Comparison between Topological Properties of HyperX and Generalized Hypercube for Interconnection Networks

Transcription

1 Joural of mathematics ad computer sciece 9 (2014), Compariso betwee Topological Properties of HyperX ad Geeralized Hypercube for Itercoectio Networks Sadoo Azizi 1*, Naser Hashemi 1, Mohammad Amiri Zaradi 2 1 Departmet of Mathematics ad Computer Sciece, Amirkabir Uiversity of Techology, Tehra, Ira 2 Departmet of Mathematics ad Computer Sciece, Shahid Bahoar Uiversity, Kerma, Ira Article history: Received August 2013 Accepted November 2013 Available olie November 2013 s.azizi@aut.ac.ir hashemi@aut.ac.ir mohammadamiri@uk.ac.ir Abstract I order to desig a itercoectio etwork, it is essetial to have a comprehesive uderstadig about properties ad limitatios of the etwork. These properties ad limitatios are characterized by the topology of the etwork. Sice a topology sets costraits ad costs, it plays a critical role i all itercoectio etworks. Differet topologies have bee proposed for itercoectio etworks i literature. The Geeralized Hypercube is oe of the oldest topologies that ca be metioed. Recetly a group of researchers at HP Lab have itroduced a ew topology for these etworks, called HyperX. Despite of may similarities betwee these two topologies, there are sigificat differeces betwee their performaces ad costs. It seems that this importat issue has bee eglected i cotexts of itercoectio etworks. I this paper, we compare HyperX ad Geeralized Hypercube topologies uder some key topological measures. We show that HyperX is somehow better tha Geeralized Hypercube i the sese of topological properties. Keywords: Topological properties, HyperX, Geeralized Hypercube, Itercoectio Networks, Performace. * Correspodig Author

2 S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), Itroductio Network topology determies the structure of the itercoectio etwork. I other words, topology shows the static order of termials, switches (or routers) ad liks. Topology is oe of the importat aspects of ay itercoectio etwork; because it sets may limitatios of the etwork icludigdegree, diameter, average distace ad the umber of liks [1]. I the literature of itercoectio etworks, differet topologies have bee preseted by researchers. For istace, we ca metio Mesh, Torus, Hyper cube ad Geeralized Hypercube topologies [1]. Durig the past few years, icreasig itegrated-circuit pi badwidth has icreased the radix of itercoectio etworks ad their switches. Exploitatio of the high-radix switches has led to higher performace ad lower cost [2]. This follows emergece of ew topologies for the itercoectio etworks. First, i 2007, Kim et al. [3] itroduced the ew high-radix topology, amed Flatteed Butterfly. Two years later, Ahet al. [4] at HP Lab proposed aother ew highradix topology called HyperX. This topology ca be cosidered as a geeral framework for other well-kow topologies icludig Hypercube ad Flatteed Butterfly [4]. It is ecessary to metio that HyperX is also a geeral framework forgeeralized Hypercube, as well. But, this issue has ot bee poited out i the referece [4] ad we will briefly show it i Sectio 2 of this paper. HyperX ad Geeralized Hypercube topological structure are slightly differet, but this slight differecehas made a huge differece o performace ad cost of them. So far i the literature of the itercoectio etworks, there is o compariso betwee these two topologies. Therefore, i this paper we are goig to focus o this issue. With this study, we are able to achieve a comprehesive uderstadig about the performace ad cost of each topology ad their compariso. Therest of the paper is orgaized as follows. The descriptio of two topologies structure is presetedisectio 2. I Sectio 3, the most importat performace ad cost measures are itroduced ad we extract them for the studied topologies.i Sectio 4these two topologies are compared together. Fially, Sectio 5 cocludes the paper ad proposes suggestios for future work. 2. Defiitios ad prelimiaries 2.1. Describig Geeralized Hypercube topology ad its model The Geeralized Hypercube topology was itroduced by Bhuya ad Agrawal i 1984 [5]. They also extracted some of its topological properties such as average distace, diameter, ad umber of liks. Geeralized Hypercubeis a kid of direct itercoectio etworks such that oe termial is exactly coected to each switch.this topology, i fact, is a geeralizatio of Hypercube. That meas, we ca put more tha two switches i each dimesio with complete coectios. If we assume that N is the umber of odes (switches), umber of dimesios ad m i the umber of odes i dimesio i, 1 i, the Geeralized Hyper cube ca be show with the symbol GHC m1,m 2,,m ad is defied as follows. Defiitio 1[5]: Each ode like A, 0 A N 1, ca be idetified as a vector of coordiates (a, a 1,, a 1 ) such that a i, 0 a i m i 1 idicatig the locatio of odes i dimesio i. I Geeralized Hypercube topology, each ode is coected to all odes that differ oly i oe dimesio.i other words, the ode (a, a 1,, a i+1, a i, a i 1,, a 1 ) i dimesio iis 112

3 S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), coected toall odeswith vector coordiates (a, a 1,, a i+1, b i, a i 1,, a 1 ) where 0 b i m i 1, b i a i. Figure 1 demostrates two-dimesioal Geeralized Hypercube. The umber of odes i the first dimesio equals 2 ad i the secod dimesio are 4. Figure 1: A 2-dimesioalGeeralized Hypercube topology. White circles are odes (switches) ad black circles are termials Describig HyperX topology ad its model Several years after itroducig Geeralized Hypercube topology, HyperX was itroduced by a group of researchers at HP Lab i 2009 [4]. HyperX, also, is a direct etwork of switches i whichthe costat umber of T termials is exactly coected to each switch. Each termial ca be a computatioal ode, cluster of computatioal odes, I/Oode, or ay other itercoectio devices. Nodes (switches), such as Geeralized Hypercube topology, are orgaized i several dimesios with complete coectios i each dimesio. Obviously, if T>1the umber of switches will be less tha the umber of termials. It is assumed that P is the umber of switches i the etwork, ad r i is the umber of switches i dimesio i. Therefore, HX r1,r 2,,r represets the -DHyperX topology. Here, it is importat to ote that Azizi et al. [9] have recetly ivestigated the topological properties of HyperX etwork i a comprehesive study. Defiitio 2[9]: I HyperX each ode such A, 0 A P 1 ca be represeted by -tuple coordiate vector(a, a 1,, a 1 )i whicha i,0 a i (r i 1) deotes the positio of the ode i dimesio i. I HyperX etwork each ode is coected to all odes that differ oly i oe dimesio. I other words, the ode(a, a 1,, a i+1, a i, a i 1,, a 1 ) i dimesio i is coected toall odes with coordiate vector (a, a 1,, a i+1, b i, a i 1,, a 1 ) i which 0 b i r i 1, b i a i. Figure 2 depicts 2-DHyperXtopology. I this figure,16 termials physically cotacted with each other through a HyperX topology. Here, the same as Figure 1,the umber of odes i the first dimesio is 2 ad the secod dimesio equal

4 S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), Figure2. HyperX topology. White circles are switches ad the black oes are termials (T = 2). As ca be iferred from two topologies structure, there is a very strog resemblace betwee them ad the oly differece is the umber of termials coected to each ode. For Geeralized Hypercube T = 1 ad this value ca be ay costat umber for HyperX, that meas T 1. It ca be clearly realized that Geeralized Hypercube is a special case of HyperX. I other words, HyperX is cosidered asa framework for Geeralized Hypercube. 3. Performace ad cost measures I [6, 10] several importat measures have bee listed i order to evaluatio of itercoectio etworks icludig diameter, average distace, umber of switches, umber of liks ad etwork degree.the diameter ad the average distace ca be cosidered asperformacemetricsad the umber of switches, the umber of liks ad the degree of a etworkare cost metrics. I this sectio wefirst begi by reviewig their defiitios ad thewe extractthemfor our studied topologies.it should be metioed that some of them have bee calculated before [4-5, 9] Evaluatio measures Defiitio 3[6] (Diameter of the etwork, D): The logest distace betwee each pair ofodes i theetwork stads for the diameter of that etwork. Defiitio 4 [5] (Average distace,d): The average distace betwee all pairs of odes i a etworkis the average distace of thatetwork adis give by D = i N i N 1 (1) Where N is the umber of etwork termials ad N i is defied by the set of odes located i distace i from the cetral ode O. Defiitio 5[6] (Network degree, ): The ode with the biggest degree i the etwork determies the degree of that etwork. Remark 1: I this paper, the ier liks (liks betwee odes ad termials are also cosidered for computig the umber of liks i the etwork to makea fair compariso betwee the two topologies. 114

5 S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), Defiitio 6[6] (Number of switches, S):The umber of switcheseeded to build a etwork of size N. Defiitio 7[6] (Number of liks, E): The liks eeded to build a etwork of sizen ad its formula is obtaied by E = S 2 (2) As oted above, Sad are the umber of switchesad the etworks degree, respectively Assessmet of measures for studied topologies Geeralized Hypercube Bhuya ad Agrawal [5] have extracted some parameters for Geeralized Hyper cube. Here, we restate them by cosiderig remark 1. Network diameter, D GHC = (3) Average distace,for the case that (m i = m, 1 m i ) D GHC =. m 1. m 1 /(N 1) (4) Network degree, GHC = 1 + (m i 1) (5) Number of switches, S GHC = N (6) Number of liks, E GHC = N(1 + ( (m i 1))/2) (7) HyperX 115

6 S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), Ah ad his collaborators [4] i their paper focused odescriptio of HyperX topology ad its compariso with folded Close topology [7]. Fially, they proposed a routig algorithm called DAL that attempt to balace the traffic load throughout the etwork. However, the topological properties of their topology have bee igored. I [9] this issue have bee doe by Azizi et al. Here, we explai some of them for our task i this paper. The diameter of HyperX etwork, like Geeralized Hypercube, is equal to the umber of etwork dimesios.that is true becauseoe of dimesio differececa be compesated with each step. Ad sice a pair of odes i HyperX i the worst case are differet i all dimesios, so the etwork diameter is the same as the umber of dimesios. D HX = (8) The followig is the average distace of HyperX. The proof is give i the appedix of this paper.here to express the average distace with aformula,its regular mode, i.e. (1 r i, r i = r), have bee cosidered. D HX =. r 1 r N. N 1 (9) Sice HyperX is a regular etwork, its degree is equal to the degree of each of ode. Each ode is coected to all odes that have differece oly i oe dimesio with it. Besides, each ode is coected exactly to T termials. The, the topology degree of HyperX is equal to HX = T + (r i 1) (10) I HyperX, ulike Geeralized Hypercube, the umber of switches is less tha or equalto the umber of termials (etwork size).because each switch hasttermials. So we have S HX = N/T (11) The umber of liks i HyperX ca be calculated with replacemet of expressios (10) ad (11) i equatio (2). Thus, we have E HX = S HX (T + ( (r i 1))/2) (12) 4. Compariso of performace ad cost of HyperX ad Geeralized Hyper cube I this sectio, we compare the topological structure of HyperX ad Geeralized Hyper cube uder measures described i subsectio Network diameter 116

7 Network degree Average Distace S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), As the equatio (3) ad (8) deotes, the diameter of two topologies is the same i.e., their diameter equal to the umber of etwork dimesios, Average distace To compare the average distace of two topologies, the size of etwork is assumed betwee 16 ad The motivatio for this choice is that we wat to measure the average distace betwee two etworks for small-scale ad large-scale oes.the simulatio resultis give i Figure 3. As ca be iferred from the result, the average distace of HyperX is less tha Geeralized Hypercube i ay size. GHC HX Network size Figure3. Compariso of HyperX ad Geeralized Hypercubeaverage distace 4.3. Network degree To compare the degree of HyperX ad Geeralized Hypercube, we assume the umber of dimesios the same, also, for HyperX T=4. As Figure 4 demostrates, HyperX degree is less tha Geeralized Hypercube. GHC HX Network size 117

8 Normalaized Average Distace S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), Figure 4. Compariso of HyperX ad Geeralized Hypercube etwork degree 4.4. Normalized average distace Topologies with low average distace usually have high etwork degree. Thus, it is importat to have a measure that take ito accout both of them as a measure. Normalized average distace is such criterio that has bee itroduced by Lee ad Gaz [11]. It is defied as the product of average distace ad etwork degree. Actually, this measure is a trade-off betwee the average distace ad the degree of a etwork. Therefore, we compare the HyperX ad Geeralized Hypercube uder it. As figure 5 idicates, HyperX is good i this measure, too. The etwork with low average distace usually has better performace. Of course, the routig algorithm play a critical role to achieve good performace ad QoS. For example, oe ca use heuristic strategies to do this [12] Network 256 size Figure 5. Compariso of HyperX ad Geeralized Hypercube etwork degree 4.5. Number of switches By comparig related expressios for the umber of switches i Geeralized Hyper cube ad HyperX topology, it ca be foud that the differece betwee them is the amout of T. It is clear that, o the oe had, the umber of switches i the etwork decrease as the amout oft icreases, ado the other had, higher T icrease the degree of switches. Therefore, i order to make a fair compariso betwee theumber ofswitches i the two topologies, we cosider theswitch degreethe same asfigure 4.As figure 6shows for buildig a fixedsize etwork, HyperX topology will require fewer switches compared with the Geeralized Hypercubetopology. It is importat to ote that the degree ad the umber of switches i HyperX topology are less tha Geeralized Hypercube, simultaeously. 118

9 Number of liks Number of switches S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), GHC HX Network size Figure 6. Compariso of the umber of HyperX ad Geeralized Hypercube switches 4.6. Number of liks I this subsectio, we compare the umber of liks used by two topologies.we assumed T=4 for HyperX topology. The result, Figure 7, shows that HyperX eeds fewer liks to compare with Geeralized Hypercube GHC HX Network size Figure 7.Compariso of the umber of HyperX ad Geeralized Hypercubeliks 5. Coclusios ad Future Works Due to progress of techology ad its effects o the computer architecture, evaluatio of this field has bee become a importat issue. The techology of sigalig ad also total badwidth of chip pis has a substatial effect o the performace of itercoectio etworks. This developmet has itroduced a ew type of etworks that are called high radix routers.hyperxis a obvious example of this architecture. Our mai cotributio i this paper is extractig some aspects of off-chip HyperX topology usig mathematical expressios. Extractig these kids of expressios i order to describe topological properties of HyperX ca have several beefits. For example, determiig the structural properties of etworks ca be very helpful i desigig graph algorithms i order to determie commuicatio patters betwee odes, desigig ad implemetig the routig algorithms, 119

10 S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), ivestigatig resource replacemet ad their allocatio, implemetig VLSI layout, examiig etwork vulerability agaist attacks ad failures ad fially aalytical modelig ad performace evaluatio of the etwork. With advet of ew topologiestothe etwork areathere is a eed to compare its structure with other topologies preseted i the literature. Oe of literature topologies is itercoectio etwork of Geeralized Hypercube that has a remarkable similarity with HyperX topology i appearace but their structural parameters are quite differet.therefore, the focus o the secod part of this paper is the compariso of HyperX ad Geeralized Hypercubeuder several importat performace ad cost measures. The results show that HyperX has better performace ad cost tha Geeralized Hypercubei terms of structural properties.aother future work ca be the evaluatio ofhyperx ad Geeralized Hypercube etwork uder importat criteria i field of etworker chips (NoCs). Also, we ca compare these two topologies with cosiderig the fuctioal measures; i.e., take ito accout the traffic egieerig withi the etwork. Appedix The proof of average distace formula forhyperx topology is as follows: Accordig to equatio (1) average distace of a topology ca obtaied by D = i N i N 1 The ukow term of this expressiois N i that is defied by the umber of termials located i distace i from the cetral of a specific ode. For example, a group of termials that have differece oly i oe dimesio will be equal to N 1 = T r i 1, i 1,2,, (13) Now, umber of termials that are located i a two hops distace from the cetral ode ca be calculated by computig the umber of a group of termials which differ i their dimesios ad cetral ode s dimesio is two. I mathematical expressio we have N 2 = T r i 1 r j 1, i, j 1,2,,, i j (14) With cotiuatio of this tred, the umber of termialslocated i distace i from the cetral of a specific ode ca be expressed as follows N i = T r i 1 r j 1 r i 1, i, j, k, 1,2,,, i j k 5) (1 120

11 S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), We assume that umber of switches i each dimesiois same as others; i.e.,1 i, r i = rthusfor N i we will havetothe followig simpler form N i = T i (r 1) i (16) With replacig the above expressio i equatio (1) we have, D = T. i. (r 1) i i N 1 7) (1 Here, with simplifyig, we wat to achieve a proper formula foraverage distace of HyperX topology.accordig to the theory of combiatorics [8] we have 1 + x = i (x) i (18) Ad with differetiatio of both sides we get 1 + x 1 = i. i (x) i 1 (19) By multiplyig Eq. (19) by x we obtai x 1 + x 1 = i. i (x) i (20) Fially, with replacig x = r 1 i the above equatio we get (r 1) r 1 = i. i (r 1) i (21) 121

12 S. Azizi, N. Hashemi, M. Amiri Zaradi/ J. Math. Computer Sci. 9 (2014), So, the average distace i a regular HyperX ca be calculated by D = T. (r 1)(r) 1 N 1 (22) Similarly Refereces D =. r 1 r N. N 1 (23) [1] W.J. Dally, B. Towles, Priciples ad practices of itercoectio etworks, Morga Kaufma Publishers, [2] J. Kim, High-Radix Itercoectio Networks, PhD Thesis, Staford Uiversity, [3] J. Kim, W.J. Dally, D. Abts, Flatteed butterfly: a cost-efficiet topology for high-radix etworks, I ISCA '07: Proceedigs of the 34th aual iteratioal symposium o Computer architecture, Sa Diego, Califoria, USA, Jue (2007) [4] J. H. Ah, N. Bikert, A. Davis, M. McLare, R. S. Schreiber, HyperX: Topology, Routig, ad Packagig of Efficiet Large-Scale Networks, Proceedigs of the Coferece o High Performace Computig Networkig, Storage ad Aalysis (2009) [5] L. N. Bhuya, D. P. Agrawal, Geeralized Hypercube ad Hyperbus Structures for a Computer Network, IEEE Trasactios o Computers, Vol.33, No. 4 (1984) [6] J. Dégila, B. Sasò, A Survey of Topologies ad Performace Measures for Large Scale Networks, IEEE Commuicatios Surveys ad Tutorials Magazie, Fourth Quarter, Vol. 6, No. 3(2004) [7] C. Leiserso, Fat-trees: Uiversal Networks for Hardware Efficiet Supercomputig, IEEE Trasactios o Computers, Vol. 34, (1985) [8] R. P. Grimaldi, Discrete ad combiatorial mathematics: a applied itroductio, 5th ed. Addiso Wesley, Readig, [9] S. Azizi, F. Safaei, N. Hashemi, O the topological properties of HyperX, The Joural of Supercomputig Vol. 66, No. 1, (2013) [10] E. Abuelrub, A Comparative Study o the Topological Properties of Hyper-Mesh Itercoectio Network, Proceedigs of World Cogress o egieerig, Lodo, UK, Vol. 1, (2008) 9-5. [11] B. Li, A. Gaz, Virtual Topologies for WDM Star LANs - the Regular Structures Approach, i INFOCOM 92. Eleveth Aual Joit Coferece of the IEEE Computer ad Commuicatios Societies, (1992)