Annales UMCS Informatica AI 2 (2004) UMCS

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1 Pobane z czasopisma Annales AI- Infomatica Annales Infomatica AI 2 (2004) Annales Infomatica Lublin-Polonia Sectio AI Embedding as a method to impove multichannel bus topology paametes Miosław Mazuek Depatment of Distibuted Systems, Faculty of Electical and Compute Engineeing, Rzeszów Univesity of Technology, W. Pola 2, Rzeszów, Poland Abstact This aticle attempts to discuss the poblems in building new topologies utilizing embedding. We popose to utilize the catesian poduct to descibe gaphs as a ight mathematical solution to adjacency matix. We applied the catesian poduct of two and sets of multiple dimensions in analysis. We poposed methodology of building logical topologies based on utilization of egula and symmetic gaphs. Fo thee paticula poducts, we pefomed the analysis to measue taffic intensity in the netwok.. Intoduction Cuently available computation o communication tasks utilized in paallel and distibuted systems show that in the ow of an accidental capacity, computing powe of cuently used computes is not sufficient. Constant gowth of computing complexity in the compute systems bings gowth of systems capacity. At the beginning this gowth was accomplished by enhancement to paametes of base elements utilized in building computing stuctues. This showed that available altenatives, offeed by base of new elements ae not sufficient. Theefoe, it becomes necessay to seach fo new diections towads impovement in tansmitting paametes of cuently available topologies. In the pocess of designing topologies, besides utilizing base topologies, meaning topologies descibed by paticula gaphs with specific abilities such as: small degee, small diamete, symmetical o a lage numbe of nodes, we should conside embedding as a ight tool to build new topologies. To achieve this goal we could use logical opeations utilized on matix of coesponding gaphs, in paticula the catesian poduct. These new developed oiginal topologies ae chaacteized by pedictable paametes descibed in base topologies. As topologies we could use those with opposite paametes (fo example bus and hypecube). Cuently conducted eseach showed that in building new multi channel communication systems egula and symmetic topologies ae of special use []. They insue minimal computation complexity of designing pocess and

2 Pobane z czasopisma Annales AI- Infomatica Miosław Mazuek liveness as well lage scalability assuing building the netwok with an unlimited numbe of nodes. 2. Embedding as a method to impove topology paametes The key ole in computing paallel and homologous computations fo example oganizing computations in the netwok of pocessos is pojection of one achitectue within anothe. The poblem with simulation of one netwok within anothe is defined as a poblem of embedding. Embedding of one topology within anothe gives new capacity of building a new communication netwok and also to allow to fomulate a poblem with finding effective epesentation of data stuctues o to allocate cicuits in the distibuto of VLSI. In ode to chaacteize opeation of embedding, we have to adopt paticula epesentation of topology. Any paticula netwok topology usually is descibed in the fomat of gaphs in which all the nodes epesent computes and all edges epesent netwok connections. Indiect gaphs ae usually used to model communication netwok. Exploiting theoy of gaphs to descibe topology could define opeation of embedding as: Definition. Any given gaph G (V, E ), known as gaph host whee V is the set of its nodes, but E is the set of its edges. Also let us have a gaph G 2 (V 2, E 2 ) known as the gaph guest, whee V 2 is the set of its nodes, and E 2 is the set of its edges. Embedding φ of gaph G (V, E ) within gaph G 2 (V 2, E 2 ) is called diect pojection of nodes of host gaph V within the guest gaph V 2 fo example φ = V V 2. The utility of embedding esults is futhe emphasized by the fact that many of the existing popula achitectues can be modeled as poduct netwoks. An embedding of a guest gaph G 2 in a host gaph G is a mapping of the vetices of G into the vetices of G 2 and the edges of G into paths in G 2. The main cost measues used in embedding efficiency ae: ) Load of embedding is the maximum numbe of nodes of G mapped to any nodes G 2. 2) Dilation of embedding is the maximum path length in G 2 epesenting an edge of G. 3) Congestion of embedding is the maximum numbe of paths that shae any edge of G 2. If G can be embedded in G 2 with load l, dilation d and congestion c, G 2 can emulate t steps of a computation unning on G in O(l+d+c)t steps. If the values l, d, and c ae constant, the slowdown intoduced by this emulation is also constant.

3 Pobane z czasopisma Annales AI- Infomatica Embedding as a method to impove multichannel Regulaity and symmety Classical communication topologies chaacteize degee of its nodes detemined by the numbe of nodes. The moment of lage incease in the numbe of nodes usually leads to the incease in a diamete, which is a geat disadvantage. Of many available topologies none of them could be consideed useful to build moden computing systems because of paametes. Achitectue of the paallel system should be chaacteized by its symmety, which allows implementing simple ules of outing and also should not be chaacteized by a lage numbe of nodes, which in esult keep the cost low. Howeve, consideing them in context of paticula attibutes, it shows that paticula systems with fixed citeia could utilize some base topologies. With egads to hypecube topologies its attention bings vey good paametes, such as: small diamete, egulaity, lage connectivity, ability to embed simple mechanisms of outing (comes fom symmety), and fault-toleance. Moeove, hypecube topologies give ability to imitate diffeent stuctues in place of stuctue nodes simplifying the embedding pocess. Fo example if we place in all nodes of cube ing of 3 nodes we get back vey popula topology CCC. The disadvantage of the above achitectue is limited scalability, which eliminates it fom being utilized in lage systems. Regulaity and symmety guaantee homogeneous topologies, which minimize cost of building and futhe exploitation [2]. By exploiting these kinds of topologies the pocess of modeling and also analysis becomes simple and it influences impovement of paametes. Based on the theoy of gaphs, a gaph is called egula, if evey pai of nodes has the same nodes degee. A gaph is called node-symmetic, if it is viewed the same fom all of its vetices o fom each banch. Fo cleane analysis of these paametes we illustate base definition of egulaity and symmety by utilizing the method of similaity. Definition 2. A topology is called egula if each node has the same node degee. If any node has diffeent node degee a topology is called iegula. In ode to descibe pecisely symmety we need to econside the definition of similaity: Definition 3. Two nodes u and v in a gaph G ae simila if fo some autompophism ρ of G, ρ(u)=v with u, v V. Two edges (u,v ) and (u 2,v 2 ) in gaph G ae simila, if fo some autompophism ρ of G, ρ((u,v ))= (u 2,v 2 ) with (u,v ), (u 2,v 2 ) E.

4 Pobane z czasopisma Annales AI- Infomatica Miosław Mazuek Definition 4. A gaph is called node-symmetic, if evey pai of nodes is simila. A gaph is called edge-symmetic, if evey pai of edges is simila. A gaph is called symmetic, if it is node-symmetic and edge-symmetic. By analyzing symmety we have to notice, that symmetic gaph by its nodes could be symmetic by its edges, although gaph which is symmetic by its edges it is not symmetic by its nodes. Definition 4 descibes only which of the gaphs is symmetic by its edges and which is symmetic by its nodes. But it is useless when compaing degee of symmety paticula high capacity netwok systems (does not allow to descibe which gaph is less o moe symmetic), theefoe it is equied to intoduce definition of s-symmety [3]. Definition 5. A gaph G(V,E) is said to be s-symmetic if the numbe of nodes YV s i i= can be divided into s subsets V,,V 2,...V s with = V and IV i = 0, and in each i= subset V i evey pai of nodes must be simila (analogous fo edges). 4. Catesian Poduct as an instument of embedding To geneate diffeent communication topologies based on those cuently existing, we exploit gaphical opeations. They allow building topologies chaacteized by incompaable paametes. If we talk about connection netwok in the context of logical opeations, we have to notice the fact that each newly built topology could be chaacteized by inheitance of outgoing gaphs paametes and all of its chaacteistics as well as by new featues. They esult fom opeation attibutes, which ae paameteized though chaacte and design of the gaphs. Of the pimay Boolean opeations such as conjunction, disjunction, symmetic diffeence, ejection, composition and catesian poduct the last one deseves a special emphasis, because of its benefit and simplicity in opeations [4]. In geneal a Boolean opeation G xg 2 esults in a gaph G with V = V xv 2, the catesian poduct of the node sets fom gaph G and gaph G 2. A node G is labeled ( u u2) with u Vi and E of G is expessed in tems of the edges in E and E 2 depending on the Boolean opeation applied. Most of the Boolean opeations could be descibed as opeations on matix of gaphs G and G2. Theefoe the catesian poduct could be defined as follows: Definition 6. A catesian poduct of two gaphs G (V, E ) and G 2 (V 2, E 2 ) is gaph G xg 2, whose set of its nodes is V xv 2 and set of its edges is defined as follows: if {u, u 2 } V and {v, v 2 } V 2 then (u v, u 2 v 2 ) is edge of G xg 2 and only then if u = u 2 and (v,v 2 ) E 2 o v = v 2 and (u, u 2 ) E. s

5 Pobane z czasopisma Annales AI- Infomatica Embedding as a method to impove multichannel 335 The example of two-dimensional catesian poduct of second degee of de Buijn s Gaph and the sta of five nodes is shown in Fig.. (4,00) (,00) (0,00) (3,00) 0 (4,0) (2,00) (4,0) (,0) (0,0) (3,0) (,0) (0,0) (3,0) (2,0) (4,) (2,0) de Buijn(2)Sta(5) (,) (0,) (3,) 3 (2,) Fig.. Example of two-dimensional catesian poduct of de Buijn(2) and Sta(5) 5. Attibutes of catesian poduct Cuent eseach shows, that the best solution fo connection netwok between all Boolean opeations is the catesian poduct. Fist of all it povides a limited numbe of solution nodes, secondly all ability to achieve a small diamete. It also allows connecting advantages povided by popula topologies, acquiing new topology that inheits advantages of both above topologies. It is possible to find topology X which is bette in scalability than topology Y and also is bette in eduction of degee o diamete but it is not bette consideing pocessing of boadcasting, fo example sta o hypecube. Theefoe of all the above opeations the catesian poduct is consideed the only ight choice within the opeations allowing building new netwoks. The catesian poduct opeation allows achieving absolute netwok components. The additional attibute of this opeation is the fact that the numbe of nodes gows in a multipliable way, but diamete and aveage distance gow in an addable way, thus cutting costs of building netwok. This new stuctue is also chaacteized by simple implementation of netwok algoithms, based on base and popula in topology

6 Pobane z czasopisma Annales AI- Infomatica Miosław Mazuek algoithms. These attibutes ae specifically useful in systems scalability and allow inceasing netwok size the same time etaining a stable numbe of nodes. Dependencies of the catesian poduct paametes in egad to input topologies descibe line of definitions which could be found in [5]. 6. -dimensional poduct Besides two-dimensional opeation, multi-dimensional opeation is of special usage in ceation of new topologies. The -dimensional poduct of G(N) is denoted PG (N), with the subscipt epesenting the numbe of dimensions. The -dimensional poduct we can identify as: Definition 7. Fo a given gaph G(N), then its dimensional catesian poduct labeled as PN (n) is equal to G (V, E ), G 2 (V 2, E 2 ),..., G N (V N, E N ) whee set of nodes V = V, V 2,V 3...V N = {u u 2...u n u V, u 2 V 2,..., u n V n } and set of edges E = {(u u 2...u n, v v 2...v n whee i exists whee (u i,v i ) V i and fo each j i we have uj= v j }. Notice that the nodes in dimension i fom copies of the gaph G i. Fo example when using the linea aay L n (V,E) defined as V = {0,,..., n } and E = {( i, i+ ) i= 0,,..., n 2} and also ing Pn(V,E) of n nodes defined as V = {0,,..., n } and E = {( i, i+ mod n) i= 0,,..., n }, simply by using the poduct opeation we could aive at -ay n cube, multidimensional gid and a multidimensional tous. Notice that gid a p x p 2 x..x p is the poduct netwok L n L n2...l n, but tous p x p 2 x..x p is the poduct netwok P n P n2...p n. Similaly the binay dimensional cube Q is a poduct of the gaph which contains one edge Q(V={0,}, E={(0,)}). One of the fundamental paametes in communication netwok is the numbe of nodes and connections in the above netwok. Compaison of pimay paametes of two- and multi-dimensional poducts is shown in Table. Table. Compaison of pimay paametes of two-and multi-dimensional poducts G(N) PG (N) Numbe of nodes N N Numbe of edges E EN - Diamete d Rd Connectivity κ κ Min. vetex degee δ Rδ Max. vetex degee Patitionability N i, dla i=0... k k

7 Pobane z czasopisma Annales AI- Infomatica Embedding as a method to impove multichannel Communication capabilities of poduct netwoks In ode to compae the communication capabilities of poduct netwoks a communication model is needed. The poposed model is a time-slotted packet switching model whee packets ae of fixed size. In each time slot, a pocesso can simultaneously send and eceive a packet. It can also geneate a single packet accoding to a Benoulli distibution of paamete p (p is the packet geneation ate of evey node). Destination nodes of geneated packets ae selected unifomly andomly. Each node contains a single unlimited buffe to accommodate incoming packets. The packets ae FIFO seviced. The links between pocessos ae full duplex. If seveal packets ae sent to the same pocesso, then a conflict occus. In this case, we assume that one of the packets to be eceived is chosen andomly, and the othes ae defeed. In ode to descibe communication ability of any given topology we intoduced concept of intensity of taffic. The taffic intensity of a node u in a netwok G(V,E) is defined to be the poduct of the load of that node by the aveage sevice time of a packet. Since the sevice time is deteministic, a pocesso can send a packet at each time slot. Thus the aveage sevice time is assumed to be one. Theefoe, the taffic intensity is educed to the total load of the node [6]. The load of node u is the total of the locally geneated load p l (defined as the aveage numbe of packets geneated by u pe time unit), the load designated to this node p d (defined as the aveage numbe of packets designated to u pe time unit) and the load that has to tansit thoughout this node p t (defined as the aveage numbe of packets tansiting u pe time unit). Theefoe, the load is defined as: p( u) = pl + pd + pt. Note that p l = p, whee p is the packet geneation ate. Let p e be equal to the sum of p t and p d. This load depends on the total numbe of paths used by the G outing algoithm, denoted by, that eithe ends at node u o have node u as an t u intemediate node and the taffic sent by each node to evey othe node. This p taffic is equal to since each node equally sends its locally geneated N taffic to evey othe node in the system (except to itself). A node u can eceive the total extenal load fom othe nodes in the system is: p G pe = tu. () N G G tu The quantity qu = N π called the extenal load facto of node u in netwok G. Theefoe, extenal load facto p e is defined as:

8 Pobane z czasopisma Annales AI- Infomatica Miosław Mazuek G N pe = qu p N. A node (u, u 2 ) in G G 2 will be denoted u u 2 and the cadinality of the set of nodes V i of a gaph G i (V i,e i ) will be denoted V i. Conside a poduct netwok G=G G 2 =(V,E) of 2 netwoks G (V,E ) and G 2 (V 2,E 2 ), the total extenal load is defined by lemma. Lemma. Let u and u 2 be two nodes in G and G 2 espectively, and G q u 2 thei coesponding extenal load facto. Then the extenal load facto of node uu 2 in G=G G 2 is: q = q q GG G G uu u u Poof. Fom fomula (), the extenal load facto of u u 2 in G is: GG 2 t GG uu 2 2 quu =. 2 N GG 2 Whee N = V V2, i.e, the numbe of nodes in G G 2 and is the total numbe of paths that eithe end at uu 2 o have node u u 2 as an intemediate node. Futhemoe, it can be easily concluded that GG 2 G G2 tuu = tu V2 + tu 2 2 Theefoe,. GG t uu 2 2 t uu 2 G q u and can be expessed as V since we have V2 copies of G and V copies of G 2. t t V + t V t t GG 2 G G2 G G2 uu u 2 u u u G G = = + = qu q u V V V V V V A coollay of this lemma fo the poduct of gaphs is that: q GG 2... G Gi uu n... u = qui i=. +. Theoem. The taffic intensity of a node u u 2 in a netwok G G 2 is defined as: N G G 2 puu = p ( ) 2 + qu + q u2 N. Poof. The poof follows fom fomula and lemma. To compute the taffic intensity of multidimensional tous P p, poduct netwoks -ay n-cube Q n and multidimensional mesh L p we fist need to detemine the extenal load facto of thei building blocks. Note that the

9 Pobane z czasopisma Annales AI- Infomatica Embedding as a method to impove multichannel 339 hypecube netwok can be obtained fom Q n by setting =2. When =2 we have K 2. Theoem 2. The extenal load facto of a node u in each building block espectively fo P n, K, L n is defined as: 2 P n p( p+ ) p p qu = p 2 2 2, K qu =, L q n u = ( p u)( 2u+ ) ( u+ ) p. The taffic intensity of each building block can be concluded fom the theoem and theoem 2. The taffic intensity of thee showed netwoks, is computed in the theoem 3. Theoem 3. The taffic intensity of a multidimensional tous N = p p p nodes, an -ay n-cube Q K of 2... n = n PP P... n n2 n of N = n nodes and a multidimensional mesh L L2... L of N = p p2... p nodes ae: N pi( pi + ) pi pi qu... u = p + N i= pi 2 2 2, N n qu... u = p + N n, N qu 2... u = p + p i ui ui + ui + N i= p i ( )( ) ( ). Poof. The poof follows fom theoem and theoem 2. Fom the theoem above, we note that the -ay n-cube and tous ae symmetic and in this case the taffic intensity does not dependent on the node. In the case of a multidimensional mesh the taffic depends on the node. The taffic intensity of a hypecube can be obtained fom that of a n-ay -cube by setting n=2. In the case of a mesh and a tous, we conside a two-dimensional netwok of size xy whee x and y ae both even. Since the mesh is not symmetic, we take the taffic intensity in the node u =y/2 and u 2 =x/2. The taffic intensity of a tous of N=xy, a hypecube of N nodes and a mesh of N=xy nodes is:

10 Pobane z czasopisma Annales AI- Infomatica Miosław Mazuek N N qu... u = p + + x 4 ( N ) x, q u... u N log2 N = p +, 2( N ) N N 2 qu ( )... u = p + + x 2 N x N. 8. Conclusions By utilizing methods of embedding in design of new topologies we allow to accomplish new oiginal topologies. These new stuctues ae chaacteized by pedictable paametes descibed above by attibutes of the base topologies. Opeation of the catesian poduct by one means guaantee of a small degee of computation nodes, by othe means it is possible to achieve a small diamete. Embedding allows connection of two popula topologies with fixed attibutes and attaining new topology, which inheit all the above topology abilities. Additional attibute of this opeation is the fact that a numbe of nodes gows by means of multiplicity and diamete gows by addict ability. The gown stuctue is also chaacteized by simple stuctue of implementation of the netwok algoithms based on fundamental and commonly known algoithms in base topologies. All these attibutes ae paticulaly useful in systems scalability and allow to incease netwok size although, etaining stabilized numbe of nodes. Refeences [] Hajde M., Dymoa P., Mazuek M., Impovement methods of multichanel bus topologies paametes, Sixth Intenational Confeence Soft Computing and Distibuted Pocessing WSIiZ Rzeszów, (200). [2] Hajde M., Mazuek M., Gafy egulane w pojektowaniu wielokanałowych optycznych systemów komunikacji, Zeszyty Naukowe Politechniki Rzeszowskiej, Infomatyka, Rzeszów (2003), in Polish. [3] Kotsis G., Inteconnection Topologies and Routing fo Paallel Pocessing Systems, Technical Repot Seies, ACPC/TR 92-9, (992). [4] Ohing S. R., Ibel M., Das S. K., Reliable Boadcasting in Poduct Netwoks in the Pesence of Faulty Nodes, Poceedings of the 7th Symposium on Paallel and Distibuted Pocessing, (995). [5] Youssef A., Design and Analysis of Poduct Netwoks, Poceedings of the 5th Symposium on the Fonties of Massively Paallel Computation, (995). [6] Bellachaachia A., Youssef A., A Unified Theoy fo a Taffic Analysis in Poduct Netwoks, Poceedings of the 9th Intenational Pocessing Symposium, (995). Poweed by TCPDF (