Conditional Fault Diagnosis of Bubble Sort Graphs under the PMC Model

Transcription

1 odtoal Fault Dagoss of Bubble Sort Graphs uder the PM Model Shumg Zhou,3, Ja Wag, Xrog Xu, ad Ju-Mg Xu 3 Key Laboratory of Network Securty ad ryptology, Fuja Normal Uversty, Fuzhou, Fuja , ha Departmet of omputer Scece, Dala Uversty of Techology, Dala, 604, ha 3 Departmet of Mathematcs, Uversty of Scece ad Techology of ha, Hefe, Ahu, 3006, ha Abstract As the sze of a multprocessor system creases, processor falure s evtable, ad fault detfcato such a system s crucal for relable computg The fault dagoss s the process of detfyg faulty processors a multprocessor system through testg The codtoal dagosablty, whch s a ew metrc for evaluatg fault tolerace of such systems, assumes that every faulty set does ot cota all eghbors of ay processor the systems Ths paper shows that the codtoal dagosablty of bubble sort graphs B uder the PM model s 4- for 4, whch s about four tmes ts ordary dagosablty uder the PM model Keywords: odtoal dagosablty, Fault dagoss, Bubble sort graphs Itroducto Wth the rapd developmet of mult-processor systems, fault dagoss of tercoecto etworks has become creasgly promet As a sgfcat crease the umber of processors, processor falure s evtable I order to esure the stable rug of the systems, we must fd out the faulty processors ad repar or replace them System-level dagoss, as a powerful tool, has bee wdely used The basc dea s to desg a effectve algorthm to fd out faulty processors through a comprehesve aalyss of test results whch are stmulated by adjacet processors Ths method does ot have to use specal equpmet to test Most of the recet research efforts system-level dagoss have focused o ehacg the applcablty of system-level dagoss-based approaches to practcal scearos such as VLSI testg, dagoss of tercoecto etworks employed parallel computers [] The PM model, proposed by Preparata et al [] for dealg wth the system's self-dagoss, assumed that each ode ca test ts eghborg odes, ad test results are "faulty" or "fault-free" Uder ths model, the dagosablty of a tercoecto etwork s the maxmum umber of faulty odes the system that ca be detfed To grat more accurate measuremet of dagosablty for a large-scale processg system, La et al [] troduce the codtoal dagosablty of a system uder the PM model, whch suppose the probablty that all adjacet odes of oe ode are faulty smultaeously Z Du (Ed): Itellgece omputato ad Evolutoary omputato, AIS 80, pp sprgerlkcom Sprger-Verlag Berl Hedelberg 03

2 54 S Zhou et al s very small That s, codtoal dagosablty s the dagoablty uder the codto that all adjacet odes of ay ode ca't be faulty smultaeously They further showed that the codtoal dagosablty of Q s 4(-)+ for 5 The codtoal dagosablttes of matchg composto etworks[3,4], Shuffle-cubes [5], folded hypercubes[6], augmeted cubes [7] are obtaed successo Ths paper establshes the codtoal dagosablty of the bubble-sort graph B uder the PM model The remader of ths paper s orgazed as follows I Secto, we troduce some termology ad prelmares used through ths paper Secto 3 cocetrates o the codtoal dagosablty of B Termologes ad Prelmares A mult-processor system, whose topologcal structure s a tercoecto etwork, ca be modeled as a smple udrected graph G= GV (, E), where a vertex ujv represets a processor ad a edge (u,v)je represets a lk betwee vertces u ad v If at least oe ed of a edge s faulty, the edge s sad to be faulty; otherwse, the edge s sad to be fault-free The coectvty of a graph G, deoted by κ ( G), s the mmum umber of vertces whose deleto results a dscoected graph or a trval graph The compoets of a graph G are ts maxmal coected subgraphs A compoet s trval f t has o edges; otherwse, t s otrval The eghborhood set of the vertex set X V( G) s defed as NG ( X) = { y V( G) x X such that (, xy) EG ( )} X For coveece, S deotes the umber of elemets the set S Ad we also use G to represet the umber of vertces the graph G The PM model requres that u ad v ca test each other for ay edge (u,v)je Whe u tests v, we call u as testg ode, ad call v as tested ode The test output s 0 (or ) whch mples that v s faulty (or faulty-free) σ (,) uv deotes the output of u testg v Ad t s assumed that the test outputs are correct f the testg ode s faultfree; otherwse the outputs are urelable The collecto of all outputs s called the sydrome σ For a gve sydrome σ, a subset of vertces F V( G) s sad to be cosstet wth σ f the sydrome σ ca be produced from the stuato that, for ay (u,v)je such that ujv-f, σ (,) uv = f ad oly f vjf It meas that F s a possble set of faulty odes Sce test output produced by a faulty ode s urelable, a gve set F of faulty odes may produce dfferet sydromes O the other had, dfferet faulty sets may produce the same sydrome Let σ ( F) represet the set of all sydromes that could be produced by F Two dstct sets F ad F of V(G) are sad to be dstgushable f σf σ F ; otherwse, F ad F are dstgushable We say that (F, F ) s a dstgushable par f σ σ = ; otherwse, (F, F ) s a dstgushable par We also use F F FF Δ = ( F F) ( F F) to deote the symmetrc dfferece of F ad F Defto [,] A system G s sad to be t-dagosable f, a gve sydrome ca be produced by a uque faulty set, provded that the umber of faulty odes does ot

3 odtoal Fault Dagoss of Bubble Sort Graphs uder the PM Model 55 exceed t The largest value of t, for whch a gve system G s t-dagosable, s called the dagosablty of system G, deoted as t(g) Lemma [,] For ay two dstct sets F ad F of V, (F, F ) s a dstgushable par ff there exst a vertex u V( G) ( F F) ad a vertex v FΔ F such that (u,v) je So, f two sets F ad F are dstgushable, the there s o edge betwee FΔ F ad VG ( ) ( F F) Defto [] A faulty set F VG () s called a codtoal fault-set, f Nv () for ay vertex vjv(g) F Defto 3 [] A system G s sad to be codtoally t-dagosable, f for ay two dstct codtoal fault-sets F ad F V( G) wth F t, F t, (F, F ) s a dstgushable par The largest value of t whch makes system G s codtoally t- dagosable s called the codtoal dagosablty of system G, deoted as tc( G ) Lemma [] A system G s t-dagosable uder the PM model, f f for ay F ad F V( G), F F wth F t, F t, (F, F ) s a dstgushable par Lemma 3 [] A system G s t-dagosable uder the PM model, f ad oly f for a dstgushable par of sets F ad F V( G), t mples that F > t or F > t Lemma 4 [] Let G(V,E) be a mult-processor system, ad (F, F ) be a dstgushable codtoal par wth F F, the the followg codtos hold: () N( u) ( V ( F F) for u ( V ( F F) ; () N( v) ( F F) ad N( v) ( F F) for v FΔ F Let (F, F ) be a dstgushable codtoal par, ad let S = F F By observato, every compoet of G-S s otrval Moreover, we have () for each compoet of G-S, f ( FΔF), the d ( ) for v ; () for each compoet of G-S, f ( FΔF), the d ( ) for v Network relablty s oe of the major factors desgg the topology of a tercoecto etwork The hypercube ad ts varats were the major class of etworks The -star graph (S for short) s a attractve alteratve to the hypercube [8] The bubble-sort graphs smlar to the -star graph, whch belogs to the class of ayley graphs, have bee attractve alteratve to the hypercubes They have some good topologcal propertes such as hghly symmetry ad recursve structure I partcular, the -dmesoal bubble-sort graph B s vertex trastve, whle t s ot edge trastve [9] The coectvty of B s - ad the dameter s (-)/ It was show that fdg a shortest path B ca be accomplshed by usg the famlar bubble-sort algorthm [8] Algorthms for ode-to- ode dsjot paths, pacyclcty ad hamltoa laceablty B are obtaed [0,,] Now, we troduce the bubble-sort

4 56 S Zhou et al graphs A -dmesoal bubble-sort graph s (-)-regular ad symmetrc It has! odes ad ( )!/ edges whle ts coectvty ad dameter are - ad ( ) /, respectvely Defto 4 A -bubble-sort graph B has! odes Each ode has a uque address, whch s a permutato of symbols =,,, A ode that has a address u= uu u s adjacet to ode whose address s u = uu u u+ u u wth A very mportat property of the bubble-sort graph s ts recursve structure [] We decompose B to subgraphs B ( =,,, ) such that each B fxes the last posto of the label strgs whch represets the vertces, ad so B s somorphc to B Let S = S Bfor =,,, For, the th elemet of the label of vertex u B s represeted by u [] A edge e=xy s called a par-edge f x [ ] = y [ ] ad x [ ] = y [ ] e=x'y' s called the coupled par-edge correspodg to e=xy, where x'[ ] = x[ ], ad y'[ ] = y[ ] We call two edge, xx' ad yy', the coupler or two par-edge e=xy ad e'=x'y' 3 odtoal Dagosablty of B We decompose B to subgraphs B ( =,,, ) such that all vertces of B have the same last bt of the label strgs, ad so B B Let S be a faulty set of V(B ) Deote A = { B Bcotas at least - odes S }, ad A = { B Bcotas at most -3 odes S } We also deote A the subgraph of B duced by the uo of subgraphs A It s easy to check that the followg lemma holds Lemma 5 A S s coected Lemma 6 t( B) 4 Proof Let e=xy be oe par-edge wth the coupled par-edge e'=x'y' These two paredges wth ther coupler costtute a cycle of legth of 4 Obvously, NB ( x, y, x', y') = 4( 3) Let F = N(, x y, x', y') {,} x y, F = N(, x y, x', y') {', x y'} It s easy to check that F ad F are two dstgushable codtoal fault-sets, ad F = F = 4( ) + = 4 0 Thus, t( B) 4 Now, we show the codtoal dagosablty of B s 4- for 5 Let F ad F V( G), F F, (F, F ) s a dstgushable par for 5 We shall show our result by provg that ether wth F 4 0 or F 4 0

5 odtoal Fault Dagoss of Bubble Sort Graphs uder the PM Model 57 Lemma 7 For ay two dstgushable codtoal fault-sets F ad F B wth 5, whch satsfes F F, we have ether F 4 0 or F 4 0 Proof Sce (F, F ) s a dstgushable codtoal-par, there exsts o edge betwee FΔ F ad B ( F F) by Lemma Defe S= F F ad let B [ F ] Δ F be the subgraph of B duced by the vertex set FΔ F We choose a maxmal compoet B [ F ] ΔF whe B [ F ] Δ F s ot coected; otherwse, let = B [ ] FΔ F By lemma 4, we have 4 Thus, we oly eed to prove / + S 4 0, whch mples that F 4 0 or F 4 0 We decompose B to subgraphs B ( =,,, ) such that each B fxes the last posto of the label strgs whch represets the vertces We also let S = S B If S 4, the we have / + S 4 0 for 4, so the lemma holds Now, we oly cosder the stuato S 4 3 Let A = { B B has at least - odes S }, ad A = { B B has at most -3 odes S } Obvously, A has at most three elemets by the fact that 4(-)> 4- If A =, by lemma 5 we have ( 3)[( )! ( 3)] > (4 ) for 5 Thus, we have / + S 4 0 Now, we cosder A as follows ase ( A S) Sce A has at most three subgraphs, we have ( 3)[( )! ( 3)] (4 0) for 5; ad we arrve at the result ase ( A S) = Obvously, A, NA ( ) S A by the maxmalty of Now, we dvde ths case to three subcases below Subcase There s exactly oe subgraph, say X, A Sce every vertex of X has exactly oe eghbor outsde of X, we have NB ( ) X S A ; ad so NB ( ) X = Obvously, NX ( ) S X Sce 4, let T be a path of legth three Obvously, NX( ) NX( T) ( T), ad so NX( ) NX( T) T Sce S= ( S X) ( S A ), we have S = S X + S A N ( ) + N ( ) 4 Thus, we have / + S 4 0 X B X

6 58 S Zhou et al Subcase There are exactly two subgraphs, say X ad Y, A We assume, wthout loss of geeralty, that X Y For ay x X, N ( x) by Lemma 4 ad every vertex of ths subgraph, the we have X Y B has exactly oe eghbor outsde of Due to NX ( X) S X, ad X has at least two vertces, S X N ( ) ( ) Smlarly, we have S Y NY ( ) ( ) Sce every par of S X, S Y, S A are dsjot, we have S = ( S X) ( S Y) ( S A) NX( X) NY( Y) S A The we have S NX( X) + NY( Y) + S A [( ) ] + S A Thus, we have / + S 4 0 Subcase 3 There are exactly three subgraphs, say X, Y, Z, A We assume, wthout loss of geeralty, that X by the fact that 4 We have F X NX ( X) ( ) Sce every vertex of B has exactly oe eghbor outsde of ths subgraph, ad every par of S X, S Y, S Z, S A are dsjot, we have S= ( S X) ( S Y) ( S Z) ( S A) NX ( X) ( Y) ( Z) S A The we have S NX ( X) + Y Z + S A ( ) + ( ) + S A 4 0 Thus, we have / + S 4 0 By Lemma 6 ad 7, we have Theorem The codtoal dagosablty of bubble sort graph B uder the PM model s t ( B ) = 4 ( 5) Theorem The codtoal dagosablty of bubble sort graph B 4 uder the PM model s t ( ) 5 B 4 = Proof Let F ad F be two dstgushable fault-sets B 4 Deote S= F F ad FΔ F be a coected compoet B 4 S By Lemma 4, we have 4 If S 4, the / + S 6 Now we suppose that S 3 I the worst case, B4 S has two compoets, oe of whch s a solated vertex u, the we have = B4 S {} u, whch mples = 0 So we have / + S 0 Thus t ( ) 5 B4, whle t ( ) 5 B4, hece t ( ) 5 B 4 = X

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