ADVANCES in the semiconductor technology have made

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1 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 6, NO. 4, APRIL The g-good-neighbor Coditioal Diagosability of k-ary -Cubes uder the PMC Model ad MM Model Ju Yua, Aixia Liu, Xue Ma, Xiuli Liu, Xiao Qi, Seior Member, IEEE, ad Jifu Zhag Abstract The diagosability of a system is defied as the maximum umber of faulty processors that the system ca guaratee to idetify, which plays a importat role i measurig of the reliability of multiprocessor systems. I the work of Peg et al. i 01, they proposed a ew measure for fault diagosis of systems, amely, g-good-eighbor coditioal diagosability. It is defied as the diagosability of a multiprocessor system uder the assumptio that every fault-free ode cotais at least g fault-free eighbors, which ca measure the reliability of itercoectio etworks i heterogeeous eviromets more accurately tha traditioal diagosability. The k-ary -cube is a family of popular etworks. I this sdy, we first ivestigate ad determie the R g -coectivity of k-ary -cube for 0 g : Based o this, we determie the g-good-eighbor coditioal diagosability of k-ary -cube uder the PMC model ad MM model for k 4; 3 ad 0 g : Our sdy shows the g-good-eighbor coditioal diagosability of k-ary -cube is several times larger tha the classical diagosability of k-ary -cube. Idex Terms PMC diagosis model, MM diagosis model, k-ary -cube, coditioal coectivity, fault diagosability Ç 1 INTRODUCTION ADVANCES i the semicoductor techology have made it possible to develop very high-performace large multiprocessor systems comprisig hudreds of thousads of processors (odes). Yet, it is almost impossible to build such a multiprocessor system without defects. Sice all the processors ru i parallel, the reliability of each processor i multiprocessor systems becomes of cetral importace for parallel computig. Therefore, i order to maitai the reliability of such multiprocessor systems, the fault processors should be foud ad replaced i time. The process of idetifyig faulty odes is called the diagosis of the system. I 1967, Preparata et al. [37] proposed a model ad a framework, called system-level diagosis, which could test the processors automatically by the system itself. It is well kow that system-level diagosis appears to be a alterative to traditioal circuit-level testig i a large multiprocessor system. I the more tha four decades followig this pioeerig work, may terms for systemlevel diagosis have bee defied ad various models (e.g., PMC, BGM, ad compariso models) have bee cosidered J. Yua, A. Liu, X. Ma, ad X. Liu are with the School of Applied Sciece, Taiyua Uiversity of Sciece ad Techology, Taiyua 03004, Shaxi, P.R. Chia. fromyuaju@tom.com, aixiayuaju@sia.com, { , }@qq.com. X. Qi is with the Departmet of Computer Sciece ad Software Egieerig, Shelby Ceter for Egieerig Techology, Aubur Uiversity, Aubur, AL qixiao@gmail.com. J. Zhag is with the School of Computer Sciece ad Techology, Taiyua Uiversity of Sciece ad Techology, Taiyua 03004, Shaxi, P.R. Chia. zjf@tyust.edu.c. Mauscript received 6 Dec. 013; revised 11 Mar. 014; accepted 8 Apr Date of publicatio 16 Apr. 014; date of curret versio 6 Mar Recommeded for acceptace by P. Sati. For iformatio o obtaiig reprits of this article, please sed to: reprits@ieee.org, ad referece the Digital Object Idetifier below. Digital Object Idetifier o /TPDS i the literare [5], [16], [3], [33], [37]. Amog the proposed models, two well-kow diagosis models, i.e., the Preparata, Metze, ad Chie (PMC) model [37] ad the Maeg ad Malek (MM) model [3], have bee widely adopted. I the PMC model, every ode u is able to test aother ode v if there is a lik that coects them, where u is called the tester ad v is called the tested ode. The outcome of a test performed by a fault-free tester is 1 (respectively, 0) if the tested ode is faulty (respectively, fault-free), whereas the outcome of a test performed by a faulty tester is ureliable. I [18], Hakimi ad Ami prove that a multiprocessor system is t-diagosable if it is t-coected with at least t þ 1 odes. They also give a ecessary ad sufficiet coditio for verifyig if a system is t-diagosable uder the PMC model. Recetly, Maik ad Gramatova [30], [31] propose a diagosis algorithm uder the PMC model which use Boolea formalizatio. Fa et al. show the disjoit cosecutive cycle (DCC, for short) liear cogruetial graphs, GðF; pþ,is t-diagosable uder the PMC model where p 3 ad t p 1 [17]. Ahlswede ad Aydiia sdy the diagosability of large multiprocessor etworks [1]. The diagosability of the well-kow itercoectio etwork hypercube ad its several variatios [4], [40] for example, the crossed cube [13], the M obius cube [14], ad the twisted cube [0] are show to be uder the PMC model. A modificatio of the PMC model, the BGM model, proposed by Barsi et al. [5], use the same testig strategy as PMC model, but it assumes that a fault ode is always tested as faulty regardless of the state of the tester. The ratioal is that tests cosist of log sequeces of stimuli ad testig a faulty ode is very likely to result i at least oe mismatch, see [], [5]. I the MM model, a ode (called a comparator) seds the same task to its two eighbors ad compares their resposes. A compariso of odes u ad v performed by ß 014 IEEE. Persoal use is permitted, but republicatio/redistributio requires IEEE permissio. See for more iformatio.

2 1166 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 6, NO. 4, APRIL 015 a ode w is deoted by ðu; vþ w : A disagreemet over a compariso performed by a fault-free comparator idicates the existece of a faulty ode, whereas the outcome of a compariso performed by a faulty comparator is ureliable. The mai advatage of this model is its simplicity sice it is easy to compare a pair of odes i multiprocessor systems. This approach seems attractive because o additioal hardware is required ad trasiet ad permaet faults may be detected before the compariso program has completed. A paper by Segupta ad Dahbura [39] revealed importat properties of a diagosable system uder this model. It suggested a special case of the MM model, called the MM model. I this model, a compariso ðu; vþ w must be performed by w if u ad v are eighbors of w i the system. It also preseted a polyomial algorithm to idetify faulty odes i a geeral system uder the MM model if the system is diagosable. The MM model was adopted i [6], [8], [15], [19], [1], [], [7], [9]. I classical measures of system-level diagosability for multiprocessor systems, it has geerally bee assumed that ay subset of processors ca potetially fail at the same time. If there is a ode v whose eighbors are faulty simultaeously, there is o way of kowig the faulty or fault-free stas of v: As a cosequece, the diagosability of a system is upper bouded by its miimum degree. However, it always uderestimates the resiliece of large etworks because the failure probability that all the eighbours of the same ode is very small i may large scale parallel/distribute system. To overcome the shortcomig, Lai et al. [8] proposed a ovel measure of diagosability, called the coditioal diagosability, for measurig the diagosability of a system uder the assumptio that for each ode u all the processors directly coected to u caot fail at the same time, i.e., at least oe of eighbors of u is fault free. They also obtaied coditioal diagosability results for hypercubes uder the PMC model i [8]. I [19], Xu et al. showed the coditioal diagosability of the -dimesioal hypercube uder the MM model is 3 5 for 5: Uder the PMC model, Zhu [47] sdied the coditioal diagosability of bijective coectio (BC) etworks, which iclude hypercubes ad a variety of hypercube variats, such as crossed cubes, twisted cubes, M obius cubes, locally twisted cubes, ad geeralized twisted cubes. I [43], Xu et al. further geeralized the previous result by sdyig this problem i a family of popular etworks, i.e., the Matchig Compositio Networks (MCNs). The coditioal diagosability of Cayley graphs geerated by traspositio trees was sdied first by Li et al. [9]. Results cocerig the coditioal diagosability of variats of the hypercube etwork uder this model were also obtaied [], [44], [46], [47]. Motivated by the cocepts of forbidde faulty sets, Peg et al. [36] the proposed the g-good-eighbor coditioal diagosability: which is defied as the maximum value t such that a graph G remais t-diagosable uder the coditio that every healthy vertex v has at least g fault-free eighborig vertices. Besides, they showed that the g-goodeighbor coditioal diagosability of the -dimesioal hypercube Q : The itercoectio etwork cosidered i this sdy is the k-ary -cube Q k ; proposed i [3], which is oe of the most commo multiprocessor systems for parallel computer/commuicatio system. The k-ary -cube is -regular with k vertices,edgesymmetric,advertex symmetric. The three most popular istaces of k-ary -cubearetherigk¼1, the hypercube k ¼ ; ad the torus ¼ : A umber of distributed memory multiprocessors are based o a k-ary -cube as the uderlyig topology, such as the iwarp [3], the J-machie [34], the Cray T3D [38] ad the Blue Gee [7]. Recetly, Chag et al. showed the coditioal diagosability of k-ary -cube uder the PMC Model is 8 7 for k 4 ad 4: Hsieh ad Lee [] showed the coditioal diagosability of k-ary -cube uder the MM Model is 6 5 for k 4 ad 4: I this paper, we sdy the g-good-eighbor coditioal diagosability uder the PMC model ad MM model, ad show that the g-good-eighbor coditioal diagosability of Q k is ð g þ 1Þ 1 for k 4; 3; 0 g uder the two models, which shows that the correspodig results based o the traditioal fault model (where g is zero) ted to substatially uderestimate etwork reliability of k-ary -cube. The remaider of this paper is orgaized as follows: Sectio provides termiology ad prelimiaries for diagosig a system. I Sectio 3, we discuss the R g -coectivity of Q k : Sectios 4 ad 5 show the g-good-eighbor coditioal diagosability of Q k uder the PMC model ad the MM model, respectively. Fially, our coclusios are give i Sectio 6. PRELIMINARIES.1 Notatios Throughout this paper, a itercoectio etwork is represeted by a udirected simple graph G with the vertex set V ðgþ ad the edge set EðGÞ. A subgraph H of G (writte H G) is a graph with V ðhþvðgþ;eðhþeðgþ ad the edpoits of every edge i EðHÞ belog to V ðhþ: Give a oempty vertex subset V 0 of V ðgþ; the iduced subgraph by V 0 i G; deoted by G½V 0 Š; is a graph, whose vertex set is V 0 ad the edge set is the set of all the edges of G with both edpoits i V 0 : The degree d G ðvþ of a vertex v is the umber of edges icidet with v: A graph G is said to be k-regular if for ay vertex v; d G ðvþ ¼k: For ay vertex v, we defie the eighborhood N G ðvþ of v i G to be the set of vertices adjacet to v: Let A G: We use N G ðaþ to deote the set ð S vv ðaþ N GðvÞÞ V ðaþ; C G ðaþ to deote the set N G ðaþ[ V ðaþ: For eighborhoods ad degrees, we will usually omit the subscript for the graph whe o cofusio arises. The coectivity kðgþ of a graph G is the miimum umber of vertices whose removal results i a discoected graph or oly oe vertex left. For graph-theoretical termiology ad otatio ot defied here we follow [4].. The PMC model A multiprocessor system is modeled as a udirected graph G ¼ðV ðgþ;eþ; whose vertices represet processors ad edges represet commuicatio liks. I the PMC model [37], two adjacet processors ca perform tests o each other. For two adjacet vertices u ad v i V ðgþ; the ordered pair ðu; vþ represets the test performed by u o v: The outcome of a test ðu; vþ is either 1 or 0 with the assumptio that the testig

3 YUAN ET AL.: THE g-good-neighbor CONDITIONAL DIAGNOSABILITY OF k-ary -CUBES UNDER THE PMC MODEL AND MM MODEL 1167 result is regarded as reliable if the vertex u is fault-free. However, the outcome of a test ðu; vþ is ureliable, provided that the tester u itself is origially a faulty processor. Suppose that the vertex u of ðu; vþ is fault-free, the the result would be 0 (respectively, 1) if v is fault-free (respectively, faulty). A test assigmet T for a system G is a collectio of tests for every adjacet pair of vertices. It ca be modeled as a directed testig graph T ¼ðV ðgþ;lþ; where ðu; vþ L implies that u ad v are adjacet i G: The collectio of all test results for a test assigmet T is called a sydrome. Formally, a sydrome is a fuctio s : L!ð0; 1Þ: The set of all faulty processors i the system is called a faulty set. This ca be ay subset of V ðgþ: The process of idetifyig all the faulty vertices is called the diagosis of the system. For a give sydrome s; a subset of vertices F V ðgþ is said to be cosistet with s if sydrome s ca be produced from the siatio that, for ay ðu; vþ L such that u V F; sðu; vþ ¼1 if ad oly if v F: This meas that F is a possible set of faulty processors. Sice a test outcome produced by a faulty processor is ureliable, a give set F of faulty vertices may produce a lot of differet sydromes. O the other had, differet fault sets may produce the same sydrome. We use otatio sðfþ to represet the set of all sydromes which could be produced if F is the set of faulty vertices. Two distict sets F 1 ad F i V ðgþ are said to be idistiguishable if sðf 1 Þ\sðF Þ 6¼ ;; otherwise, F 1 ad F are said to be distiguishable. Besides, we say ðf 1 ;F Þ is a idistiguishable pair if sðf 1 Þ\sðF Þ 6¼ ;; else, ðf 1 ;F Þ is a distiguishable pair..3 The MM model I the MM model, to diagose a system, a vertex seds the same task to two of its eighbors, ad the compares their resposes. To be cosistet with the MM model, we have the followig assumptios: 1) All faults are permaet. ) A faulty processor produces icorrect outputs for each of its give tasks. 3) The output of a compariso performed by a faulty processor is ureliable. 4) Two faulty processors give the same iput ad task do ot produce the same output. The compariso scheme of a system G is modeled as a multigraph, deoted by MðV ðgþ;lþ; where L is the labeled-edge set. A labeled edge ðu; vþ w L represets a compariso i which two vertices u ad v are compared by a vertex w; which implies ðu; wþ; ðv; wþ EðGÞ: The collectio of all compariso results i MðV ðgþ;lþ is called the sydrome, deoted by s ; of the diagosis. The result of the compariso ðu; vþ w i r is deoted by rððu; vþ w Þ: If the compariso ðu; vþ w disagrees, the s ððu; vþ w Þ¼1; otherwise, s ððu; vþ w Þ¼0: Hece, a sydrome is a fuctio from L to 0; 1: The MM model is a special case of the MM model. I the MM model, all comparisos of G are i the compariso scheme of G; i.e., if ðu; wþ; ðv; wþ EðGÞ; the ðu; vþ w L: For a give sydrome s ; a set of faulty vertices F V ðgþ is said to be cosistet with s if s ca be produced from F; i.e., if the followig coditios are satisfied, accordig to the assumptios of the MM model: 1) if u; v F ad w V ðgþ F; the s ððu; vþ w Þ¼1; ) if u F ad v; w V ðgþ F; the s ððu; vþ w Þ¼1; ad 3) if u; v; w V ðgþ F; the s ððu; vþ w Þ¼0: Sice a faulty comparator ca lead to a ureliable result, oe set of faulty vertices may produce differet sydromes. Let s ðf Þ deote the set of all sydromes which F is cosistet with. Two distict sets F 1 ;F V ðgþ are said to be distiguishable if s ðf 1 Þ\s ðf Þ¼;: Otherwise, they are said to be idistiguishable. ðf 1 ;F Þ is a distiguishable pair (respectively, a idistiguishable pair) if F 1 ad F are distiguishable (respectively, idistiguishable)..4 Diagosability I this sectio, some kow coceptios ad results about the diagosability of system are listed as follows. Defiitio.4.1 [11]. A system of processors is t-diagosable if all faulty processors ca be idetified without replacemet, provided that the umber of faults preseted does ot exceed t: The diagosability of a system G deoted as tðgþ; is the maximum value of t such that G is t-diagosable. I [8], Lai et al. preset a sufficiet ad ecessary coditio for a system to be t-diagosable as follows. Theorem.4. [8]. A system G ¼ðV; EÞ is t-diagosable if ad oly if F 1 ad F are distiguishable, for ay two distict subsets F 1 ad F of V with jf 1 jt; jf jt: Note that the diagosability of a system always uderestimates the resiliece of large etworks because the failure probability that all the eighbours of the same ode is very small i may large scale parallel/distribute system. For this reaso, Lai et al. [8] itroduce the coditioal diagosability. They cosider the siatio that ay faulty set caot cotai all the eighbors of ay vertex i a system. Defiitio.4.3 [8]. A system G ¼ðV; EÞ is coditioally t-diagosable if F 1 ad F are distiguishable, for each pair of distict faulty sets F 1 ;F V with jf 1 jt; jf jt ad F 1 + NðvÞ;F + NðvÞ for ay vertex v V: The coditioal diagosability t c ðgþ of a graph G is the maximum value of t such that G is coditioally t-diagosable. Motivated by these cocepts of coditioally t-diagosability ad forbidde faulty sets [8], [4], Peg et al. [36] the propose the g-good-eighbor coditioal diagosability by claimig that for every fault-free vertex i a system, it has at least g fault-free eighbors. Defiitio.4.4 [36]. A faulty set F V is called a g-goodeighbor coditioal faulty set if jnðvþ\ðv FÞj g for every vertex v i V F: Defiito.4.5 [36]. A system G ¼ðVðGÞ;EÞ is g-goodeighbor coditioal t-diagosable if each distict pair of g-good-eighbor coditioal faulty sets F 1 ad F of V with jf 1 jt; jf jt are distiguishable. Defiito.4.6 [36]. The g-good-eighbor coditioal diagosability t g ðgþ of a graph G is the maximum value of t such that G is g-good-eighbor coditioally t-diagosable..5 k-ary -cube The k-ary -cube Q k (k ad 1) is a graph cosistig of k vertices, each of which has the form u 1 u u ; where 0 u i k 1 for 1 i : Two vertices u ¼ u 1 u u ad

4 1168 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 6, NO. 4, APRIL 015 Fig ary -cube Q 4 ad 4-ary -dimesioal hypercube Q(,4). Fig. v ¼ v 1 v v are adjacet if ad oly if there exists a iteger j; 1 j ; such that u j ¼ v j 1 (mod k) ad u l ¼ v l for every l 6¼ j; 1 l : Such a edge ðu; vþ is called a j-dimesioal edge. For brevity, we omit writig (mod k) i similar expressios for the remaider of the paper. Note that each vertex of Q k has degree whe k 3; ad whe k ¼ : Obviously, Q k 1 is a cycle of legth k; Q is a -dimesioal hypercube, ad Q k is a k k wrap-aroud mesh. Fig. 1a shows the four-ary two-cube Q 4 : We ca partitio Q k alog the dimesio j; by deletig all the j-dimesioal edges, ito k disjoit subcubes, Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š (abbreviated as Q½0Š;Q½1Š;...; Q½k 1Š; if there is o ambiguity). It is clear that Q j ½iŠ is a subgraph of Q k iduced by fu : u ¼ u 1 u u j u V ðq k Þ ad u j ¼ ig ad each Q j ½iŠ is isomorphic to Q k 1 for every 0 i k 1: Moreover, there is a perfect matchig betwee Q j ½iŠ ad Q j ½i þ 1Š for 0 i k 1. Let X be a do t care symbol ad let X t ¼ XX fflfflfflfflffl{zfflfflfflfflffl} X : t For coveiece of represetatio, we deote by a -legth strig of symbols X m a l X m l the subgraph i Q k iduced by the vertex set fv ¼ u 1 u u V ðq k Þju mþ1;u mþ ;...; u mþl ¼ ag: Let Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š be a partitio of Q k alog some dimesio j: Clearly, Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š ca be deoted by X j 1 0X j ;X j 1 1X j ;...;X j 1 ðk 1ÞX j ; respectively. The k-ary -dimesioal hypercube Qð; kþ is a graph cosistig of k vertices, each of which has the form u 1 u u ; where 0 u i k 1 for 1 i : Two vertices u ¼ u 1 u u ad v ¼ v 1 v v are adjacet if ad oly if there exists exactly a dimesio j; 1 j ; such that u j 6¼ v j ad u l ¼ v l for every l 6¼ j: Note that each vertex of Qð; kþ has degree ðk 1Þ: Obviously, Qð; Þ is a -dimesioal hypercube Q,adQð; 3Þ is a three-ary -cube Q 3 : Fig. 1b shows the four-ary two-dimesioal hypercube Qð; 4Þ: 3 THE R g -CONNECTIVITY OF k-ary -CUBE Q k I order to get the g-good-eighbor coditioal diagosability of Q k ; we first eed to discuss the R g-coectivity of Q k ; which is closely related to g-good-eighbor coditioal diagosability, proposed by Latifi et al. [5]. A g-goodeighbor coditioal cut of a graph G is a g-good-eighbor coditioal faulty set F such that G F is discoected. The cardiality of the miimum g-good-eighbor coditioal cut is said to be the R g -coectivity of G; deoted by. The subgraph iduced by the eighbors N Q k ðhþ of H. k g ðgþ: As a more refied idex tha the traditioal coectivity, the R g -coectivity ca be used to measure of coditioal fault tolerace of etworks. There are may results cocerig the R g -coectivity for particular classes of itercoectio etworks ad small g s, such as [9], [1], [5], [6], [35], [41], [4], [45]. But, with regard to geeral iteger g; little iformatio has bee foud. I this sectio, we determie the R g -coectivity of k-ary -cube Q k for k 4; 3 ad 0 g : The followig lemma discusses some properties of the subgraph of Q k ; which is isomorphic to the g-dimesioal hypercube Q g ; 0 g : By this Lemma, a upper boudary of the R g -coectivity of Q k is obtaied. Lemma 3.1. Let H be a iduced subgraph of k-ary -cube Q k such that H is isomorphic to the g-dimesioal hypercube Q g ; ad let C Q k ðhþ ¼N Q k ðhþ[vðhþ; where 0 g ; k 5 ad 3: The jn Q k ðhþj ¼ ð gþ g ad the miimum degree of Q k C Q k ðhþ is ot less tha : Proof. By the symmetry of Q k ; without loss of geerality, let H ¼ X g 0 g ; where X f0; 1g: We first prove ay pair of vertices i H have o commo eighbors i N Q k ðhþ: Suppose, o the cotrary, there are two vertices u; v V ðhþ such that they have commo eighbors i N Q k ðhþ: Say u ¼ u 1 u u g 0 g ;v¼ v 1 v v g 0 g : The, by the defiitio of k-ary -cube, we have either there is precise oe dimesio i f1; ;...;gg such that u i 6¼ v i or there are precise two dimesios i; j f1; ;...;gg such that u i 6¼ v i ;u j 6¼ v j : Cosider the followig two cases. Case 1. There is precise oe dimesio i f1; ;...;gg such that u i 6¼ v i : I this case, we have u i ¼ v i ðmodkþ: Sice u; v V ðhþ; it follows that u i ;v i f0; 1g ad hece u i ¼ v i 1; cotradictig u i ¼ v i ðmodkþ: Case. There are precise two dimesios i; j f1; ;...;gg such that u i 6¼ v i ad u j 6¼ v j : Without loss of geerality, let i ¼ 1;j¼ : The, u ad v precisely have two commo eighbors w 1 ¼ u 1 v u 3... u g 0 g ad w ¼ v 1 u u 3 u g 0 g i Q k : Sice u; v V ðhþ; it follows that u l ;v l f0; 1g for ay 1 l g: Therefore, both w 1 ad w belog to V ðhþ; that is u ad v have o commo eighbors i N Q k ðhþ: Therefore, ay pair of vertices i H have o commo eighbors i N Q k ðhþ: Combiig this with H ¼ X g 0 g ; X f0; 1g; it is ot difficult to see that (the subgraph iduced by the eighbors of H is showed i Fig. ):

5 YUAN ET AL.: THE g-good-neighbor CONDITIONAL DIAGNOSABILITY OF k-ary -CUBES UNDER THE PMC MODEL AND MM MODEL 1169 NQ k ðhþ ¼jVðX g 10 g 1 Þ[VðX g 010 g Þ [[VðX g 0 g 1 1Þj þ jv ðx g ðk 1Þ0 g 1 Þ [ V ðx g 0ðk 1Þ0 g Þ[[VðX g 0 g 1 ðk 1ÞÞj þjvðx g 1 0 g Þ[VðXX g 0 g Þ [[VðX g 1 0 g Þj þ jv ððk 1ÞX g 1 0 g Þ [ V ðxðk 1ÞX g 0 g Þ[[VðX g 1 ðk 1Þ0 g Þj ¼jVðX g 10 g 1 Þj þ jv ðx g 010 g Þjþ þjvðx g 0 g 1 1Þj þ jv ðx g ðk 1Þ0 g 1 Þj þjvðx g 0ðk 1Þ0 g Þjþ þjvðx g 0 g 1 ðk 1ÞÞj þ jv ðx g 1 0 g Þj þjvðxx g 0 g Þþ þjvðx g 1 0 g Þj þ jv ððk 1ÞX g 1 0 g Þj þjvðxðk 1ÞX g 0 g Þjþ þjvðx g 1 ðk 1Þ0 g Þj ¼ g ð gþþ g ð gþþ g 1 g þ g 1 g ¼ g ð gþ: We ow shall show the miimum degree of Q k C Q k ðhþ is ot less tha : For ay vertex x V ðq k C Q k ðhþþ; assume that x ¼ðu 1u u g u gþ1 u Þ: Cosider the followig six cases: Case 1. There is precise oe elemet, which is or k 1; i fu 1 ;u ;...;u g g; ad the others are 0 or 1; there is precise oe elemet, which is 1 or k 1 i u gþ1 ; u gþ ;u gþ3 ;...;u ; ad the others are all 0: Without loss of geerality, let x ¼ðu u g 10 g 1 Þ: Note k 5: The the eighbors of x i C Q k ðhþ are ð1u u g 10 g 1 Þ ad ðu u g 0 g Þ: So d Q k C Q k ðhþ ðxþ ¼ : Case. Each of u 1 ;u ;...;u g is 0 or 1; ad precise oe of u gþ1 ;u gþ ;...;u is or k ; ad the others are all 0: Without loss of geerality, let x ¼ðu 1 u u g 0 g 1 Þ: The the eighbor of x i C Q k ðhþ is ðu 1 u u g 10 g 1 Þ: So d Q k C Q k ðhþðxþ ¼ 1: Case 3. All of u 1 ;u ;...;u g are 0 or 1; ad there are precise two elemets, which are 1 or k 1 i fu gþ1 ; u gþ ;...;u g; ad the others are all 0: Without loss of geerality, let x ¼ðu 1 u u g 110 g Þ: The the eighbors of x i C Q k ðhþ are ðu 1 u u g 10 g 1 Þ ad ðu 1 u u g 010 g Þ: So d Q k C Q k ðhþðxþ ¼ : Case 4. Precise two elemets of fu 1 ;u ;...;u g g are or k 1; ad the others are 0 or 1; ad all of u gþ1 ; u gþ ;...;u are 0: Without loss of geerality, let x ¼ðu 3 u g 0 g Þ: The the eighbors of x i C Q k ðhþ are ð1u 3 u g 0 g Þ ad ð1u 3 u g 0 g Þ: So d Q k C Q k ðhþðxþ ¼ : Case 5. Precise oe elemet of u 1 ;u ;...;u g is 3 or k ; ad the others are 0 or 1; all of u gþ1 ;u gþ ;...;u are 0: Without loss of geerality, let x ¼ð3u u g 0 g Þ: If k ¼ 5; the the eighbors of x i C Q k ðhþ are ðu u g 0 g Þ ad ð4u u g 0 g Þ: Otherwise, the eighbor of x i C Q k ðhþ is ðu u g 0 g Þ: So d Q k C Q k ðhþðxþ : Case 6. x does ot satisfy ay oe of the above five cases. The x is ot adjacet to ay vertex of C Q k ðhþ: So d Q k C Q k ðhþðxþ ¼: The proof is complete. Corollary 3.. The R g -coectivity k g ðq k Þð gþg for 0 g ; k 5 ad 3: Proof. Let H be a iduced subgraph of k-ary -cube such that H ffi Q g : By Lemma 3.1, jn Q k ðhþj ¼ ð gþ g ad the miimum degree dðq k C Q k ðhþþ : Sice 3; it follows that : Thus dðq k N Q k ðhþþ ¼ dðhþ¼ g: By the defiitios of g-good-eighbor coditioal cut ad R g -coectivity, we have N Q k ðhþ is a g-good-eighbor coditioal cut of Q k ad hece k g ðq k ÞjN Q k ðhþj ¼ ð gþg : The proof is complete. Next, we shall show ð gþ g is also the lower boudary of k g ðq k Þ for 0 g ; k 5 ad 3: Before doig this, we eed to have some useful topological properties of Q k : Lemma 3.3. Let H be a coected subgraph of k-ary -cube such that the miimum degree dðhþ of H is ot less tha g; where 0 g ; 3;k 4: The jv ðhþj g : Proof. The proof is by iductio o g: Clearly, the result is true for the base cases g ¼ 0 ad 1: Assume the result is true for g 1: We ow cosider g : Sice dðhþ g ;Q k ca be partitioed ito Q½0Š;Q½1Š;...;Q½k 1Š alog some dimesio l such that H is ot a subgraph of ay Q½iŠ for i ¼ 0; 1;...;k 1: By symmetry, without loss of geerality, we may assume V ðhþ \ V ðq½išþ 6¼ ; for i ¼ 0; 1;...;p 1; where p k: For 0 i p 1; let H i be the iduced subgraph Q½iŠ½V ðhþ \ V ðq½išþš ad u be a arbitrary vertex i H i : It is sufficiet to discuss the followig two cases. Case 1. p k 1: Clearly, for i ¼ 0 or p 1;d Hi ðuþ d H ðuþ 1 g 1; ad for i ¼ 1; ;...;p ; d Hi ðuþ d H ðuþ g : It follows that dðh 0 Þg 1; dðh p 1 Þg 1; ad dðh i Þ g for i ¼ 1; ;...;p : By iductio, we have jv ðh 0 Þj g 1 ; jv ðh p 1 Þj g 1 ; ad jv ðh i Þj g for i ¼ 1; ;...;p : So, jv ðhþj ¼ jv ðh 0 Þj þ jv ðh 1 ÞjþþjVðH p 1 Þj g 1 ¼ g : Case. p ¼ k: Clearly, d Hi ðuþ d H ðuþ g : It follows that dðh i Þg for i ¼ 0; 1;...;k 1: By iductio, jv ðh i Þj g ; ad hece jv ðhþj ¼ jv ðh 0 Þj þ jv ðh 1 ÞjþþjVðH k 1 Þj g k g : The proof is complete. Lemma 3.4. Let Q i ½0Š;Q i ½1Š;...;Q i ½k 1Š be the decompositio of Q k ðk 5; 3Þ alog some dimesio i ad let A be a

6 1170 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 6, NO. 4, APRIL 015 coected subgraph of Q k ad A Qk ½T i¼1 ððv ðq i½0šþ [ V ðq i ½1ŠÞ [ V ðq i ½ŠÞÞŠ: If 0 g dðaþ ; the jn Q k ðaþj g ð gþ: Proof. The proof proceeds by iductio o g: Sice jn Q k ðaþj kðq k Þ¼; the statemet is true for the base case g ¼ 0: Cosider the case g 1: Let R ¼ Q k ½T i¼1 ððv ðq i½0šþ [ V ðq i ½1ŠÞ [ V ðq i ½ŠÞÞŠ: Sice A R ad dðaþg 1; there exists a dimesio i f1; ;...;g such that at least two of A \ Q i ½0Š;A\ Q i ½1Š;A\ Q i ½Š are ot empty. Case 1. Exactly two of A \ Q i ½0Š;A\ Q i ½1Š;A\ Q i ½Š are ot empty. Without loss of geerality, say A 0 ¼ A \ Q i ½0Š 6¼ ;; A 1 ¼ A \ Q i ½1Š 6¼ ;: By the costructio of Q k ad dðaþ g; we ca deduce that dða 0 Þg 1 ad dða 1 Þg 1: Furthermore, by iductio, we have jn Qi ½0ŠðA 0 Þj g 1 ð g 1Þ ad jn Qi ½1ŠðA 1 Þj g 1 ð g 1Þ: Sice dða 0 Þg 1 ad dða 1 Þg 1; by Lemma 3.3, we ca deduce that jv ða 0 Þj g 1 ; jv ða 1 Þj g 1 : Combiig this with the costructio of Q k ; we ca deduce jn Q i ½k 1ŠðV ða 0 ÞÞj¼jV ða 0 Þj g 1 ; jn Qi ½ŠðV ða 1 ÞÞj ¼ jv ða 1 Þj g 1 : Therefore, NQ k ðaþ jnqi ½k 1ŠðV ða 0 ÞÞj þ jn Qi ½0ŠðA 0 Þj þjn Qi ½1ŠðA 1 Þj þ jn Qi ½ŠðV ða 1 ÞÞj g ð gþ: Case. All of A \ Q i ½0Š;A\ Q i ½1Š;A\ Q i ½Š are ot empty.deote A 0 ¼ A \ Q i ½0Š;A 1 ¼ A \ Q i ½1Š;A ¼ A\ Q i ½Š: By the costructio of Q k ad dðaþ g; we ca deduce that dða 0 Þg 1; dða 1 Þg ad dða Þ g 1: By iductio, we have jn Qi ½0ŠðA 0 Þj g 1 ð g 1Þ; jn Qi ½1ŠðA 1 Þj g ð gþ ad jn Qi ½ŠðA Þj g 1 ð g 1Þ: Sice dða 0 Þg 1 ad dða Þg 1; by Lemma 3.3, we ca deduce that jv ða 0 Þj g 1 ; jv ða Þj g 1 : Clearly, jn Qi ½k 1ŠðV ða 0 ÞÞj ¼ jv ða 0 Þj g 1 ; jn Qi ½3Š ðv ða ÞÞj ¼ jv ða Þj g 1 : Therefore, NQ 3 ðaþ jnqi ½k 1ŠðV ða 0 ÞÞj þ jn Qi ½0ŠðA 0 Þj þjn Qi ½1ŠðA 1 Þj þ jn Qi ½ŠðA Þj þjn Qi ½3ŠðV ða ÞÞj > g ð gþ: The proof is complete. The g-restricted coectivity of G is closed related to the k g -coectivity. A vertex cut of G is called a g-restricted cut if G S is discoected ad every compoet of G S has more tha g vertices. The g-restricted coectivity of G; deoted by ek g ðgþ; is defied as the cardiality of a miimum g-restricted cut. Clearly, k 0 ðgþ ¼ek 0 ðgþ ¼kðGÞ ad k 1 ðgþ ¼ek 1 ðgþ: I 004, Day determied the 1-restricted coectivity of Q k ðk 4Þ [10]. Lemma 3.5 [3], [10]. The g-restricted coectivity of Q k ðk 4Þ, ek g ðq k Þ¼ðgþ1Þ gfor ad g ¼ 0; 1: The followig Lemma shows ð gþ g is the lower boudary of k g ðq k Þ for 0 g ; k 5 ad 3: Lemma 3.6. The R g -coectivity k g ðq k Þð gþg for 0 g ; k 5 ad 3: Proof. Let F be a arbitrary R g -cut of k-ary -cube Q k, where 0 g ; k 5 ad 3: It is sufficiet to show that jf j g ð gþ: We shall prove jf j g ð gþ by iductio o g: By Lemma 3.5, the statemet is true for the base cases g ¼ 0; 1: We ow cosider g : Let Q½0Š;Q½1Š;...;Q½k 1Š be a decompositio of Q k alog dimesio lð1 l Þ ad let F i ¼ F \ V ðq½išþ for ay i ¼ 0; 1;...;k 1 (F i is allowed to be empty.). We discuss the followig two cases. Case 1. Q½iŠ F i is discoected for ay i ¼ 0; 1;...;k 1: Sice F is a R g -cut of Q k ; we have dðqk F Þg: Note that the l-dimesioal edge set betwee Q½iŠ ad Q½i þ 1Š is a perfect matchig of V ðq½išþ ad V ðq½i þ 1ŠÞ. Thus, for ay i ¼ 0; 1;...;k 1; dðq½iš F i Þg : It follows that F i is a R g -cut of Q½iŠ: By iductio, jf i j g ð gþ for i ¼ 0; 1;...;k 1: Therefore, jf j¼ jf 0 jþjf 1 jþþjf k 1 jk g ð gþ g ð gþ: Case. There exists a Q½iŠ such that Q½iŠ F i is coected. Assume that G i0 ¼ Q½i 0 Š F i0 is coected ad G is the compoet of Q k F such that G i 0 G: Without loss of geerality, assume that V ðgþ\vðq½jšþ 6¼ ; for j ¼ 0; 1;...;p 1ð p kþ: Clearly, 0 i 0 p 1: Case.1. p k 1: Let x be a arbitrary vertex i G i0 ad let x i V ðq½išþði 6¼ i 0 Þ be the vertex such that just the ith coordiate of x i is differet from that of x: Sice p k 1; it follows that there exists a iteger i such that x i F F i0 : Thus jf j¼jf F i0 jþjf i0 j jv ðg i0 Þj þ jf i0 j¼jvðq½i 0 ŠÞj ¼ k 1 g ð gþ: Case.. p ¼ k: For i ¼ 0; 1;...;k 1; deote G \ Q½iŠ by G i : Case..1. There are at least four of Q½iŠ F i for i ¼ 0; 1;...;k 1; such that each of them is discoected. Say Q½i 1 Š F i1 ;Q½i Š F i ;Q½i 3 Š F i3 ;Q½i 4 Š F i4 are discoected. Clearly, dðq k F Þg: Note that the l-dimesioal edge set betwee Q½iŠ ad Q½i þ 1Š is a perfect matchig of V ðq½išþ ad V ðq½i þ 1ŠÞ. We ca deduce F i1 ¼ F \ Q½i 1 Š;F i ¼ F \ Q½i Š;F i3 ¼ F \ Q½i 3 Š;F i4 ¼ F \ Q½i 4 Š are R g cuts of Q½i 1 Š;Q½i Š;Q½i 3 Š;Q½i 4 Š; respectively. By iductio, jf i1 j g ð gþ; jf i j g ð gþ; jf i3 j g ð gþ; jf i4 j g ð gþ: Thus jfj jf i1 jþjf i jþjf i3 jþjf i4 j g ð gþ: Case... There are at most three of Q½iŠ F i ;i¼ 0; 1;...;k 1 such that each of them is discoected. By cotradictio, suppose jfj < g ð gþ: We ow shall show two importat claims. Claim 1. jv ðgþj >k 13 g ð gþ: Suppose there are exactly three of Q½iŠ F i ;i¼ 0; 1;...;k 1 such that each of them is discoected. Say Q½i 1 Š F i1 ;Q½i Š F i ;Q½i 3 Š F i3 are discoected. If i 1 ;i ;i 3 are three cotiuous itegers(mod k), the without loss of geerality, assume i 1 ¼ 0;i ¼ 1;i 3 ¼ : Clearly, F 0 ;F 1 ;F are R g -cuts of Q½0Š;Q½1Š ad Q½Š; respectively. By iductio, jf i j g ð gþ for i ¼ 0; 1; : Let H i ¼ Q½iŠ F i G i for i ¼ 0; 1; : The, by

7 YUAN ET AL.: THE g-good-neighbor CONDITIONAL DIAGNOSABILITY OF k-ary -CUBES UNDER THE PMC MODEL AND MM MODEL 1171 the costructio of Q k ; we have jv ðh 0Þj jf k 1 j; jv ðh Þj jf 3 j ad jv ðh 1 Þj mifjv ðh 0 Þj þ jf 0 j; jv ðh Þj þ jf jg jv ðh 0Þj þ jf 0 jþjvðh Þj þ jf j jf k 1jþjF 0 jþjf 3 jþjf j jf j jf 1j : Sice jf j < g ð gþ; jf i j g ð gþ for i ¼ 0; 1; ; ad jf jjf 0 jþjf 1 jþjf jþjf 3 jþjf k 1 j; we have that jf 3 jþjf k 1 j < g ð gþ: Therefore, jv ðgþj ¼ jv ðq k Þj jv ðh 0Þj jv ðh 1 Þj jv ðh Þj jf j k jf k 1 j jf j jf 1j jf 3 j jfj >k g ð gþ g ð gþ g ð gþ g ð gþ ¼ k 13 g ð gþ: If i 1 ;i are cotiuous itegers(mod k), but i 1 ;i ;i 3 are ot cotiuous, the without loss of geerality, assume i 1 ¼ 0;i ¼ 1: Clearly, we ca show that F 0 ;F 1 ;F i3 are R g -cuts of Q½0Š;Q½1Š ad Q½i 3 Š; respectively. By iductio, jf i j g ð gþ for i ¼ 0; 1;i 3 : Let H i ¼ Q½iŠ F i G i for i ¼ 0; 1;i 3 : The jv ðh 0 Þj jf k 1 j; jv ðh 1 Þj jf j; jv ðh i3 Þj mifjf i3 1j; jf i3 þ1jg jf i 3 1jþjF i3 þ1j jfj jf 0 j jf 1 j jf i3 j : Therefore, jv ðgþj ¼ V ðq k Þ jvðh0 Þj jv ðh 1 Þj jv ðh i3 Þj jf j k jf k 1 j jf j jfj jf 0j jf 1 j jf i3 j jfj >k g ð gþ g ð gþ 3 g ð gþ g ð gþ ¼ k 13 g ð gþ: Suppose that i 1 ;i ad i 3 are ot cotiuous itegers pairwise. Clearly, we ca show that F i1 ;F i ;F i3 are R g -cuts of Q½i 1 Š;Q½i Š ad Q½i 3 Š; respectively. By iductio, jf ij j g ð gþ for j ¼ 1; ; 3: Let H ij ¼ Q½i j Š F ij G ij for j ¼ 1; ; 3: The jv ðh i1 Þj mifjf i1 1j; jf i1 1jg jf i 1 1jþjF i1 1j ; jv ðh i Þj mifjf i 1j; jf i 1jg jf i 1jþjF i 1j ad jv ðh i3 Þj mifjf i3 1j; jf i3 þ1jg jf i 3 1jþjF i3 þ1j : Sice jfj < g ð gþ ad jf ij j g ð gþ for j ¼ 1; ; 3; we have that Therefore, X 3 j¼1 X 3 V ðhij Þ 1 Fij 1 þ Fij þ1 Þ j¼1 1 jf j X3 jf ij j jv ðgþj ¼ V ðq k Þ X 3 j¼1 < 7 g 3 ð gþ: j¼1 jv ðh ij Þj jf j >k g 3 ð gþ g ð gþ >k 13 g ð gþ: Usig a similar discussio, we ca deduce that the Claim is true whe there are at most two of Q½iŠ F i ;i¼ 0; 1;...;k 1 such that each of them is discoected. The proof of Claim 1 is complete. Claim. For ay decompositio Q j ½0Š;Q j ½1Š;...; Q j ½k 1Š of Q k alog some dimesio 1 j ; G itersects with every oe of Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š: Suppose o the cotrary that there exists a decompositio Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š of Q k alog some dimesio 1 j such that G does ot itersect with every oe of Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š: Without loss of geerality, assume that G \ Q j ½0Š;G\ Q j ½1Š;...;G\ Q j ½l 1Š are ot empty. Clearly, l k 1: Deote G \ Q j ½iŠ by G i for 0 i l 1: Let F j ½iŠ ¼F \ V ðq j ½iŠÞ for i ¼ 0; 1;...; k 1: If Q j ½iŠ F j ½iŠ is discoected for ay i ¼ 0; 1;...;k 1; the by Case 1, we have jf j g ð gþ; a cotradictio. So suppose that there exists a Q j ½iŠ such that Q j ½iŠ F j ½iŠ is coected. Let Q j ½i 0 Š F j ½i 0 Š be coected, ad H be the compoet of Q k F such that Q j ½i 0 Š F j ½i 0 Š is a subgraph of H: If H does ot itersect with every oe of Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š; the similar to Case.1, we ca coclude jf j g ð gþ; a cotradictio. So suppose H itersects with every oe of Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š: By the assumptio that G does ot itersect with every oe of Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š; we ca coclude H 6¼ G: If there are at least four of Q j ½iŠ F j ½iŠ;i¼ 0; 1;...;k 1; such that each of them is discoected, the similar to Case..1, we have jfj g ð gþ; a cotradictio. Therefore, there are at most three of Q j ½iŠ F j ½iŠ;i¼ 0; 1;...;k 1; such that each of them is discoected. Similar to the Claim 1 of Case.., we have jv ðhþj k 13 g ð gþ: Sice F is a R g 1 -cut of Q k ; by iductio, we have jfj g 1 ½ g þ 1Š: So k ¼jVðQ k Þj jv ðgþj þ jv ðhþj þ jfj > k 13 g ð gþ þ g 1 ½ g þ 1Š: It is easy to show k < k 13 g ð gþ þ g 1 ½ g þ 1Š;

8 117 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 6, NO. 4, APRIL 015 Fig. 3. Illustratio of a distiguishable pair ðf 1 ;F Þ uder the PMC model. a cotradictio. The proof of Claim is complete. Let A be a aother compoet of Q k F except G: The by Claim ad the above discussio, we ca deduce that for ay decompositio Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š of Q k alog some dimesio 1 j ; A itersects at most three of Q j ½0Š;Q j ½1Š;...;Q j ½k 1Š: Without loss of geerality, let A Q k ½T j¼1 ðv ðq j½0šþ [ V ðq j ½1ŠÞ ðq j ½ŠÞÞŠ: Sice F is a R g -cut of Q k ; it follows that dðaþ g: The by Lemma 3.4, jf jjn Q 3 ðaþj g ð gþ; a cotradictio. Latifi et al. [5], ad Wu ad Guo [4] have sdied the R g -coectivity of the hypercube ad got the followig result. Theorem 3.7 [5], [4]. Assume that 3 ad 0 g : The the R g -coectivity of -dimesioal hypercube Q ; k g ðq Þ¼ð gþ g : By the defiitio of k-ary -cube, we have Q 4 ¼ Q : So, by Theorem 3.7, we ca deduce the followig corollary. Corollary 3.8. Assume that ad 0 g : The the R g -coectivity of four-ary -cube Q 4 ; k gðq 4 Þ¼ð gþg : Combiig Corollaries 3. ad 3.8, ad Lemma 3.6, we ca obtai the R g -coectivity of Q k for 0 g ; 3 ad k 4: Theorem 3.9. Assume that k 4; 3 ad 0 g : The the R g -coectivity of k-ary -cube Q k ; k gðq k Þ¼ð gþg : 4 THE g-good-neighbor CONDITIONAL DIAGNOSABILITY OF k-ary -CUBE Q k UNDER THE PMC MODEL I this sectio, we shall show the g-good-eighbor coditioal diagosability of k-ary -cube Q k uder the PMC model. Let G ¼ðV; EÞ be a udirected graph of a system G. Let F 1 ad F be two distict subsets of V; ad let the symmetric differece F 1 ~F ¼ðF 1 [ F Þ ðf 1 F Þ: I 1984, Dahbura ad Masso [11] proposed a sufficiet ad ecessary coditio for two distict subsets F 1 ad F to be a distiguishable-pair uder the PMC model. Theorem 4.1 [11]. For ay two distict subsets F 1 ad F of V; ðf 1 ;F Þ is a distiguishable pair uder the PMC model if ad oly if there is a vertex u V ðf 1 [ F Þ ad there is aother vertex v ðf 1 ~F Þ such that ðu; vþ E (see Fig. 3). The followig lemma follows from Defiitio.4.5 ad Theorem 4.1. Lemma 4.. A system G is g-good-eighbor coditioal t-diagosable uder the PMC model if ad oly if there is a edge ðu; vþ EðGÞ with u V ðf 1 [ F Þ ad v ðf 1 ~F Þ for each distict pair of g-good-eighbor coditioal faulty subsets F 1 ad F of V with jf 1 jtad jf jt: Fig. 4. A illustratio about the proofs of Theorem 4.3 ad Lemma 5.5. Let g be a positive iteger with 0 g : To fid the g-good-eighbor coditioal diagosability t g ðq k Þ uder the PMC model, we first show that t g ðq k Þ is o more tha ð g þ 1Þ g 1 for k 5; 3 ad 0 g : Theorem 4.3. Assume that k 5; 3 ad 0 g : The the g-good-eighbor coditioal diagosability of k-ary -cube Q k uder the PMC model, t gðq k Þð gþ1þg 1: Proof. Let A ffi Q g be a subgraph of Q k ad let F 1 ¼ N Q k ðaþ;f ¼ C Q k ðaþ (See Fig. 4). The by Lemma 3.1, jf 1 j¼ð gþ g ; jf j¼ð gþ1þ g ; ad the miimum degree of Q k F is ot less tha ; i.e., F 1 ad F are two g-good-eighbor coditioal faulty sets of V ðq k Þ with F 1 ð gþ1þ g ad F ð gþ1þ g : O the other had, sice V ðaþ ¼F 1 DF ad N Q k ðaþ ¼F 1 F ; there is o edge ðu; vþ EðQ k Þ with u V ðf 1 [ F Þ ad v ðf 1 ~F Þ: By Lemma 4. ad Defiitio.4.5, the g-good-eighbor coditioal diagosability t g ðq k Þ ð g þ 1Þ g 1 uder the PMC model. The proof is complete. Next, we show that t g ðq k Þ is o less tha ð g þ 1Þg 1 for k 5; 3 ad 0 g : Theorem 4.4. Assume that k 5; 3 ad 0 g : The the g-good-eighbor coditioal diagosability of k-ary -cube Q k uder the PMC model, t gðq k Þð gþ1þg 1: Proof. By Defiitio.4.6, it is sufficiet to show Q k is g-good-eighbor coditioal ð g þ 1Þ g 1-diagosable. By Defiitio.4.5, to prove Q k is g-good-eighbor coditioal ð g þ 1Þ g 1-diagosable, it is equivalet to prove that F 1 ad F must be distiguishable for every two distict g-good-eighbor coditioal faulty sets F 1 ad F of Q k with jf 1jð gþ1þ g 1 ad jf jð gþ1þ g 1: We prove this statemet by cotradictio. Suppose that there are two distict g-good-eighbor coditioal faulty sets F 1 ad F with jf 1 jð gþ1þ g 1 ad jf jð gþ1þ g 1; but ðf 1 ;F Þ is idistiguishable. Now we cosider all the possible cases such that F 1 ad F are idistiguishable. By Theorem 4.1, there are two cases such that F 1 ad F are idistiguishable: V ðq k Þ¼F 1 [ F or V ðq k Þ 6¼ F 1 [ F but there is o edge from V ðq k Þ ðf 1 [ F Þ to F 1 DF : Without loss of geerality, assume that F F 1 6¼;: We shall show that each of the above cases has a cotradictio with our assumptio. Case 1. V Q k ¼ F1 [ F :

9 YUAN ET AL.: THE g-good-neighbor CONDITIONAL DIAGNOSABILITY OF k-ary -CUBES UNDER THE PMC MODEL AND MM MODEL 1173 Sice k 5; 3;g ad V ðq k Þ¼F 1 [ F ; we deduce that 5 k ¼ V Q k ¼jF1 [ F jjf 1 jþjf j ½ð gþ1þ g 1Š < ð þ 1Þ þ1 ; which is a cotradictio. Case. V ðq k Þ 6¼ F 1 [ F : Sice F 1 ad F are idistiguishable, by Theorem 4.1 (See Fig. 3), there are o edges betwee V ðq k Þ ðf 1 [ F Þ to F 1 DF : By the assumptio that F F 1 6¼;ad F 1 is a g-good-eighbor coditioal faulty set, ay vertex i F F 1 has at least g good eighbors i the subgraph iduced by F F 1 : By Lemma 3.3, we have jf F 1 j g : Sice F 1 ad F are both g-good-eighbor coditioal faulty sets, F 1 \ F is also a g-good-eighbor coditioal faulty set. I additio, sice there are o edges betwee V ðq k Þ ðf 1 [ F Þ to F 1 DF ;Q k F 1 \ F is discoected, that is F 1 \ F is a R g -cut of Q k : By Theorem 3.9, the cardiality of the miimum R g -cut of Q k is ð gþg : Thus, we obtai that jf 1 \ F j ð gþ g : As a result, jf j¼jf F 1 jþjf 1 \ F j g þð gþ g ; which cotradicts with jf j g þð gþ g 1: Based o these discussios above, we coclude that t g ðq k Þð gþ1þg 1: The proof of this theorem is complete. Recetly, the g-good-eighbor coditioal diagosability of hypercube t g ðq Þ uder the PMC model is show by Peg et al. [36]. Theorem 4.5 [36]. The g-good-eighbor coditioal diagosability of Q uder the PMC model is t g ðq Þ¼ g ð g þ 1Þ 1; if g 3; 1 1; if g 1: Note that Q 4 ¼ Q : So, by Theorem 4.5, we ca deduce the followig result. Corollary 4.6. The g-good-eighbor coditioal diagosability of uder the PMC model is Q 4 ( t g ðq 4 Þ¼ g ð g þ 1Þ 1; if g 3; 1 1; if g 1: Combiig Theorems 4.3, 4.4 ad Corollary 4.6, we ca obtai the g-good-eighbor coditioal diagosability of Q k for 0 g ; 3 ad k 4: Theorem 4.7. Assume that k 4; 3 ad 0 g : The the g-good-eighbor coditioal diagosability of k-ary -cube Q k ; uder the PMC model, t gðq k Þ¼ ð g þ 1Þ g 1: 5 THE g-good-neighbor CONDITIONAL DIAGNOSABILITY OF k-ary -CUBE Q k UNDER THE MM MODEL I this sectio, we shall show the g-good-eighbor coditioal diagosability of k-ary -cube Q k uder the MM model. Fig. 5. Illustratio of a distiguishable pair ðf 1 ;F Þ uder the MM model. Let G ¼ðV; EÞ be a udirected graph of a system G. I 199, Dahbura ad Masso [39] proposed a sufficiet ad ecessary coditio for two distict subsets F 1 ad F to be a distiguishable pair uder the MM model. Theorem 5.1 [39]. For ay two distict subsets F 1 ad F of V ðgþ; ðf 1 ;F Þ is a distiguishable pair uder the MM model if ad oly if oe of the followig coditios is satisfied (see Fig. 5): (1) There are two vertices u; w V ðgþ F 1 F ad there is a vertex v F 1 DF such that ðu; wþ ad ðv; wþ E: () There are two vertices u; v F 1 F ad there is a vertex w V ðgþ F 1 F such that ðu; wþ ad ðv; wþ E: (3) There are two vertices u; v F F 1 ad there is a vertex w V ðgþ F 1 F such that ðu; wþ ad ðv; wþ E: The followig lemma follows from Defiito.4.5 ad Theorem 5.1. Lemma 5.. A system G is g-good-eighbor coditioal t-diagosable uder the MM model if ad oly if for each distict pair of g-good-eighbor coditioal faulty subsets F 1 ad F of V with jf 1 jt ad jf jt satisfies oe of the followig coditios: (1) There are two vertices u; w V ðgþ F 1 F ad there is a vertex v F 1 DF such that ðu; wþ ad ðv; wþ E: () There are two vertices u; v F 1 F ad there is a vertex w V ðgþ F 1 F such that ðu; wþ ad ðv; wþ E: (3) There are two vertices u; v F F 1 ad there is a vertex w V ðgþ F 1 F such that ðu; wþ ad ðv; wþ E: Let g be a positive iteger with 0 g : To fid the g-good-eighbor coditioal diagosability t g ðq k Þ uder the MM model, we first show that t g ðq k Þ is o more tha ð g þ 1Þ g 1 for k 5; 3 ad 0 g : Theorem 5.3 [45]. Let g ; 3; Q g be a g-dimesioal subcube of Q ; C Q ðq g Þ¼N Q ðq g Þ[VðQ g Þ: The Q ½N Q ðq g ÞŠ is the uio of g disjoit g-dimesioal subcubes of Q ad Q C Q ðq g Þ is coected ad the miimum degree of Q C Q ðq g Þ is ot less tha : Corollary 5.4. Let the g-dimesioal hypercube Q g be a subgraph of Q k ; ad C Q k ðq gþ¼n Q k ðq g Þ[V ðq g Þ; where 0 g ; k 4 ad 3: The jn Q k ðq g Þj ¼ ð gþ g ad the miimum degree of Q k C Q k ðq gþ is ot less tha : Proof. Whe k ¼ 4; the by Theorem 5.3, we have jn Q 4 ðq g Þj ¼ jn Q ðq g Þj ¼ ð gþ g ad the miimum degree of Q 4 C Q 4 ðq gþ¼q C Q ðq g Þ is ot less tha : Whe k 5; by Lemma 3.1, the result is also true.

10 1174 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 6, NO. 4, APRIL 015 Lemma 5.5. Assume that k 4; 3 ad 0 g : The the g-good-eighbor coditioal diagosability of k-ary -cube Q k uder the MM model, t g ðq k Þð gþ1þg 1: Proof. Let A ffi Q g be a subgraph of Q k ad let F 1 ¼ N Q k ðaþ;f ¼ C Q k ðaþ (see Fig. 4). The by Corollary 5.4, jf 1 j¼ð gþ g ; jf j¼ð gþ1þ g ; ad the miimum degree of Q k F is ot less tha : That is F 1 ad F are two g-good-eighbor coditioal faulty sets of V ðq k Þ with jf 1jð gþ1þ g ad jf jð g þ 1Þ g : O the other had, by the defiitios of F 1 ad F ; either oe of the three coditios of Lemma 5. is satisfied. By Lemma 5,, k-ary -cube Q k is ot g-good-eighbor coditioal ð g þ 1Þ g -diagosable. The proof is complete. Next, we show that uder the MM model, t g ðq k Þ is o less tha ð g þ 1Þ g 1 for k 4; 3 ad 0 g : Lemma 5.6. Assume that k 4; 3 ad 0 g : The the g-good-eighbor coditioal diagosability of k-ary -cube Q k uder the MM model, t g ðq k Þð gþ1þg 1: Proof. Suppose, by cotradictio, that t g ðq k Þ is less tha ð g þ 1Þ g 1 uder the MM model. By Lemma 5., there are two distict g-good-eighbor coditioal faulty sets F 1 ad F with jf 1 jð gþ1þ g 1 ad jf j ð g þ 1Þ g 1; but the vertex set pair ðf 1 ;F Þ is ot satisfied with ay oe coditio i Lemma 5.. Without loss of geerality, assume that F F 1 6¼;: We shall discuss the followig two cases. Case 1. V ðq k Þ¼F 1 [ F : Sice k 4; 3;g ad V ðq k Þ¼F 1 [ F ; we obtai the followig iequality: 4 k ¼ V Q k ¼jF1 [ F jjf 1 jþjf j ½ð gþ1þ g 1Š ðþ1þ þ1 4 ; which is a cotradictio. Case. V ðq k Þ 6¼ F 1 [ F : I this case, we first prove a useful claim. Claim 3. Q k F 1 F has o isolated vertex. We show the claim by cosiderig the followig two subcases. Subcase A. g ¼ 1: Suppose, by cotradictio, that Q k F 1 F has at least oe isolated vertex. Let w be a isolated vertex i Q k F 1 F : Sice F 1 is a 1-good eighbor coditio faulty set, there is a vertex u F F 1 such that u is adjacet to w: O the other had, sice the vertex set pair ðf 1 ;F Þ is ot satisfied with ay oe coditio i Lemma 5., by Lemma 5.(3) (see Fig. 5), there is at most oe vertex u F F 1 such that u is adjacet to w: Thus, there is just a vertex u F F 1 such that u is adjacet to w: Similarly, we ca deduce there is just a vertex v F 1 F such that v is adjacet to w: Let W V ðq k Þ F 1 F be the set of isolated vertices, ad let H be the iduced subgraph by the vertex set V ðq k Þ F 1 F W: The for ay vertex w W; there are eighbors i F 1 \ F : By jf j ð g þ 1Þ g 1 ad g ¼ 1; we have jf j4 1: Therefore, X jn F1 \F ðwþj ¼ jwjð Þ X d Q k ðvþ ww vf 1 \F jf 1 \ F j ðjf j 1Þ ð4 Þ: It follows that jwj 4 þ 3: Assume H ¼;: The 4 k ¼ V ðq k Þ ¼jF1 [ F jþjwj jf 1 jþjf j jf 1 \ F jþjwj ½ð4 1Þ 1Šþ4þ3 ¼ 1 1: It follows that <3; cotradicts 3: So H 6¼ ;: Sice the vertex set pair ðf 1 ;F Þ is ot satisfied with the coditio (1) of Lemma 5. (see Fig. 5), ad ay vertex of V ðhþ is ot isolated, we have there is o edge betwee H ad F 1 DF : Thus, F 1 \ F is a cut of Q k ad dðqk ðf 1 \ F ÞÞ 1; i.e., F 1 \ F is a 1-good-eighbor coditioal cut of Q k : By Theorem 3.9, jf 1 \ F j4 : Note that jf 1 jð gþ1þ g 1 ¼ 4 1; jf jð gþ1þ g 1 ¼ 4 1 ad either F 1 F or F F 1 is empty. Thus, jf 1 F j¼jf F 1 j¼1: Say F 1 F ¼fv 1 g;f F 1 ¼fv g: The for ay vertex w W; w are adjacet to v 1 ad v : Sice there are at most two commo eighbors for ay pair of vertices i Q k ; it follows that there are at most two isolated vertices i V F 1 F : Assume there is exactly oe isolated vertex v: The v 1 ;v are adjacet to v i V F 1 F : Clearly, N Q k ðvþ fv 1 ;v gf 1 \ F : Sice Q k cotais o triagle, it follows that N Q k ðv 1 Þ fvg F 1 \ F ;N Q k ðv Þ fvg F 1 \ F ; ½N Q k ðvþ fv 1 ;v gš \ ½N Q k ðv 1 Þ fvgš ¼ ;; ad ½N Q k ðvþ fv 1 ;v gš \ ½N Q k ðv Þ fvgš¼;: Sice there are at most two commo eighbors for ay pair of vertices i Q k ; it follows that j½n Q k ðv 1Þ fvgš\ ½N Q k ðv Þ fvgšj 1: Therefore, jf 1 \ F j NQ k ðvþ fv 1 ;v g þ NQ k ðv 1 Þ fvg þ NQ k ðv Þ fvgš 1 ¼ þ 1 þ 1 1 ¼ 6 5: It follows that jf j¼jf F 1 jþjf 1 \ F j1þ6 5 ¼ 6 4 > 4 1; cotradictig jf jð gþ1þ g 1 ¼ 4 1: Assume there is aother isolated vertex v 0 6¼ v i V F 1 F : The v 1 ;v are adjacet to v 0 : Similarly, sice Q k cotais o triagle ad there are at most two commo eighbors for ay pair of vertices i Q k ; it follows that the four vertex sets N Q k ðvþ fv 1 ;v g; N Q k ðv 0 Þ fv 1 ;v g;n Q k ðv 1 Þ fv; v 0 g ad N Q k ðv Þ fv; v 0 g do ot itersect pairwise. Therefore, jf 1 \ F j NQ k ðvþ fv 1 ;v g þ NQ k ðv 0 Þ fv 1 ;v g þ NQ k ðv 1 Þ fv; v 0 g þ NQ k ðv Þ fv; v 0 g ¼ þ þ þ ¼ 8 8:

11 YUAN ET AL.: THE g-good-neighbor CONDITIONAL DIAGNOSABILITY OF k-ary -CUBES UNDER THE PMC MODEL AND MM MODEL 1175 It follows that jf j¼jf F 1 jþjf 1 \ F j1þ8 8 ¼ 8 7 > 4 1; TABLE 1 The t g ðq 5 Þ of Small a cotradictio. Subcase B. g : Sice F 1 is a g-good-eighbor coditio faulty set, jn Q k F 1 ðxþj g for ay x V F 1 : Note the vertex set pair ðf 1 ;F Þ is ot satisfied with ay oe coditio i Lemma 5.. By Lemma 5.(3) (see Fig. 5), for ay pair of vertices u; v F F 1 ; there is o vertex w V F 1 F such that ðu; wþ; ðv; wþ EðQ k Þ: Thus, ay vertex w i V F 1 F has at most oe eighbor i F F 1 : Therefore, for ay vertex w V F 1 F ; jn Q k F 1 F ðwþj g 1 1; i.e., every vertex of Q k F 1 F is ot a isolated vertex. The proof of Claim 3 is complete. Let u be a vertex i Q k F 1 F : By Claim 3, u has at least oe eighbor i Q k F 1 F : Sice the vertex set pair ðf 1 ;F Þ is ot satisfied with ay oe coditio i Lemma 5., by Lemma 5.(1) (see Fig. 5), for ay pair of adjacet vertices u; w V F 1 F ; there is o vertex v F 1 DF such that ðu; vþ or ðv; wþ EðQ k Þ: It follows that u has o eighbor i F 1 DF : By the arbitrariess of u; there is o edge betwee V F 1 F ad F 1 DF : Sice F F 1 6¼;ad F 1 is a g-good-eighbor coditioal faulty set, dðq k ½F F 1 ŠÞ g: By Lemma 3.3, jf F 1 j g : Sice both F 1 ad F are g-good-eighbor coditioal faulty sets ad there is o edge betwee V F 1 F ad F 1 DF ;F 1 \ F is a g-good-eighbor coditioal cut of Q k : By Theorem 3.9, we have jf 1 \ F jð gþ g : Therefore, jf j¼jf F 1 jþjf 1 \ F j g þð gþ g ¼ð g þ 1Þ g ; cotradictig jf jð gþ1þ g 1 The proof is complete. Combiig Lemmas 5.5 ad 5.6, the g-good-eighbor coditioal diagosability of k-ary -cube Q k shows below. Theorem 5.7. Assume that k 4; 3 ad 0 g : The the g-good-eighbor coditioal diagosability of k-ary -cube Q k uder the MM model, t g ðq k Þ¼ð gþ1þg 1: Proof. O the oe had, by Lemma 5.5, the g-good-eighbor coditioal diagosability of k-ary -cube Q k uder the MM model, t g ðq k Þð gþ1þg 1: O the other had, by Lemma 5.6, the g-good-eighbor coditioal diagosability of k-ary -cube Q k uder the MM model, t g ðq k Þð gþ1þg 1: Therefore, the g-good-eighbor coditioal diagosability of k-ary -cube Q k uder the MM model, t g ðq k Þ¼ð gþ1þg 1: The proof is complete. Table 1 shows the g-good-eighbor coditioal diagosability of five-ary -cube t g ðq 5 Þ of small ð 3Þ where 0 g : 6 CONCLUSIONS The g-good-eighbor coditioal diagosability ca measure diagosability for a large-scale processig system more accurately tha classical diagosability because the classical diagosability always assumes that all eighbors of each processor i a system ca potetially fail at the same time regardless of the probability. I fact, if there are exactly faulty processors i a system of miimum degree ; however, the probability of the faulty set cotaiig all the eighbors of ay vertex is statistically low for large multiprocessor systems. Therefore, it is worthy to the determiig the g-good-eighbor coditioal diagosability of itercoectio etwork for multiprocessor systems. I the area of diagosability, the PMC model ad the MM model are two well-kow ad widely chose fault diagosis models. I this paper, we sdy the g-good-eighbor coditioal diagosability of k-ary -cube uder the these models, ad demostrate the g-good-eighbor coditioal diagosabilitis of k-ary -cube Q k uder the PMC model ad MM model are both ð g þ 1Þ g 1 for k 4; 3 ad 0 g : Observig that whe g ¼ 0; there is o restrictio o the faulty sets ad we have the traditioal diagosability o the hypercube as : I

12 1176 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 6, NO. 4, APRIL 015 additio, i the special case of g ¼ 1; our result is slightly differet from the measure of coditioal diagosability give by Lai et al. [8]. The differece betwee these two measures is that we oly cosider the coditio of the faultfree vertices i the etwork. The geeralizatio of coditioal diagosability by requirig every vertex to have at least g good eighbors is also a iterestig problem to ivestigate i the fure. For further discussio, it is a attractive work to develop differet measures of these coditioal diagosabilities based o applicatio eviromet, etwork topology, etwork reliability, ad statistics related to fault patters. ACKNOWLEDGMENTS This work was supported by the the Natioal Naral Sciece Foudatio of Chia ( , , , 61763) ad the Naral Sciece Foudatio of Shaxi Provice ( ). Xiao Qi s research was supported by the US Natioal Sciece Foudatio (NSF) uder Grats CCF (CAREER), CNS (CSR), CNS (CSR), CCF (CPA), CNS (CyberTrust), CNS (CRI), OCI (CITEAM), DUE (CCLI), ad DUE (SFS). Ju Yua is the correpodig author. REFERENCES [1] R. Ahlswede ad H. Aydiiaa, O diagosability of large multiprocessor etworks, Discr. Appl. Math., vol. 156, o. 18, pp , Nov [] L. C. P. Albii, S. Chessa, ad P. Maestrii, Diagosis of symmetric graphs uder the BGM model, Comput. J., vol. 47, o. 1, pp. 85 9, Ja [3] B. Bose, B. Broeg, Y. Kwo, ad Y. Ashir, Lee distace ad topological properties of k-ary -cubes, IEEE Tras. Comput., vol. 44, o. 8, pp , Aug [4] J. A. Body ad U. S. R. Murty, Graph Theory with Applicatios. New York, NY, USA: Macmilla, New York, [5] F. Barsi, F. Gradoi, ad P. Maestrii, A theory of diagosability of digital systems, IEEE Tras. Comput., vol. 5, o. 6, pp , Ju [6] G. Y. Chag, G. H. Che, ad G. J. Chag, ðt; kþ-diagosis for matchig compositio etworks uder the MM model, IEEE Tras. 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13 YUAN ET AL.: THE g-good-neighbor CONDITIONAL DIAGNOSABILITY OF k-ary -CUBES UNDER THE PMC MODEL AND MM MODEL 1177 [40] D. B. West, Itroductio to Graph Theory. Eglewood Cliffs, NJ, USA: Pretice-Hall, 001. [41] M. Wa ad Z. Zhag, A kid of coditioal vertex coectivity of star graphs, Appl. Math. Lett., vol., o., pp , Feb [4] J. Wu ad G. Guo, Fault tolerace measures for m-ary -dimesioal hypercubes based o forbidde faulty sets, IEEE Tras. Comput., vol. 47, o. 8, pp , Aug [43] M. Xu, K. Thulasirama, ad X. D. Hu, Coditioal diagosability of matchig compositio etworks uder the PMC model, IEEE Tras. Circuits Syst., vol. 56, o. 11, pp , Nov [44] M. C. Yag, Coditioal diagosability of matchig compositio etworks uder the MM model, If. Sci., vol. 33, o. 1, pp , Ju [45] W. H. Yag ad J. X. Meg, Geeralized measures of fault tolerace i hypercube etworks, Appl. Math. Lett., vol. 5, o. 10, pp , Oct. 01. [46] S. Zhou, The coditioal diagosability of crossed cubes uder the compariso model, It. J. Comput. Math., vol. 87, o. 15, pp , Dec [47] Q. Zhu, O coditioal diagosability ad reliability of the BC etworks, J. Supercomput., vol. 45, o., pp , Aug Ju Yua received the BS, MS, ad PhD degrees from Shaxi Uiversity, Chia, i 00, 005, ad 008, respectively, all i mathematics. Curretly, he is a associate professor i the School of Applied Scieces, Taiyua Uiversity of Sciece ad Techology. His research iterests iclude graph theory ad itercoectio etworks, computatioal complexity, ad algorithms. Aixia Liu received the BS degree i mathematics from Taiyua Normal Uiversity, Chia, i 00, ad the MS degree i mathematics from Shaxi Uiversity, Chia, i 006. Curretly, she is a faculty member of the School of Applied Scieces, Taiyua Uiversity of Sciece ad Techology. Her research iterests iclude graph theory ad itercoectio etworks, ad parallel ad distributed systems. Xiuli Liu received the BS degree i geographic iformatio system from Harbi Normal Uiversity, Chia, i 01. Curretly, she is workig toward the postgraduate degree at the Taiyua Uiversity of Sciece ad Techology. Her research iterests iclude complex etworks system ad geographic iformatio system. Xiao Qi received the BS ad MS degrees i computer sciece from the HUST, Chia, ad the PhD degree i computer sciece from the Uiversity of Nebraska-Licol i 199, 1999, ad 004, respectively. Curretly, he is a associate professor i the Departmet of Computer Sciece ad Software Egieerig, Aubur Uiversity. His research iterests iclude parallel ad distributed systems, storage systems, fault tolerace, real-time systems, ad performace evaluatio. He received the US Natioal Sciece Foudatio (NSF) Computig Processes ad Artifacts Award, the NSF Computer System Research Award i 007, ad the NSF CAREER Award i 009. He is a seior member of the IEEE. Jifu Zhag received the BS ad MS degrees i computer sciece ad techology from the Hefei Uiversity of Techology, Chia, ad the PhD degree i patter recogitio ad itelligece systems from the Beijig Istite of Techology, i 1983, 1989, ad 005, respectively. He is curretly a professor i the School of Computer Sciece ad Techology at TYUST. His research iterests iclude data miig ad artificial itelligece, ad parallel ad distributed systems. " For more iformatio o this or ay other computig topic, please visit our Digital Library at Xue Ma received the BS degree i mathematics from Tagsha Normal Uiversity, Chia, i 011, ad the MS degree i mathematics from the Taiyua Uiversity of Sciece ad Techology, Chia, i 014. Her research iterests iclude graph theory ad itercoectio etworks, ad parallel ad distributed systems.