Iteratioal Joural of Egieerig ad Applied Scieces (IJEAS) ISSN: 394-366 Volume-4 Issue-8 August 07 The Nature Diagosability of Bubble-sort Star Graphs uder the PMC Model ad MM* Model Mujiagsha Wag Yuqig Li Shiyig Wag Abstract May multiprocessor systems have itercoectio etworks as uderlyig topologies ad a itercoectio etwork is usually represeted by a graph where odes represet processors ad liks represet commuicatio liks betwee processors No fault set ca cotai all the eighbors of ay fault-free vertex i the system which is called the ature diagosability of the system Diagosability of a multiprocessor system is oe importat study topic As a famous topology structure of itercoectio etworks the -dimesioal bubble-sort star graph has may good properties I this paper we prove that the ature diagosability of is 4 7 uder the PMC model for 4 the ature diagosability of model for 5 is 4 7 uder the MM* Idex Terms Bubble-sort star graph Diagosability Itercoectio etwork I INTRODUCTION May multiprocessor systems have itercoectio etworks (etworks for short) as uderlyig topologies ad a etwork is usually represeted by a graph where odes represet processors ad liks represet commuicatio liks betwee processors Some processors may fail i the system so processor fault idetificatio plays a importat role for reliable computig The first step to deal with faults is to idetify the faulty processors from the fault-free oes The idetificatio process is called the diagosis of the system A system G is said to be t -diagosable if all faulty processors ca be idetified without replacemet provided that the umber of preseted faults does ot exceed t The diagosability tg ( ) of G is the maximum value of t such that G is t -diagosable For a t -diagosable system Dahbura ad Masso [] proposed a algorithm with time 5 complex O ( ) which ca effectively idetify the set of faulty processors Several diagosis models (eg Preparata Metze ad Chie's (PMC) model [] Barsi Gradoi ad Maestrii's (BGM)model [3] ad Maeg ad Malek's (MM) model [4] have bee proposed to ivestigate the diagosability of multiprocessor systems I particular two of the proposed models the PMC model ad the MM model are well kow ad widely used I the PMC model the diagosis of the system is achieved through two liked processors testig each other I the MM model to diagose a system a ode seds the same task to two of its eighbors ad the compares their resposes For this reaso the MM model is Mujiagsha Wag School of Electrical Egieerig ad Computer Sciece The Uiversity of Newcastle NSW 308 Australia Yuqig Li School of Electrical Egieerig ad Computer Sciece The Uiversity of Newcastle NSW 308 Australia *Shiyig Wag School of Mathematics ad Iformatio Sciece Hea Normal Uiversity Xixiag PR Chia +86-037333648 also said to be the compariso model Segupta ad Dahbura [] proposed a special case of the MM model called the MM* model i which each ode must test its ay pair of adjacet odes Numerous studies have bee ivestigated uder the PMC model ad MM model or MM* model I the traditioal diagosis of a multiprocessor system oe geerally assumes that ay subset of processors may simultaeously fail If all the eighbors of some ode v are faulty simultaeously it is impossible to determie whether v is faulty or fault-free As a cosequece the diagosability of the system is less tha its miimum ode degree However i some large-scale multiprocessor systems we ca safely assume that all eighbors of ay ode do ot fail at the same time Based o this assumptio i 005 Lai et al [5] itroduced the restricted diagosability of the system called the coditioal diagosability They cosider the situatio that o fault set ca cotai all the eighbors of ay ode i the system Sice the probability that the all eighbors of a fault ode fail ad create faults is more to the probability that the all eighbors of a fault-free ode fail ad create faults i the system we cosider the situatio that o fault set ca cotai all the eighbors of ay fault-free ode i the system which is called the ature diagosability of the system I 0 Peg et al [6] proposed a measure for fault diagosis of the system amely the g-good-eighbor diagosability (which is also called the g-good-eighbor coditioal diagosability) which requires that every fault-free ode cotais at least g fault-free eighbors I [6] they studied the g-good-eighbor diagosability of the -dimesioal hypercube uder the PMC model I [7] Wag ad Ha studied the g-good-eighbor diagosability of the -dimesioal hypercube uder the MM* model I 06 Re ad Wag [8] gave some properties of the g-good-eighbor diagosability of a multiprocessor system I 07 Wag et al [9] studied that the -good-eighbor diagosability of bubble-sort star graph etworks uder the PMC model ad MM* model Yua et al [0] studied that the g-good-eighbor diagosability of the k-ary -cube ( k 3) uder the PMC model ad MM* model As a favorable topology structure of itercoectio etworks the Cayley graph C geerated by the traspositio tree has may good properties I [] Wag et al studied the -good-eighbor diagosability of C uder the PMC model ad MM* model I 06 Zhag et al [3] proposed a ew measure for fault diagosis of the system amely the g-extra diagosability which restrais that every fault-free compoet has at least (g+) fault-free odes I [3] they studied the g-extra diagosability of the -dimesioal hypercube uder the PMC model ad MM* model I 06 Wag et al [4] studied that the -extra diagosability of the -dimesioal bubble-sort star graph uder the PMC model ad MM* model I [5] Ha ad Wag studied that the 55 wwwijeasorg
The Nature Diagosability of Bubble-sort Star Graphs uder the PMC Model ad MM* Model g -extra diagosability of folded hypercubes I 07 Wag ad Yag [6] studied the -good-eighbor (-extra) diagosability of alteratig group graph etworks uder the PMC model ad MM* model I [7] Wag et al studied the ature diagosability of C uder the PMC model ad MM* model ad proved that the ature diagosability of the system is less tha or equal to the coditioal diagosability of the system Therefore the ature diagosability of the system is ature ad oe importat study topic I 06 Bai ad Wag [8] studied the ature diagosability of M o bius cubes; Hao ad Wag [9] studied the ature diagosibility of augmeted k-ary -cubes; Jirimutu ad Wag [0] studied the ature diagosability of alteratig group graph etworks; Ma ad Wag [] studied the ature diagosability of crossed cubes; Zhao ad Wag [] studied the ature diagosability of augmeted 3-ary -cubes The star graph ad the bubble-sort graph have bee proved to be a importat viable cadidate for itercoectig a multiprocessor system The feature of the star graph iclude low degree of ode small diameter symmetry ad high degree of fault-tolerace The diagosabilities of the star graph uder the PMC model ad MM model were studied i [34] Li et al [5] showed that the coditioal diagosability of the star graph uder the compariso diagosis model is 3 7 I this paper the ature diagosability of the -dimesioal bubble-sort star graph uder the PMC model ad MM* model has bee studied It is proved that the ature diagosability of is 4 7 uder the PMC model for 4 the ature diagosability of is 4 7 uder the MM* model for 5 II PRELIMINARIES I this sectio some defiitios ad otatios eeded for our discussio the bubble-sort star graph the PMC model ad MM* model are itroduced A Defiitios ad Notatios A multiprocessor system is modeled as a udirected simple graph G ( V E) whose vertices (odes) represet processors ad edges (liks) represet commuicatio liks Give a oempty vertex subset V of V the subgraph iduced by V i G deoted by GV [ ] is a graph whose vertex set is V ad the edge set is the set of all the edges of G with both edpoits i V The degree dg () v of a vertex v is the umber of edges icidet with v We deote by ( G) the miimum degrees of vertices of G For ay vertex v we defie the eighborhood NG () v of v i G to be the set of vertices adjacet to v u is called a eighbor or a eighbor vertex of v for u N () v Let S V We use N ( ) G S to deote the set vs NG ( v) \ S For eighborhoods ad degrees we will usually omit the subscript for the graph whe o cofusio arises A graph G is said to be k -regular if for ay vertex v dg () v k Let G be a coected graph The coectivity ( G) of a graph G is the miimum umber of vertices whose removal results i a discoected graph or oly oe vertex left whe G is complete A fault set F V G is called a ature faulty set if N( v) ( V \ F) for every vertex v i V\ F A ature cut of G is a ature faulty set F such that G F is discoected The miimum cardiality of ature cuts is said to be the ature coectivity of G deoted by * ( G) For graph-theoretical termiology ad otatio ot defied here we follow [6] B The PMC model ad MM* model For the PMC model ad MM* model we follow [0] I a system G ( V E) a faulty set F V is called a coditioal faulty set if it does ot cotai all of eighbors of ay vertex i G A system G is coditioal t -diagosable if every two distict coditioal faulty subsets F F V with F t F t are distiguishable The coditioal diagosability tc ( G ) of G is the maximum umber of t such that G is coditioal t -diagosable By [7] t ( G) t( G) c Theorem ([7]) For a system G ( V E) t( G) t0( G) t ( ) ( ) G tc G I [7] Wag et al proved that the ature diagosability of the Bubble-sort graph B uder the PMC model is 3 for 4 I [8] Zhou et al proved the coditioal diagosability of B is 4 for 4 uder the PMC model Therefore t ( B ) t ( B ) whe 5 ad t ( B ) t ( B ) whe 4 c c C The bubble-sort star graph The bubble-sort star graph has bee kow as a famous topology structure of itercoectio etworks I this sectio its defiitio ad some properties are itroduced Let [ ] { } ad let S be the symmetric group o [ ] cotaiig all permutatios p pp p of [ ] It is well kow that {( i) : i } is a geeratig set for S So {( i) : i } {( i i ) : i } is also a geeratig set for S The -dimesioal bubble-sort star graph [930] is the graph with vertex set V ( ) = S i which two vertices u v are adjacet if ad oly if u v( i) i or u v( i i ) i It is easy to see from the defiitio that graph o! vertices Note that is a special Cayley graph is a ( 3) -regular has the followig useful properties Propositio For ay iteger is ( 3) - regular vertex trasitive Propositio For ay iteger is bipartite Propositio 3 For ay iteger 3 the girth of is 4 Theorem ([3]) Let H be a simple coected graph with V ( H ) 3 If H ad H are two differet labelled graphs obtaied by labellig H with { } Cay H S is isomorphic to Cay( H S ) ( ) the We ca partitio ito subgraphs where every vertex u xx x V ( ) has a fixed iteger i i the last positio x for i [ ] It is obvious that 56 wwwijeasorg
i i is isomorphic to for i [ ] Let v V( ) The v( ) ad v( ) are called outside eighbors of v i Propositio 3 ([9]) Let be defied as above The there are ( )! idepedet cross-edges betwee two differet H i 's Propositio 4 ([9]) Let be the bubble-sort star graph If two vertices uv are adjacet there is o commo eighbor vertex of these two vertices ie N( u) N( v) 0 If two vertices uv are ot adjacet there are at most three commo eighbor vertices of these two vertices ie N( u) N( v) 3 * Lemma ([9]) The ature coectivity ( ) of the bubble-sort star graph 4 is 8 A coected graph G is super ature coected if every miimum ature cut F of VG ( ) isolates oe edge If i additio G F has two compoets oe of which is a edge the G is tightly F super ature coected Theorem 3 ([4]) For 5 the bubble-sort star graph is tightly (4 8) super ature coected Lemma Let A {()()} If 4 F N ( A) F A N ( A) the F 4 8 F 4 6 ( F ) ad ( F ) Proof By A {()()} we have [ A ] K Sice has ot 3-cycles N ( A) 4 8 Thus from calculatig we have F 4 8 F A F 4 6 Claim For ay x S \ F N ( x) F ) 4 Sice is a bipartite graph there is o 5-cycle ()( ki) x()( lj )()() of where ( ki)( lj) S \ () Let u N (()) \ () If u is adjacet to x the x is ot adjacet to each of N (()) \ () Sice N (()) \ () 4 we have that x is adjacet to at most ( 4) vertices i F By Claim N ( x) F ) 4 for ay x S \ F Therefore ( F ) 3 ( 4) F has two compoets F ad Note that ( ) Therefore ( F ) III THE NATURE DIAGNOSABILITY OF THE BUBBLE-SORT STAR GRAPH UNDER THE PMC MODEL I this sectio we shall show the ature diagosability of the bubble-sort star graph uder the PMC model Let F ad F be two distict subsets of V for a system G ( V E) Defie the symmetric differece F F ( F \ F ) ( F \ F ) Yua et al [0] preseted a sufficiet ad ecessary coditio for a system to be ature t -diagosable uder the PMC model Theorem 4 ([0]) A system G ( V E) is ature t -diagos -able uder the PMC model if ad oly if there is a edge uv E with u V \ ( F F) ad v F F for each distict Iteratioal Joural of Egieerig ad Applied Scieces (IJEAS) ISSN: 394-366 Volume-4 Issue-8 August 07 4 pair of ature faulty subsets F ad F of V with F t ad F t Lemma 3 A graph of miimum degree has at least two vertices The proof of Lemma 3 is trivial Lemma 4 Let 4 The the ature diagosability of the bubble-sort star graph uder the PMC model is less tha or equal to 4 7 ie t ( ) 4 7 Proof Let A be defied i Lemma ad let F N ( A) F A N ( A) By Lemma F 4 8 F 4 6 ( F ) ad ( F ) Therefore F ad F are both ature faulty sets of with F 4 8 ad F 4 6 Sice A F F ad N ( A) F F there is o edge of betwee V ( ) \ ( F F) ad F F By Theorem 4 we ca deduce that is ot ature (4 6) -diagosable uder the PMC model Hece by the defiitio of ature diagosability we coclude that the ature diagosability of is less tha 4 6 ie t ( ) 4 7 Lemma 5 Let 4 The the ature diagosability of the bubble-sort star graph uder the PMC model is more tha or equal to 4 7 ie t ( ) 4 7 Proof By the defiitio of ature diagosability it is sufficiet to show that is ature (4 7) -diagosable By Theorem 4 to prove is ature (4 7) -diagosable it is equivalet to prove that there is a edge uv E( ) with u V ( ) \ ( F F) ad v F F for each distict pair of ature faulty subsets F ad F of V ( ) with F 4 7 ad F 4 7 We prove this statemet by cotradictio Suppose that there are two distict ature faulty subsets F ad F of V ( ) with F 4 7 ad F 4 7 but the vertex set pair ( F F ) is ot satisfied with the coditio i Theorem 4 ie there are o edges betwee V ( ) \ ( F F) ad F F Without loss of geerality assume that F \ F Suppose V ( ) F F By the defiitio of F F S! It is obvious that! 8 4 for 4 Sice 4 we have that! V ( ) F F F F F F F F (4 7) 8 4 a cotradictio Therefore V ( ) F F Sice there are o edges betwee V ( ) \ ( F F) ad F F ad F is a ature faulty set F has two parts F F ad [ F \ ] F (for coveiece) Thus ( F F) ad ( [ F \ F]) Similarly ( [ F \ F]) whe F \ F Therefore F F is also a ature faulty set Whe F \ F F F F is also a ature faulty setsice there are o edges betwee V ( F F) ad F F F F is a ature cut Sice 4 by Theorem 3 F F 4 8 By Lemma 3 F \ F Therefore F F \ F F F 4 57 wwwijeasorg
The Nature Diagosability of Bubble-sort Star Graphs uder the PMC Model ad MM* Model 8 4 6 which cotradicts with that F 4 7 So is ature (4 7) -diagosable By the defiitio of t ( ) t ( ) 4 7 Combiig Lemmas 4 ad 5 we have the followig theorem Theorem 5 Let 4 The the ature diagosability of the bubble-sort star graph uder the PMC model is 4 7 IV THE NATURE DIAGNOSABILITY OF THE BUBBLE-SORT STAR GRAPH UNDER THE MM* MODEL Before discussig the ature diagosability of the bubble-sort star graph uder the MM* model we first give a existig result Theorem 6 ([0]) A system G ( V E) is ature t -diagosable uder the MM* model if ad oly if each distict pair of ature faulty subsets F ad F of V with F t ad F t satisfies oe of the followig coditios () There are two vertices u wv \ ( F F) ad there is a vertex F F such that uw E ad vw E () There are two vertices u v F \ F ad there is a vertex wv \ ( F F ) such that uw E ad vw E (3) There are two vertices u v F \ F ad there is a vertex wv \ ( F F) such that uw E ad vw E Lemma 6 Let 4 The the ature diagosability of the bubble-sort star graph uder the MM* model is less tha or equal to 4 7 ie t ( ) 4 7 Proof Let A F ad F be defied i Lemma By Lemma F 4 8 F 4 6 ( F ) ad ( F ) So both F ad F are ature faulty sets By the defiitios of F ad F FF A Note F \ F F \ F A ad ( V ( ) \ ( F F)) A Therefore both F ad F are ot satisfied with ay oe coditio i Theorem 6 ad is ot ature (3 6) -diagosable Hece t ( ) 4 7 The proof is complete Lemma 7 Let 5 The the ature diagosability of the bubble-sort star graph uder the MM* model is more tha or equal to 4 7 ie t ( ) 4 7 Proof By the defiitio of ature diagosability it is sufficiet to show that is ature (4 7) -diagosable By Theorem 6 suppose o the cotrary that there are two distict ature faulty subsets F ad F of with F 4 7 ad F 4 7 but the vertex set pair ( F F ) is ot satisfied with ay oe coditio i Theorem 6 Without loss of geerality assume that F \ F Similarly to the discussio o V ( ) F F i Lemma 5 we ca deduce that V ( ) F F Therefore V ( ) F F Claim I F F has o isolated vertex Suppose o the cotrary that F F has at least oe isolated vertex w Sice F is a ature faulty set there is a vertex u F \ F such that u is adjacet to w Sice the vertex set pair ( F F ) is ot satisfied with ay oe coditio i Theorem 6 there is at most oe vertex u F \ F such that u is adjacet to w Thus there is just a vertex u F \ F such that u is adjacet to w Similarly we ca deduce that there is just a vertex v F \ F such that v is adjacet to w whe F \ F Let W S \ ( F F) be the set of isolated vertices i [ S \ ( F F)] ad let H be the subgraph iduced by the vertex set S \ ( FF W ) The for ay w W there are ( 5) eighbors i F F Sice F 4 7 we have ww N ( w) W ( 5) d ( v) [( F F ) W ] vf F F F ( 3) ( F )( 3) (4 8)( 3) 8 84 8 8 4 It follows that W 4 3 5 for 5 Note F F F F F F (4 7) ( 5) 6 9 Suppose V( H) The! S V ( ) F F W 6 9 4 3 0 This is a cotradictio to 5 So V( H) Sice the vertex set pair ( F F ) is ot satisfied with the coditio () of Theorem 6 ad ay vertex of V( H ) is ot isolated i H we iduce that there is o edge betwee V( H ) ad F F Thus F F is a vertex cut of ad ( ( F F)) ie F F is a ature cut of By Theorem 3 F F 4 8 Because F 4 7 F 4 7 ad either F \ F or F \ F is empty we have F \ F F \ F Let F \ F { v} ad F \ F { v } The for ay vertex w W w are adjacet to v ad v Accordig to Propositio 5 there are at most three commo eighbors for ay pair of vertices i it follows that there are at most three isolated vertices i F F ie W 3 Suppose that there is exactly oe isolated vertex v i F F Let v ad v be adjacet to v The N ( v) \{ v v } F F Sice cotais o triagle it follows that N ( v ) \{ v} F F ; N ( v ) \{ v} F F ; [ N ( v) \{ v v }] [ N ( v ) \{ v}] ad [ N ( v) \{ v v }] [ N ( v ) \{ v}] By Propositio 5 [ N ( v ) \{ v}] [ N ( v ) \{ v}] Thus FF N ( v) \{ v v } N ( v ) \{ v} N ( v ) \{ v} ( 5) ( 4) ( 4) 6 5 It follows that F F \ F F F 6 5 6 4 4 7 ( 5) which cotradicts F 4 7 Suppose that there are exactly two isolated vertices v ad w i F F Let v ad v be adjacet to v ad w respectively The N ( v) \{ v v} F F Sice cotais o triagle it follows that N ( v ) \{ v w} F F N ( v ) \{ v w } 58 wwwijeasorg
[ N ( v) \{ v v }] [ N ( v ) \{ v w}] ad F F [ N ( v) \{ v v }] [ N ( v ) \{ v w}] By Propositio 5 there are at most two commo eighbors for ay pair of vertices i Thus it follows that [ N ( v ) \{ v w}] N v v w Thus F F N v v v [ ( ) \{ }] ( ) \{ } N ( w) \{ v v } N ( v ) \{ v w} N ( v ) \{ v w} ( 5) ( 5) ( 5) ( 5) 8 It follows that F F \ F F F 8 8 4 7 ( 5) which cotradicts F 4 7 Suppose that there are exactly three isolated vertices uv ad w i F F Let v ad v be adjacet to uv ad w respectively The N ( v) \{ v v} F F Sice cotais o triagle it follows that N ( v ) \{ u v w } F F N ( v ) \{ u v w} F F [ N ( v) \{ v v}] [ N ( v ) \{ u v w}] ad [ N ( v) \{ v v}] [ N ( v ) \{ u v w}] By Propositio 5 there are at most three commo eighbors for ay pair of vertices i Thus it follows that [ N ( v ) \{ u v w}] [ N ( v ) \{ u v w}] 0 Thus F F N ( u) \{ v v } N ( v) \{ v v } N ( w) \{ v v } N ( v ) \{ u v w} N ( v ) \{ u v w} ( 5) ( 5) ( 5) ( 6) ( 6) 3 0 30 It follows that F F \ F F F 0 30 0 9 4 7 ( 5) which cotradicts F 4 7 Suppose F \ F The F F Sice F is a ature faulty set F F F has o isolated vertex The proof of Claim I is complete Let u V ( ) \ ( F F) By Claim I u has at least oe eighbor i F F Sice the vertex set pair ( F F ) is ot satisfied with ay oe coditio i Theorem 6 by the coditio () of Theorem 6 for ay pair of adjacet vertices u wv ( ) \ ( F F) there is o vertex v F F such that uw E( ) ad vw E( ) It follows that u has o eighbor i F F By the arbitrariess of u there is o edge betwee V ( ) \ ( F F) ad F F Sice F \ F ad F is a ature faulty set ([ F \ F]) By Lemma 3 F \ F Sice both F ad F are ature faulty sets ad there is o edge betwee V ( ) \ ( F F) ad F F F F is a ature cut of By Theorem 3 we have F F 4 8 Therefore F F \ F F F (4 8) 4 6 which cotradicts F 4 7 Therefore is ature (4 7) -diagosable ad t ( ) 4 7 The proof is complete Combiig Lemmas 6 ad 7 we have the followig theorem Theorem 7 Let 5 The the ature diagosability of the bubble-sort star graph uder the MM* model is 4 7 Iteratioal Joural of Egieerig ad Applied Scieces (IJEAS) ISSN: 394-366 Volume-4 Issue-8 August 07 ACKNOWLEDGMENT This work is supported by the Natioal Natural Sciece Foudatio of Chia (NO 67700) REFERENCES [] Dahbura AT Masso GM A 5 O ( ) Fault idetificatio algorithm for diagosable systems IEEE Trasactios o Computers 33 (6) 486-49 984 [] Preparata FP Metze G Chie RT O the coectio assigmet problem of diagosable systems IEEE Trasactios o Computers EC-6 848-854 967 [3] Barsi F Gradoi F Maestrii P A Theory of Diagosability of Digital Systems IEEE Trasactios o Computers 5 (6) 585-593 976 [4] Maeg J Malek M A compariso coectio assigmet for self-diagosis of multiprocessor systems Proceedig of th Iteratioal Symposium o Fault-Tolerat Computig 73-75 98 [5] Lai P-L Ta JJM Chag C-P Hsu L-H Coditioal Diagosability Measures for Large Multiprocessor Systems IEEE Trasactios o Computers 54 () 65-75 005 [6] Peg S-L Li C-K Ta JJM Hsu L-H The g-good-eighbor coditioal diagosability of hypercube uder PMC 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