer Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in

Computer Science

June 2008

Hsinchu, Taiwan, Republic of China

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在 PMC 模式下超立方體之

g

-good-neighbor 條件式

診斷能力

研究生:彭紹綸 指導教授:譚建民 博士

國立交通大學

資訊科學與工程研究所

摘要

在多處理器系統當中,為維持處理器在計算上的可靠度,處理器偵錯一直

是很重要的議題。對於許多著名的連結網路,已經有相關的處理器偵錯之

診斷能力的研究結果。 舉例而言,n 維的超立方體(hypercubes)、n 維的交

叉立方體(crossed cubes)、n 維的梅氏立方體(möbius cubes)、n 維的雙扭立

方體(twisted cubes)之處理器偵錯之診斷能力皆為 n。而 n 維的超立方體在

PMC 模式下條件式處理器偵錯之診斷能力為 4(n-2)+1。在本文中我們將探

討 n 維的超立方體在 PMC 模式下的 g-good-neighbor 條件式處理器偵錯之

診斷能力,並証明其為 2

g

(n-g)+2

g

-1,其中 0 ≤ g ≤ n – 3。 在 g-good-neighbor

條件式下處理器偵錯之診斷能力為傳統的處理器偵錯之診斷能力的數倍。

關鍵字:超立方體、PMC 診斷模式、t-可診斷性、診斷能力、g-good-neighbor

條件式診斷能力。

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The g-good-neighbor Conditional Diagnosability of

Hypercube under PMC Model

Student: Shao-Lun Peng

Advisor: Jimmy J. M. Tan

Institute of Computer Science and Engineering

National Chiao Tung University

Abstract

Processor fault diagnosis plays an important role in multiprocessor systems for reliable computing, and the diagnosability of many well-known networks has been explored. For example, hypercubes, crossed cubes, möbius cubes, and twisted cubes of dimension n all have diagnosability n. The conditional diagnosability of n- dimensional hypercube Qn is proved to be 4(n - 2) + 1 under the PMC model.

In this thesis, we study the g-good-neighbor conditional diagnosability of Qn under

the PMC model and show that it is 2g(n - g) + 2g -1 for 0 ≤ g ≤ n - 3. The

g-good-neighbor conditional diagnosability of Qn is several times larger than the

classical diagnosability of Qn.

Keywords: hypercube, PMC diagnosis model, t-diagnosable, diagnosability, g-good-neighbor conditional diagnosability.

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誌 謝

在計算機理論實驗室的兩年研究生涯,首先要感謝指導教授譚建民教授,不論是

論文指導或是研究的態度與方法,都給我很多的啟發和勉勵,讓我能夠順利的在時間

內完成碩士論文。同時也感謝我的論文口試委員徐力行教授、梁婷教授以及蔡錫鈞教

授,在口試時提出了寶貴的意見,也使本論文能夠更加完整,讓我受益良多。

感謝實驗室的博班學長:政寬學長、倫閔學長、玠峰學長、圜鋼學長、宗翰學長、

自良學長、國晃學長和元翔學長,在學習過程中的分享、照顧與幫助,讓我平日在實

驗室的研究生活能在輕鬆的氣氛中參與討論、一同學習,讓我研究成果的更加充實,

也排解了研究生活中不時的煩悶與壓力。感謝同學孟宏時常提醒修業上的細節,他也

是和我一同跟論文奮戰、相互扶持的好夥伴。還要感謝研究所的室友,陳俊宇、沈宣

佐、蘇郁恭、許多大學朋友以及在系壘一塊在球場上奔馳競技的隊友們,平日生活的

相互關照,在精神上給我莫大的鼓勵與支持,是你們豐富我這兩年的生活。

最後,要感謝我摯愛的家人,感謝父母親以他們的血汗、青春,使我日漸茁壯,

願將來我能夠回報他們的是更豐盛的佳餚;感謝我的哥哥姐姐,對於我的研究生活全

面的支持與鼓勵,讓我能夠無後顧之憂的攻讀碩士;也感謝我的女友瑞謙,給予我無

微不至的照料與關懷,自始至終的陪伴我,並為我加油打氣,讓我能心無旁鶩地完成

碩士學位。對於他們的苦心栽培也在此表達最深的感謝。

彭紹綸 謹誌

中華民國 九十七年 六月

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Contents

1 Introduction ………...1

2 Terminology and Notations ………..3

3 The n-dimensional Hypercube ……….4

4 Diagnosability ………6

5 The g-good-neighbor Conditional Diagnosability of Hypercube ……12

6 Conclusions ………..16

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List of Figures

1 Illustration of the hypercube of dimension 1, 2, 3 and 4. . . 4

2 Illustration of hypercube of dimension n consisted of two Q

n-1

. . . 5

3 Illustration of the directed testing graph T produced from graph G. . . . 7

4 An example of a faulty vertex x

1

embedded in Q

2

. . . 7

5 Illustration of a distinguishable pair (F

1,

F

2

). . . 9

6 Illustration of F

1

and F

2

. . . 13

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List of Tables

1 All possible results of v tested by u . . . 9

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1

Introduction

With the rapid development of technology, the need for high-performance large multipro-cessor systems has been continuously increasing day by day. Since all the promultipro-cessors run in parallel, the reliability of each processor in multiprocessor systems becomes an important issue for parallel computing. In order to maintain the reliability of such multiprocessor systems, whenever a processor (node or vertex) is found faulty, it should be replaced by a fault-free processor.

The hypercube [25, 32] is a well-known interconnection network for multiprocessor systems. Fault-tolerant computing for the hypercube has been of interest to many re-searchers. The process of identifying faulty vertices is called the diagnosis of the system. System diagnosis can be done in two different approaches, that is, circuit-level diagnosis and system-level diagnosis. In circuit-level diagnosis, the processors must be tested one after one by the human labor, which induces diagnosis complicated and possibly inac-curate. On the other hand, system-level diagnosis could be done automatically by the system itself. Thus, system-level diagnosis appears to be an alternative to circuit-level testing in a large multiprocessor system. Many terms for system-level diagnosis have been defined and various models have been proposed in literature [1, 11, 28, 31]. If all allowable fault sets can be diagnosed correctly and completely based on a single syndrome, then the diagnosis is referred to as one-step diagnosis or diagnosis without repairs.

In this study, we use the widely adopted PMC model [31] as the fault diagnosis model. In [15], Hakimi and Amin proved that a multiprocessor system is t-diagnosable if it is

t-connected with at least 2t + 1 vertices. Besides, they gave a necessary and sufficient

condition for verifying if a system is t-diagnosable under the PMC model. Reviewing the previous papers, there are several variations of the hypercube [19], for example, the crossed cube [9], the m¨obius cube [10], and the twisted cube [16]. For each of these cubes, an n-dimensional cube can be constructed from two copies of (n − 1)-dimensional cubes

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by adding a perfect matching between them. One of the common property among them is that all these variations have diagnosability n under the PMC model.

In classical 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. If there is a vertex v whose neighboring vertices are faulty simultaneously, there is no way of knowing the faulty or faut-free status of v. As a consequence, the diagnosability of a system is upper bounded by its minimum degree. Motivated by the deficiency of the classical measurement of diagnosability, Lai et al. [21] introduced a measure of conditional

diagnosability by claiming the property that any faulty set cannot contain all neighbors

of any processor. Under this condition, they showed that the conditional diagnosability of

the n-dimensional hypercube Qn is 4(n−2)+1. We are then led to the following question:

how large the maximum value t can be such that a graph G remains t-diagnosable under the condition that every vertex v has at least g fault-free neighboring vertices. More precisely, we assume the faulty set F satisfies the condition that each vertex v in G−F has at least g good neighbors. We notice that, considering the situation that all the neighbors of each vertex cannot fail simultaneously, many properties of the network would be much better, including the connectivity and diagnosability. The aim of this thesis is to study more of these better properties.

In this thesis, we extend the concept of conditional diagnosis and propose a new measure of diagnosability. We define g-good-neighbor conditional diagnosability as the maximum number of faulty vertices that the system can guarantee to identify under the condition that every fault-free vertex has at least g fault-free neighbors. We show that the g-good-neighbor conditional diagnosability of Qnis 2g(n − g) + 2g− 1 under the PMC

model, which is several times larger than the classical diagnosability of Qn.

The rest of this thesis is organized as follows: Section 2 provides terminology and pre-liminaries for a multiprocessor system. In Section 3, we introduce the n-dimensional hy-percube. We introduce system diagnosis and propose the concept of the g-good-neighbor conditional diagnosis in Section 4. In Section 5, we show the proof of the g-good-neighbor conditional diagnosability of Qn. Finally, our conclusions are given in Section 6.

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2

Terminology and Notations

A multiprocessor system or a network is usually represented as an undirected graph where vertices represent processors and edges represent communication links. Throughout this thesis, we follow [33] for the graph definitions and notations, and we focus on the undirected graph without loops (simply abbreviated as graph).

Let G = (V, E) be a graph where V is a finite set and E is a subset of {(u, v) | (u, v) is an unordered pair of V }. We say that V is the vertex set and E is the edge set. We use n(G) = |V | to denote the cardinality of V . The degree of a vertex v in a graph G,

written as degG(v) or deg(v), is the number of edges incident to v. The maximum degree

of graph G is denoted by ∆(G) = max{degG(v) | v ∈ V (G)}, and the minimum degree

of graph G is δ(G) = min{degG(v) | v ∈ V (G)}. The graph G is regular if ∆(G)=δ(G).

It is k-regular if the common degree is k. The neighborhood of a vertex v, written NG(v)

or N(v), is the set of vertices adjacent to v. We use N(A) = {x | y ∈ A, x ∈ G − A, and (x, y) ∈ E(G)} to denote the neighborhood of a vertex subset A of G.

Two vertices u and v are adjacent if (u, v) ∈ E. A path is a sequence of adjacent vertices, written as hv0, v1, v2, . . . , vki, in which all the vertices v0, v1, . . . , vk are distinct

except possibly v0 = vk. A graph G is connected if for any two vertices, there is a path

joining them, otherwise it is disconnected. For a set S of V , the notation G−S represents the graph obtained by removing the vertices in S from G and deleting those edges with at least one end vertex in S. If G − S is disconnected, then S is called a separating set (or a vertex cut). A graph H is a subgraph of G if V (H) ⊆ V (G) and E(H) ⊆ E(G). A component of a graph G is its maximal connected subgraph. A component is trivial if it has no edges; otherwise, it is nontrivial. The connectivity κ(G) of a graph G is the minimum number of vertices whose removal results in a disconnected graph or only one vertex left. A graph G is k-connected if its connectivity is at least k.

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3

The n-dimensional Hypercube

An n-dimensional hypercube, Qn, is an undirected n-regular graph containing 2nvertices

and n2n−1 edges (See Figure1). Let u = u

n−1un−2. . . u1u0 be an n-bit binary string. In a

sense, the hypercube Qn consists of all n-bit binary strings as its vertices. Two vertices u

and v are adjacent if their binary string representations differ in exactly one bit position.

For 0 ≤ i ≤ n − 1, we use ui to denote the i-th neighbor of u, i.e., the binary string

vn−1vn−2. . . v1v0 where vi = 1 − ui and vk = uk if k 6= i.

Q

2 00 01 10 0 11 0 1 1

Q

3 000 001 010 011 110 101 111 100 0 0 0 0 2 2 2 2 1 1 1 1

Q

4 0000 0100 0001 0010 0011 0110 0101 0111 0 0 0 0 2 2 2 2 1 1 1 1 1000 1001 1010 1011 1110 1101 1111 1100 0 0 0 0 2 2 2 2 1 1 1 1 3 3 3 3 3 3 3 3

Q

1 0 1 0

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The Hamming weight of u, denoted by w(u), is the number of i such that ui = 1.

The hypercube Qn is a bipartite graph with bipartition {u | w(u) is odd} and {u | w(u)

is even}. We use black vertices to denote those vertices of odd weight and white vertices to denote those vertices of even weight. For i ∈ {0, 1}, we set Qi

n to be the subgraph of

Qn which is induced by {u ∈ V (Qn) | un−1 = i}. The hypercube of n-dimension Qn is

consisted of two Qn−1, and Qinis isomorphic to Qn−1 for i = 0, 1, as shown in Figure 2. It

is well known that Qn is vertex transitive and edge transitive [14, 25]. Furthermore, the

permutation on the coordinate of Qn and the componentwise complement operations are

graph isomorphisms.

Q

n

Q

n-1

Q

n-1

.

.

.

.

.

.

Figure 2: Illustration of hypercube of dimension n consisted of two Qn−1.

The topological properties of Qnhas been studied extensively in recent years. Readers

can refer [25] for a survey on the properties of hypercubes. For further study of the hypercube Qn, we need to use the following property of hypercube.

Lemma 1. For every two distinct vertices u and v in hypercube, they have at most 2 common neighbors. That is |N(u) ∩N(v)| ≤ 2.

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4

Diagnosability

Under the classical PMC model [31], adjacent processors are capable of performing tests on each other. For two adjacent vertices u and v in V , the ordered pair (u, v) represents the test performed by u on v. In this situation, u is called the tester and v is called the

tested vertex. The outcome of a test (u, v) is either 1 or 0 with the assumption that the

testing result is regarded as reliable if the tester u is fault-free. However, the outcome of a test (u, v) is unreliable, provided that the tester u itself is originally a faulty processor. Suppose that the tester u is fault-free, then the result would be 0 (respectively, 1) if v is fault-free (respectively, faulty). In this thesis, for each pair of adjacent vertices (u, v), u and v can perform the test to each other. All the possible results of v tested by u are listed in the following table.

u v (u, v)

fault-free fault-free 0

fault-free faulty 1

faulty fault-free 0/1

faulty faulty 0/1

Table 1: All possible results of v tested by u

A test assignment T for a system G is a collection of tests for every adjacent pairs of vertices. It can be modeled as a directed testing graph T = (V, L) where (u, v) ∈ L implies that u and v are adjacent in G. See Figure 3 for an illustration of T produced from G. Throughout this thesis, we assume that each vertex tests the other whenever there is an edge between them and all these tests are gathered in the test assignment. The collection of all test results for a test assignment T is called a syndrome. Formally, a syndrome is a function σ : L → {0, 1}. The set of all faulty processors in the system is called a faulty set. This can be any subset of V . The process of identifying all the faulty vertices is called the diagnosis of the system. The maximum number of faulty vertices that the system G

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Graph G(V, E) directed testing graph T(V, L)

Figure 3: Illustration of the directed testing graph T produced from graph G.

can guarantee to identify is called the diagnosability of G, written as t(G). For a given syndrome σ, a subset of vertices F ⊆ V is said to be consistent with σ if syndrome σ can be produced from the situation that, for any (u, v) ∈ L such that u ∈ V − F, σ(u, v) = 1 if and only if v ∈ F . Because a faulty tester can lead to an unreliable result, a given set

F of faulty vertices may produce different syndromes. We use notation σ(F ) to

repre-sent the set of all syndromes which could be produced if F is the set of faulty vertices. Two distinct sets F1 and F2 in V are said to be indistinguishable if σ(F1) ∩ σ(F2) 6= ∅,

otherwise, F1 and F2 are said to be distinguishable. Besides, we say (F1, F2) is an indistinguishable pair if σ(F1) ∩ σ(F2) 6= ∅, else (F1, F2) is a distinguishable pair.

Fig-ure 4 illustrates an example of a faulty vertex embedded in Q2. In this example, F =

{x1} and σ(F ) = {(x1, x2), (x2, x1), (x1, x4), (x4, x1), (x2, x3), (x3, x2), (x3, x4), (x4, x3)} = {{0, 1, 0, 1, 0, 0, 0, 0}, {0, 1, 1, 1, 0, 0, 0, 0}, {1, 1, 0, 1, 0, 0, 0, 0}, {1, 1, 1, 1, 0, 0, 0, 0}} with 4 pos-sible syndromes. x1 x2 x4 x 3 faulty

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

Definition 1. [31] A system of n units is t-diagnosable if all faulty units can be identified without replacement, provided that the number of faults presented does not exceed t.

Let F1 and F2 be two distinct subsets of V , and let the symmetric difference F1∆F2 =

(F1− F2) ∪ (F2− F1). DahBura and Masson [5] proposed a polynomial time algorithm to

check whether a system is t-diagnosable.

Theorem 1. [5] A system G = (V, E) is t-diagnosable if and only if, for any two distinct subsets F1 and F2 of V with |F1| ≤ t and |F2| ≤ t, there is at least one test from V − (F1∪ F2) to F1∆F2.

The following two results related to t-diagnosable systems are due to Hakimi and Amin [15], and Preparata et al. [31], respectively.

Theorem 2. [31] Let G(V, E) be the graph representation of a system G with V repre-senting the processors and E the interconnection among them. Let |V | = n. The following two conditions are necessary for G to be t-diagnosable under PMC model:

1. n ≥ 2t + 1, and

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

Theorem 3. [15] The following two conditions are sufficient for a system G of n pro-cessors to be t-diagnosable under PMC model:

1. n ≥ 2t + 1, and

2. κ(G) ≥ t.

For a directed graph G and a vertex v ∈ V (G), the notation din(v) is used to denote

the number of edges directed toward v in G. Let Γ(v) = {vi|(v, vi) ∈ E} and Γ(X) =

∪v∈XΓ(v) − X, where X ⊂ V . Hakimi and Amin [15] presented a necessary and sufficient

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Theorem 4. [15] Let G(V, E) be the directed graph of a system G with n vertices. Then G is t-diagnosable under PMC model if and only if (i) n ≥ 2t + 1, (ii) din(v) ≥ t for all

v ∈ V , and (iii) |Γ(X)| > p for each integer p with 0 ≤ p ≤ t − 1 and for each X ⊂ V with |X| = n − 2t + p.

In this thesis, we propose some new viewpoints on diagnosis, and we will focus on undirected graph (simply abbreviated as graph). Let G(V, E) be an undirected graph of a system G. The following result follows directly from Theorem 1.

Theorem 5. For any two distinct subsets F1 and F2 of V , (F1, F2) is a distinguishable-pair under PMC model if and only if there is a vertex u ∈ V − (F1 ∪ F2) and there is another vertex v ∈ F1∆F2 such that (u, v) ∈ E. (See Figure 5.)

F

1

F

2

or

F

1

F

2

u

v

u

v

Figure 5: Illustration of a distinguishable pair (F1, F2).

Duo to Definition 1, the following lemma holds.

Lemma 2. A system is t-diagnosable under PMC model if and only if for each distinct pair of subsets F1 and F2 of V with |F1| ≤ t and |F2| ≤ t, F1 and F2 are distinguishable.

An equivalent way of stating the above lemma is the following:

Lemma 3. A system is t-diagnosable under PMC model if and only if for each indistin-guishable pair of sets F1 and F2 of V , it implies that |F1| > t or |F2| > t.

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

Corollary 1. [31] Let G(V, E) be an undirected graph with n vertices. The following two conditions are necessary for G to be t-diagnosable under PMC model:

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1. n ≥ 2t + 1, and

2. δ(G) ≥ t.

For classical measurement of diagnosability, it is usually assumed that processor fail-ures are statically independent. It does not reflect the total number of processors in the system and the probabilities of processor failures. In [29], Najjar and Gaudiot proposed fault resilience as the maximum number of failures that can be sustained while the net-work remains connected with a reasonably high probability. For hypercube, the fault

resilience is shown as 25 percent for Q4 and it increases to 33 percent for Q10. More

particularly, for the 10-dimensional hypercube Q10, 33 percent of processors can fail and

the network still remains connected with a probability of 99 percent. They also gave a conclusion that large-scale systems with a constant degree are more susceptible to failures by disconnection than smaller networks. With the observation of Theorem 5, a connected network gives higher probability to diagnose faulty processors and has better ability to distinguish any two sets of processors.

In an n-dimensional hypercube, Qn has (2

n

n ) vertex subsets of size n, among which

there are only 2n vertex subsets which contains all the neighbors of some vertex. Since

the ratio 2n/(2n

n ) is very small for large n, the probability of a faulty set with size n

containing all the neighbors of any vertex is very low. For this reason, Lai et al. [21] introduced a new restricted diagnosability of multiprocessor systems called conditional

diagnosability. They consider the situation that any faulty set cannot contain all the

neighbors of any vertex in a system.

Motivated by this concept [21], we extend this idea about conditional diagnosis. In this thesis, we introduce g-good-neighbor condition by claiming that for every fault-free vertex in a system, it has at least g fault-free neighbors. We now give some formal terms related to the g-good-neighbor conditional diagnosis in the following. A faulty set F ⊂ V is called a g-good-neighbor conditional f aulty set if |N(v) ∩ (V − F )| ≥ g for every vertex

v in V − F . A system G is g-good-neighbor conditional t-diagnosable if F1 and F2 are

distinguishable, for each distinct pair of g-good-neighbor conditional faulty subsets F1

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g-good-neighbor property of H, Pg(H), holds for H if and only if every vertex in H has

at least g neighbors in H.

Definition 2. The g-good-neighbor conditional diagnosability tg(G) of a graph G is the

maximum value of t such that G is g-good-neighbor conditional t-diagnosable

Follow from this definition, we observe that under the stronger g-good-neighbor con-dition, the diagnosability of G is greater as well.

Lemma 4. For any given graph G, tg(G) ≤ tg0(G) if g ≤ g0.

The g-good-neighbor conditional diagnosability is a new concept, so there are not

many known results. Finding the exact value of tg(G) for general graphs would be a

difficult problem. So we study a popular topology, the hypercube Qn, as an example. In

the next section, we will derive the the g-good-neighbor conditional diagnosability tg(Qn)

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5

The g-good-neighbor Conditional Diagnosability of

Hypercube

Before discussing the g-good-neighbor conditional diagnosability of hypercube, we have some useful observations as follows:

Theorem 6. [30] Let n ≥ 3 and 1 < p ≤ n. Suppose that F is a minimum cardinality cut of Qn such that |NQn(x) ∩ F | ≤ p for all x ∈ V (Qn) − F . Then |F | = p2

n−p.

In the above theorem, we note that F is a g-good-neighbor conditional faulty set if

p = n − g.

Theorem 7. [24] Let H be a subgraph of Qn satisfying Pg(H). Then |V (H)| ≥ 2g, for

0 < g ≤ n.

To find the g-good-neighbor conditional diagnosability of the hypercube Qn, we first

give an example to show that tg(Qn) is no more than 2g(n − g) + 2g− 1. We are going

to show that there exist two g-good-neighbor conditional faulty sets F1 and F2 of V (Qn)

with |F1| ≤ 2g(n − g) + 2g and |F2| ≤ 2g(n − g) + 2g, but F1 and F2 are indistinguishable.

Thus, we know Qn is not g-good-neighbor conditional (2g(n − g) + 2g)-diagnosable.

We set A = {y = yn−1yn−2. . . y0 | yi = 0 for i ∈ {g, g + 1, . . . , n − 1} and yj ∈ {0, 1}

for j ∈ {0, 1, . . . , g − 1}} and Vk = {yn−k | y ∈ A for every 1 ≤ k ≤ n − g}. Then we set

F1 = ∪n−gi=1Vi and F2 = A ∪ F1. Since |A| = 2g and |Vi| = 2g for every 1 ≤ i ≤ n − g, we

obtain that |F1| = 2g(n − g) and |F2| = 2g+ 2g(n − g). By Theorem 5, we conclude that

(F1, F2) is an indistinguishable pair because A = F1∆F2 and N(A) = F1. See Figure 6.

Now we verify that both F1 and F2 are g-good-neighbor conditional faulty sets. Let X

be the set V (Qn) − (F1∪ F2). Since F1 is the subset of F2, X = V (Qn) − F2. Therefore,

it is sufficient to verify both Pg(A) and Pg(X) are satisfied. For every vertex u in A, it

is easy to see that ui in A for every i ∈ {0, 1, . . . , g − 1}. Thus, P

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Q

n

A

. . . n-g neighbors

V

i

n-g

i=1

1

F =

. . . g neighbors

Figure 6: Illustration of F1 and F2.

consider the vertices in X. By the definition of X, we know that for every x in X, xi ∈ X

for every 0 ≤ i ≤ g − 1. Thus, the property Pg(X) also holds. Therefore, both F1 and F2

are g-good-neighbor conditional faulty sets of Qn.

Since (F1, F2) is an indistinguishable pair with |F1| = 2g(n − g) and |F2| = 2g(n − g)+2g, we conclude that the g-good-neighbor conditional diagnosability of Q

nis less than

2g(n − g) + 2g. The following lemma states the fact.

Lemma 5. For 0 ≤ g ≤ n − 3, tg(Qn) ≤ 2g(n − g) + 2g− 1.

The following theorem shows that the g-good-neighbor conditional diagnosability of hypercube tg(Qn) is exactly 2g(n − g) + 2g − 1.

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Theorem 8. For 0 ≤ g ≤ n − 3, tg(Qn) = 2g(n − g) + 2g − 1.

Proof:

By Lemma 5, we already know that tg(Qn) ≤ 2g(n − g) + 2g − 1. Therefore, we

only need to verify that tg(Qn) ≥ 2g(n − g) + 2g − 1. To prove Qn is g-good-neighbor

conditional (2g(n − g) + 2g− 1)-diagnosable, it is equivalent to prove that F1 and F

2 must

be distinguishable for every two distinct g-good-neighbor conditional faulty sets F1 and F2 of Qn, provided that both the cardinality of F1 and cardinality of F2 are no more than

2g(n − g) + 2g− 1.

We prove this theorem by contradiction. Suppose that there are two distinct g-good-neighbor conditional faulty sets F1 and F2, which are indistinguishable with |F1| ≤ 2g(n − g) + 2g− 1 and |F2| ≤ 2g(n − g) + 2g− 1. Now we consider all the possible cases such that

F1 and F2 are indistinguishable. By Theorem 5, there are two cases such that F1 and F2

are indistinguishable: V (Qn) = F1∪ F2 or V (Qn) 6= (F1∪ F2) but there is no test from V (Qn) − (F1 ∪ F2) to F1∆F2. Without loss of generality, we assume that F2 − F1 6= ∅.

We show that each case has contradiction with our assumption. Case 1: V (Qn) = F1∪ F2.

Since g ≤ n − 3 and all the vertices of Qn are in F1 ∪ F2, we obtain the following

equation with contradiction:

2n = |V (Q n)| = |F1| + |F2| − |F1∩ F2| ≤ |F1| + |F2| ≤ 2(2g(n − g) + 2g− 1) ≤ 2(2n−3(n − (n − 3) + 1)) − 2 = 2n− 2 which is a contradiction. Case 2: V (Qn) 6= (F1∪ F2).

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regardless F1 ⊂ F2 or not. Since F1 and F2 are indistinguishable, there are no edges

between V (Qn) − (F1∪ F2) and F1∆F2. By the assumption that F2− F1 6= ∅ and F1 is a g-good-neighbor conditional faulty set, any vertex in F2−F1 has at least g good neighbors

in subgraph F2 − F1. By Theorem 7, the size of F2− F1 is characterized, and thus we

have |F2− F1| ≥ 2g. In addition, since V (Qn) − (F1 ∪ F2) and F1∆F2 are disconnected, Qn would be divided into several components by deleting F1∩ F2. Besides, since F1 and

F2 are both g-good-neighbor conditional faulty sets, F1 ∩ F2 is also a g-good-neighbor

conditional faulty set. By Theorem 6, the minimum cardinality cut of Qn with

g-good-neighbor condition is p2n−p = (n − g)2g. Thus, we obtain that |F

2 ∩ F1| ≥ (n − g)2g.

As a result, |F2| = |F2− F1| + |F2 ∩ F1| ≥ 2g + (n − g)2g which contradicts with that |F2| ≤ 2g+ (n − g)2g− 1.

Based on these two cases above, we conclude that tg(Qn) ≥ 2g(n − g) + 2g− 1. Thus,

the g-good-neighbor conditional diagnosability tg(Qn) = 2g(n−g)+2g−1. This completes

the proof of this theorem. ¤

The following table shows the g-good-neighbor conditional diagnosability of n-dimensional hypercube tg(Qn) of small n where 0 ≤ g ≤ n − 3.

n g |V (Qn)| tg(Qn) ratio 3 0 8 3 0.375 4 0 16 4 0.25 4 1 16 7 0.4375 5 0 32 5 0.15625 5 1 32 9 0.28125 5 2 32 15 0.46875 6 0 64 6 0.09375 6 1 64 11 0.171875 6 2 64 19 0.296875 6 3 64 31 0.484375 7 0 128 7 0.0546875 7 1 128 13 0.1015625 7 2 128 23 0.1796875 7 3 128 39 0.3046875 7 4 128 63 0.4921875 Table 2: tg(Qn) of small n

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6

Conclusions

In probabilistic models of multiprocessor systems, processors fail independently, but with different probabilities. The probability that all faulty processors are neighbors of one pro-cessor is very small. In this thesis, we propose the concept of g-good-neighbor conditional diagnosis with any fault-free vertex has at least g neighboring fault-free vertices. To grant more accurate measurement of diagnosability for a large-scale processing system, we intro-duce the g-good-neighbor conditional diagnosability of a system under the PMC model.

The g-good-neighbor conditional diagnosability of the hypercube Qn is demonstrated to

be 2g(n − g) + 2g− 1.

Observing that when g = 0, there is no restriction on the faulty sets and we have the traditional diagnosability on the hypercube as n. In addition, in the special case of g = 1, our result is slightly different from the measure of diagnosability given by Lai et al. [21]. The difference between these two measures is that we only consider the condition of the fault-free vertices in the network. A thorough investigation of the diagnosability with the requirement of having at least g good neighbors for all vertices would be an interesting problem to study in the future.

In the area of diagnosability, the comparison model is another well-known and widely chosen fault diagnosis model. Hence, for further discussion, it is worthy to determining the g-good-neighbor conditional diagnosability of a system under comparison model.

The classical diagnosability of a system is small owing to the assumption that all neighbors of each processor can potentially fail at the same time regardless of the proba-bility. If there are exactly n faulty processors in a system of minimum degree n, however, the probability of the faulty set containing all the neighbors of any vertex is statistically low for large multiprocessor systems. Therefore, it is an attractive work to develop more different measures of g-good-neighbor conditional diagnosability based on application en-vironment, network topology, network reliability, and statistics related to fault patterns.

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數據

Figure 1: Illustration of the hypercube of dimension 1, 2, 3 and 4.
Figure 2: Illustration of hypercube of dimension n consisted of two Q n−1 .
Figure 4: An example of a faulty vertex x 1 embedded in Q 2 .
Figure 5: Illustration of a distinguishable pair (F 1 , F 2 ).
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