The Counterchanged Crossed Cube Interconnection Network and Its Topology Properties

Transcription

1 WSEAS TRANSACTIONS o COMMUNICATIONS Wag Xiyag The Couterchaged Crossed Cube Itercoectio Network ad Its Topology Properties WANG XINYANG School of Computer Sciece ad Egieerig South Chia Uiversity of Techology Higher Educatio Mega Cetre Payu District Guagzhou Guagdog Provice PR CHINA wxyyuppie@39com / wxyyuppie@hotmailcom Abstract: I this paper by combiig with the etwork structures of the twisted cube ad the crossed cube the couterchaged crossed cube etwork is proposed a rigorous recursive defiitio is made ad the etwork topology structure graph is offered Basig o the defiitio of the couterchaged crossed cube this paper also aalyzes the basic properties of the etwork proves that the etwork is -regular ad its coectivity is illustrates the recursio characteristics of the etwork ad two importat corollaries are obtaied Through the recursive properties of the couterchaged crossed cube the we discuss the relatioship betwee ay two vertexes i the etwork ad fially prove that the etwork diameter is D ( CCQ ) = ( + ) / 2 Key words: the couterchaged crossed cube; itercoectio etwork; etwork diameter; coectivity; etwork topology Itroductio I a parallel processig system the etwork itercoectio structure ofte determies the the system performace Amog may of the existig itercoectio etwork structures the hyper cube [] is oe of the most classic ad most widely applied etwork The hypercube etwork has very excellet structure properties ad is a hotspot of research o etwork topology However with the developmet of practical applicatio ad eeds it has bee foud that the structure of the hypercube is ot always optimal I order to give full play to the hypercube its superior properties ad to avoid the defects i the structure of its ow a variety of variat structures basig o the hypercube is proposed ie the crossed cube [2-4] the twisted cube [5 6] the Möbius cube [7 8] the twisted -cube [9] the locally twisted cube [0 ] etc These variatios optimize the performaces of a certai aspects o the basis of the origial hypercube ad were applied to solve practical problems These etwork variatios always achieve the optimizatio about some parameters o the basis of the origial etwork model such as etwork diameter coectivity etc For example amog the variatios of the hypercube the etwork diameters of the crossed cube ad the twisted -cube are smaller tha that i the hypercube Ad they themselves also have some excellet properties for example the twist -cube reduces the etwork commuicatio diameter by removig E-ISSN: Volume 4 205

2 WSEAS TRANSACTIONS o COMMUNICATIONS disjoit edges i the hypercube ad crossig them to form the twisted edges i the etwork I documet [9] Esfahaia ad others have prove that there are two disjoit - dimesioal hypercubes i the twisted -cube the twisted -cube is -regular ad routig time ca be reduced from to - i the worst case; I additio Kemal Efe i [2] itroduced the routig algorithm ad the broadcastig algorithm i the crossed cube obtaied the crossed cube etwork diameter ( + ) / 2 approximately half of the hypercube ad proved the embeddig properties of basic etworks ito the crossed cube Combiig with the atures of the above two etworks a ew etwork structure the couterchaged crossed cube is proposed ad its primary etwork properties are carefully researched i this paper This paper is orgaized as follows: sectio 2 itroduces the related basic cocepts of the graph theory ad itercoectio etwork; Sectio 3 defies the couterchaged crossed cube etwork structure ad proves its edge coectig property; Sectio 4 presets the mai topology properties of the couterchaged crossed cube; Sectio 5 proves the diameter of the couterchaged crossed cube; Sectio 6 carries o a comparative aalysis o the atures of the couterchaged crossed cube call the legth of the shortest path from vertex u to v the distace betwee u ad v deoted by d(u v)ad call max{d(uv) u v V(G)} the diameter of graph G deoted by D(G); deg(u) deotes the degree of vertex u; δ ( G) ad ( G) represet the maximum degree ad miimum degree of graph G; κ(g) ad λ ( G) represet the vertex coectivity ad edge coectivity of graph G respectively; σ(u)=i deotes vertices u that u =u - = u i+ =0 ad u i =; for a biary strig u = uu uu 2 ui represets the i-th complemetary bit of u i Let x=x 2 x y=y 2 y be two biary strigs ad we say x ad y are pair-related if ad oly if (x y) {(0000)(00)} deoted by x~y If x ad y are ot pair-related it is deoted by x /~ y Defiitio [9] For a 4-legth cycle <u v y x u> i -dimesioal hypercube Q if delete the edges (u y) ad (v x) ad add edges (u x) ad (v y) the obtaied etwork structure is called twisted -cube deoted as TQ The etwork structures of Q 3 ad TQ 3 are show i Fig u v u Wag Xiyag v 2 Relative Defiitios y x y x This paper uses ormative termiologies ad otatios i the graph theory to represet relative cocepts ad formulas Let G be a udirected graph where V(G) ad E(G) represet the vertex set ad edge set of graph G respectively Vertices i graph G are deoted by biary umbers e(u v) deotes the edge that coects two adjacet vertices u ad v; we Q 3 TQ 3 Fig Q 3 ad TQ 3 Defiitio 2 [2] The -dimesioal crossed cube (CQ ) is a -label graph it ca be defied iductively as follows: CQ is K 2 the complete graph of two vertices with labels 0 ad ; for > CQ cosists of two (-)-dimesioal crossed cube CQ () ad CQ where E-ISSN: Volume 4 205

3 WSEAS TRANSACTIONS o COMMUNICATIONS Wag Xiyag () i V( CQ )={x x - x x =i}(i=0) The vertex x=0x - x -2 x i ad the vertex y=y - y -2 y i adjacet i CQ if ad oly if: () x - =y - if is eve ad CQ () CQ are (2) For i ( ) / 2 x 2i x 2i- ~ y 2i y 2i- The etwork structures of CQ 3 ad CQ 4 are show i Fig CQ 3 CQ 4 Fig 2 CQ 3 ad CQ 4 Defiitio 3 u v V(CQ ) if u ad v are adjacet ad σ ( u+ v) = i the we say u ad v are adjacet i i-th dimesio Defiitio 4 [2] I graph G two edges without public vertex are called idepedet edges For two idepedet edges (u v) ad (x y) i edge set E delete the coectios betwee the two edges ad add ew edges (u y) ad (v x) the call the operatio a X-couterchage The process of X-couterchage is show i Fig 3 u x v y Fig 3 X-couterchage It is observed that the degree of vertexes i the graph remais uchaged after X-couterchage 3 Defiitio of the Network The defiitio ad etwork structure of the couterchaged crossed cube is as follows Defiitio 5 The -dimesioal couterchaged crossed cube (CCQ ) is a -label graph CCQ is K 2 the complete u x v y graph of two vertices with labels 0 ad ; CCQ 2 is a 4-legth cycle < >; for 3 CCQ cosists of two (-)-dimesioal couterchaged crossed () cube CCQ ad CCQ where () i V( CCQ )={x x - x x =i}(i=0) The vertex x=0x - x -2 x i ad the vertex y=y - y -2 y i are adjacet i CCQ if ad oly if: () x - =y - if is eve ad CCQ () CCQ (2) for i ( ) / 2 x 2i x 2i- ~y 2i y 2i- Fig 4 shows the etwork structures of -4 dimesio CCQ Accordig to Defiitio 5 the followig theorem holds Theorem For 2 ay edge e = ( x y) i CCQ where x = xx xx 2 ( xi {0} i= 2 -) y = y y yy 2 ( yi {0} i= 2 -) there exists m( 2 m ) such that at least oe of the followig situatios holds: () If x = y the xx xm+ = yy ym+ ad x m = y m ; (2) Whe i ( m) / 2 there is x2ix2i y2iy2i Proof: we make a iductive proof o () For situatio () whe =2 m= E-ISSN: Volume 4 205

4 WSEAS TRANSACTIONS o COMMUNICATIONS Wag Xiyag the theorem is true Suppose the theorem is true for =k the whe =k+ there is xk+ = yk+ if σ(x+y)=l ad let m=k+-l+ apparetly xm = ymad xk+ xk+ xm+ = yk+ yk+ ym+ ie situatio () is established; (2) For situatio (2) if x y the m=accordig to Defiitio 5 the coclusio holds; 2 if x = y from situatio () we kow there exists the positive iteger m ( 2 m ) such that xx xm+ = yy y = ad x m+ m = y m it is easy to kow that ad ( m ) m m 2 ( m) x x x xx m m 2 = y = y y yyare two vertexes of ( m) ( m) the edge ( x y ) i the couterchaged crossed cube Accordig to the coclusio i whe i ( m) / 2 x2ix2i y2iy2i is true I coclusio the theorem is prove CCQ CCQ 2 CCQ 3 CCQ 4 From the proof of the above theorem if vertexes x ad y differs from the -m+-th bit from the left the call the m-th bit the leftmost differet bit betwee x ad y say u ad v are adjacet i m-th dimesio deoted by u=n m (v) ad call the edge the i-dimesioal edge betwee u ad v deoted by e m (uv) 4 Structure Properties The etwork properties are determied by its structure ad these properties have importat impacts o the performace of the etwork This sectio will study properties such as the regularity recursiveess ad coectivity 4Regularity I a regular graph each vertex has the same degree If the degree of every vertex is k the graph is a k regular graph A regular etwork has a good symmetry ature the trasmissio speed ad etwork loads are balaced From the defiitio of CCQ it is observed that the degrees of vertexes i the Fig 4-4 dimesio CCQ etwork remai uchaged after the X-couterchage The proof of CCQ regularity is show as follows Theorem 2 CCQ is a -regular graph Proof: Accordig to the defiitio of CCQ ad Theorem vertexes i the etwork have oe ad oly oe i-dimesioal eighbor vertex where i= 2 3 Apparetly the degree of ay vertex u i CCQ is deg(u)= O the defiitio of regular graph CCQ is -regular 42Recursiveess From the CCQ etwork defiitio CCQ is formed of lower dimesioal subets recursively ad the etwork structure is of recursiveess The followig will further research the subet characteristics of the couterchaged crossed cube Γ ( G) deote the subets which cosist of all the vertices that prefix with E-ISSN: Volume 4 205

5 WSEAS TRANSACTIONS o COMMUNICATIONS or i G Γ ( G) is abbreviated as Γ ( G) These isomorphic subets costitute higher dimesioal CCQ subets Theorem 3 For 2 there are 0 ad ( CCQ ) Γ CCQ Proof: Accordig to the defiitio of CCQ -dimesioal CCQ cosist of two () (-)-dimesioal CCQ ad CCQ ad CCQ CCQ CCQ CCQ () the theorem is true Let be two biary strigs of the same legth if there exists a positive iteger l such that u Γ ( CCQ ) Nl( u) Γ ( CCQ) ad v Γ ( CCQ ) Nl( v) Γ ( CCQ) the call Γ ( CCQ ) ad Γ ( CCQ ) two l-dimesioal adjacet subets deoted by Γ ( CCQ) = Nl( Γ ( CCQ)) Theorem 4 Let ad be two biary strigs of the same legth ad = = s if two subets Γ ( CCQ ) ad Γ ( CCQ ) i CCQ satisfy Γ ( CCQ ) = N ( Γ ( CCQ )) l the ()For l s = ad ; l (2)For l > s ad s+ Proof: Sice Γ ( CCQ ) = N ( Γ ( CCQ )) l If l s accordig to the adjacet subet defiitio for u Γ ( CCQ ) ad v Γ ( CCQ ) there is σ ( u+ v) = l so = ad u u u u = v v v v s+ s l+ s+ s l+ the i terms of the defiitio of CCQ we have ; l If l > s let the mappig : Γ ( CCQ ) CCQ such that ϕ s+ u Γ ( CCQ ) ϕ( u) = uu u (I) l s If u v Γ ( CCQ ) ad u=n j (v) the for j -s u v Γ ( CCQ ) or u v Γ ( CCQ ) ad hece ϕ( u) = N ( ϕ( v)) ; for j>-s let u Γ ( CCQ ) v Γ ( CCQ ) ϕ( u) N ( ϕ( v)) j accordig to formula (I) we have = ie s+ ϕ ( u) ad ϕ ( v) are adjacet i CCQ -s+ so s+ The followig corollaries ca be easily obtaied from Theorem 4 Corollary If ( 4) is eve the Corollary 2 If ( 5) is odd the We ca see the recursive characters of CCQ from Corollary ad 2: whe ( 4) is eve CCQ is formed of four subets prefixig with ad ; whe ( 5) is odd CCQ is formed of eight subets prefixig with If we imagie these subets as a ode the for ( 4) is eve ad ( 5) is odd the coectio methods of these subet are show i Fig 5 (a) ( 4) is eve (b) ( 5) is odd Fig 5 the recursive coectio methods of CCQ subets 43Vertex/edge Coectivity Wag Xiyag I the actual etwork there always exist failure odes or edges due to various reasos Coectivity is the miimum umber of vertices or edges that eed to be removed to make a graph ucoected Coectivity reflects the tolerace degree towards fault amely the ability of ormal commuicatio whe there are failures i the etwork which thus is a very importat E-ISSN: Volume 4 205

6 WSEAS TRANSACTIONS o COMMUNICATIONS property Defiitio 6 [9] Let G ad G 2 be two graphs with the same vertex umber: V(G )= {u u 2 u p } V(G 2 )={v v 2 v p } Let H=G G 2 where V(H)=V(G ) V(G 2 ) E(H)=E(G ) E(G 2 ) {(u i v i ) u i V(G ) v i V(G 2 ) i p} Lemma [9] Let G ad G 2 be two coected graphs defied i Defiitio 6 ad H=G G 2 the: κ( H) + mi( κ( G ) κ( G )) 2 Lemma 2 [3] For ay graph G κ( G) λ( G) δ( G) Theorem 5 κ( CCQ ) = λ( CCQ ) = Proof: Accordig to Defiitio 5 CCQ costitutes of two subets CCQ ad () CCQ the from Defiitio 6 ad Lemma we have CCQ CCQ CCQ = ad () κ( CCQ ) + κ( CCQ ) + ( + κ( CCQ )) 2 ( ) + κ ( CCQ ) Sice κ ( CCQ ) = the κ ( CCQ ) Ad by Theorem 2 ad Lemma 2we kow κ( CCQ ) λ( CCQ ) δ( CCQ ) = I coclusio κ( CCQ ) = λ( CCQ ) = the theorem is prove 5 Network Diameter Network diameter embodies the etwork commuicatio delay therefore a good etwork should keep the diameter as small as possible o the premise of coectivity The followig will study the diameter of CCQ etwork Theorem 6 Whe 2 for xy V(CCQ ) the relatioship betwee x ad y must satisfy oe of the followig situatios: () Whe is eve x ad y exist i the same CCQ subet G where G CCQ or G CCQ 2 ; (2) Whe is odd x ad y exist i the same CCQ subet G where G CCQ 2 or G CCQ 3 ; (3) The vertex x (or y) has a adjacet vertex x ' ' ' ' (or y ) such that x (or y ) ad y (or x) exist i the same CCQ subet G where whe is eve G CCQ ; whe is odd G CCQ 2 Proof: Accordig to the recursiveess of CCQ ad Corollaries ad 2 whe is eve CCQ is formed of four itercoected subets which are isomorphic to CCQ 2 ; whe is odd CCQ is formed of eight itercoected subets which are isomorphic to CCQ 3 The followig will make correspodig discusses about the above situatios: () Whe is eve accordig to Fig 5(a) if x ad y exist i the same subet G the G CCQ 2 ; if x ad y exist i two adjacet subets the these two subets costitute a ew subset G by itercoectig ad it is easy to kow G CCQ situatio () is true; (2) Whe is odd accordig to Fig 5(b) if x ad y exist i the same subet G the G CCQ 3 ; if x ad y exist i two adjacet subets the these two subets costitute a ew subset G by itercoectig ad it is easy to kow G CCQ 2 situatio (2) is true; Wag Xiyag (3) From Fig 5 we ca see that the maximum distace amog subets is 2 ie ay two o-adjacet subets eed at most oe itermediate subet to coect with each other If x ad y exist i two o-adjacet subets there must be a eighbor vertex x ' ' ' ' (or y ) such that x (or y ) ad y (or x) are i the same subet G Whe is eve from Fig 5(a) we have G CCQ ; whe is odd from Fig 5(b) E-ISSN: Volume 4 205

7 WSEAS TRANSACTIONS o COMMUNICATIONS Wag Xiyag we have G CCQ 2 situatio (3) is true that CCQ has the followig superiorities tha the twisted -cube: To sum up the theorem holds + Theorem 7 D( CCQ ) = () κ( CCQ ) = λ( CCQ ) = δ( CCQ )= 2 compared to the twisted -cube CCQ has better coectivity ad fault tolerace; Proof: Accordig to Fig 4 we have + (2) D( CCQ ) = the diameter D( CCQ ) = D( CCQ ) = D( CCQ ) = 2 it meets of CCQ is almost half of those of the the theorem We make a iductive proof o hypercube ad the twisted -cube which k Suppose whe k m ( 2k + ) the greatly reduces the etwork commuicatio theorem is true the diameter ad icreases the etwork commuicatio efficiecy D( CCQ ) = D( CCQ ) = m + The ext 2m 2m+ will discuss the situatios whe k=m+ () We first discuss the diameter of CCQ 2m+2 If vertexes a ad b exist i the same subet it coforms to situatio () i Theorem 6 the from the assumptio we kow dab ( ) = m+ ; if a ad b exist i two o-adjacet subets the a must has a eighbor vertex a ' ' such that a ad b are i the same subet which coforms to situatio (3) i Theorem 6 the similarly from the assumptio we have ' ' d( a b) = m+ ad daa ( ) = Accordig to the defiitio of etwork diameter we have D( CCQ ) = m + 2; 2m+ 2 (2) For the diameter of CCQ 2m+3 like the proof process i () we similarly have D( CCQ ) = m + 2 2m+ 3 I coclusio the assumptio stads ie D( CCQ ) = D( CCQ ) = k + By 2k 2k+ trasformig the express we obtai the + equivalet formula D( CCQ ) = 2 the theorem holds 6 Compariso ad Aalysis Through the researches o CCQ etwork structure ad properties we ca see 7 Coclusio I this paper by combiig with the twisted cube ad the crossed cube we propose a ew etwork structure - the couterchaged crossed cube (CCQ ) etwork ad make detailed discussios about the etwork structure characteristics ad basic properties of CCQ for example CCQ is -regular it has excellet recursiveess ad its coectivity is Through the recursive properties of the couterchaged crossed cube we prove that the etwork diameter is D ( CCQ ) = ( + ) / 2 which is early half of that of the hypercube Through compariso it is foud that CCQ is also a favorable etwork structure The subsequet researches will focus o problems such as the etwork embeddig properties the routig algorithm the fault-tolerace strategies ad the optimal paths Refereces [] Saad Y Schultz M H Topological properties of hypercube [J] IEEE Tras Comput (7): [2] Kemal Efe "The Crossed Cube Architecture for Parallel Computatio" IEEE Tras Parallel ad Distributed E-ISSN: Volume 4 205

8 WSEAS TRANSACTIONS o COMMUNICATIONS Wag Xiyag Systems vol 3 o 5 pp [3] D J Wag "O Embeddig Hamiltoia Cycles i Crossed Cubes" IEEE Tras Parallel ad Distributed Systems vol 9 o 3 pp [4] J Xi Fa X L Li ad X H Jia "Optimal Path Embeddig i Crossed Cubes" IEEE Tras Parallel ad Distributed Systems vol 6 o 2 pp [5] J Fa X ad Li X Jia "Optimal Embeddigs of Paths with Various Legths i Twisted Cubes" IEEE Tras Parallel ad Distributed Systems vol 8 o 4 pp [6] Xi Wag J X Fa X H Jia S K Zhag ad J Yu "Embeddig Meshes ito Twisted cubes" Iformatio Scieces vol 8 pp [7] Paul Cull ad Shaw M Larso "The Möbius Cubes" IEEE Tras Comput vol 44 o 5 pp [8] S Y Hsieh ad N W Chag "Hamiltoia Path Embeddig ad Pacyclicity o the Möbius Cube with Faulty Nodes ad Faulty Edges" IEEE Tras Comput vol 55 o 7 pp [9] Abdol-Hossei Esfahaia Lioel M Ni ad Bruce E Saga "The Twisted N-Cube with Applicatio to Multiprocessig" IEEE Tras Comput vol 40 o pp [0] XYag DJ Evas ad GM Megso "The Locally Twisted Cubes" Iteratioal Joural of Computer Mathematics vol 82 o 4 pp []T K Li C J Lai ad C H Tsai "A Novel Algorithm to Embed a Multi-Dimesioal Torus ito a Locally Twisted Cube" Theoretical Computer Sciece vol 42 o 22 pp [2] Wag De-qiag The costructio of the twisted-cube coected etwork Joural of Dalia Maritime Uiversity 998 4(): [3] Gray Chartrad Zhag Pig Itroductio to Graph Theory [M] POST & TELECOM PRESS Beijig 2007 E-ISSN: Volume 4 205