Structural Properties of Torus-Butterfly Interconnection Network

Transcription

1 Innaional Jounal of Compu Applicaions ( ) Volum 46 No.16, May 212 Sucual Popis of Tous-Bufly Inconncion Nwok Laifah STMIK Jakaa STI&K Jl.BRI Radio Dalam No. 17 Jakaa1347 Indonsia Enasui Gunadama Univsiy Jl. Magonda Raya no. 1 Dpok, Indonsia Djai Kami Univsiy of Indonsia Dpok Indonsia ABSTRACT This pap inoducd nw inconncion nwok namd as Tous-Bufly. Th nwok is gnad by a poduc of nwok fom Tous and Enhancd Bufly inconncion nwoks which is suiabl fo paalll compus. W hav analyzd and povd ha h sucual popis such as nwok diam and nod dg of h Tous-Bufly inconncion nwoks is mo scalabl han oh inconncion nwoks. In addiion o hm, h nwok cos is psnd. Th sul is also mo scalabl. Gnal Tms Dsign, pfomanc Kywods: Tous Nwok, Enhancd Bufly Nwok, Casian poduc nwok. Cayly Gaph. 1. INTRODUCTION Cun compu inconncion nwoks hav bn widly applid in vaious aas, such as paalll compuing sysm, mulipocsso sysms, and woksaion nwoks [1]. Accoding o Zhang [2] modl (opology) of inconncion nwoks is an impoan pa fo paalll pocssing o disibud sysm. Zhang [3] sas ha a good modl of h inconncion nwok mus hav h symmy popis, masud (scalabl), has a small diam, and also has a consan and a limid dg [4, 5]. In pacic i is mo dsiabl ha h nwok modl has a high connciviy and a small diam. Conncdnss is widly usd o masu h faul-olanc capaciy of h nwok, whas h diam showd h fficincy of ouing (snding daa) [6]. To valua an inconncion nwok modl, h sachs compad h inconncion nwok modls hough an analysis of vaious paams, such as, fo sucual popis: dg and diam. In cn yas, h is sach on a class gaph as a modl of inconncion nwoks calld h Cally gaph [2]. This is du o Cayly gaphs hav many dsiabl popis as good as h nwok modl ha has bn mniond, which has h popis of symmy, small diam, conncdnss and high faul olanc [2]. Cayly gaph also has a gula chaac, i a ach nod has a consan dg [7]. Som modls of inconncion nwoks ha a Cayly gaph is a hypcub, bufly, msh and ous [3]. Hypcub o n-cub, is a nwok modl ha has chaacisics of small diam and ponially in h inconncion nwok modl, howv hypcub hav limiaions on h dg ha a no consan, bu volv accoding o is siz. In od o hav consan dg, i has bn dsignd a nwok modl calld Bufly. Som sach dvlopd Wap aound Bufly [8]. I has bn also dvlopd a nwok modl calld Enhancd Bufly [4]. Enhancd Bufly nwok modl has h popis of small diam, consan dg and also of symmy. Oh modls of inconncion nwoks ha a Cayly gaph is calld Tous, which is widly usd in paalll compuing sysm [9]. In cn yas, dmand fo high spd and high houghpu compuing machins has ld o h dvlopmn of a nw inconncion nwok modl wih a lo numb of pocssos [1, 11]. Fo som modls i is a modifid vsion of xising modls o o combin wo poposd modls o bnfi fom h popis ownd by boh modls [1]. Th a many mhods fo combining wo xising modls, on of hm is h muliplicaion (poduc) such as h Casian poduc [12, 3]. Rsach has bn don using a mhod of poduc of wo nwok modls such as sa-cub [13], hypbufly [14], h Tous mbddd Hypcub [15] and Scalabl wisd Hypcub [16], bu h suls of hs sudis indica h dg of hypcub is no consan, and consqunly a lag nough nwok cos. This siuaion is pcisly h wish o avoid. In his pap, w inoducd anoh poduc nwok of Tous and Enhancd Bufly opology, namd as Tous-Bufly inconncion nwok. Th advanags of Tous and Enhancd Bufly a usd fo his poduc nwok. Th Ida of his sach is o dsign h inconncion nwok modl calld Tous-Bufly, which is h Casian poduc of h inconncion nwok modl Tous and Enhancd Bufly. Thn analyz h popis of modls of Tous-Bufly inconncion nwok hough sucual paam such as dg and diam and valua h nwok cos, symy and gulaiy. 2. THEORETICAL REVIEW I is known ha h sucu of h modl of a nwok can b dscibd as a conncd gaph G = (V, E), wih V h s of vics is h s of pocssos and E is h s of dg o link in h nwok [8]. An dg is an odd pai of x, y {x, y} of disinc vics in G. Th s of vics V in a gaph G is dnod as V G and h s of dg E in a gaph G is dnod by E G. Dfiniion : A gaph G = (V, E) is calld conncd if fo any wo nods of a gaph G h is always h pah ha conncs h scond nod [17]. 31

2 Innaional Jounal of Compu Applicaions ( ) Volum 46 No.16, May 212 Dfiniion : Th dg of a nod x V G, dnod by dg (x), is a conncd ac fom x o nods y V G, y x [18]. Dfiniion: Th diam of h conncd gaph G (VG, EG) is h maximum disanc of all pais of vics [19]. In inconncion nwok modl, diam can b inpd as h maximum numb of links ha should b acd whn snding a mssag o any pocsso along h shos pah [2]. Th small h diam h sho h im h nwok o snd mssags fom on pocsso o anoh pocsso [21]. Diam is ndd o spd h im quid a h im of on pocsso snds a mssag o anoh pocsso, h small h diam h fas im of dlivy of h mssag. Th small diam will also b bnficial o h ducion of nwok cos and will impov h pfomanc of h pocsso. As alady mniond in h inoducion, on of h modls ha a widly usd inconncion nwoks a Cayly gaphs, his is causd by h Cayly gaph has h popis fini, connc, undicd and symmy [22]. Bcaus of h inconncion nwok modl is considd as a gaph hn h m inconncion nwok modl and a gaph can b plac ach oh. H is h dfiniion of h Cayly gaph. Dfiniion : Suppos H is a goup and S H foming a s of H such ha S S -1. A Cayly gaph of H agains S is undicd gaph Cay (H, S) wh h s of vics is H and h dg conncing g o gs fo vy slcion g H and s S. A Cayly gaph Cay (H, S) is a gula gaph S of od H [23]. Now w giv dfiniion of Bufly nwok. Dfiniion: Suppos d N (N = s of posiiv ings). Bufly inconncion nwok modl of dimnsion d is dnod by B(d) is a gaph wih vx s V = [d +1] [2] d and h s of dg E = E 1 E 2 wih E1 {{(i, α), (i +1, α) / i [d], α [2] d and E2 {(i, α), (i +1, β) / i [d], α, β [2] d, α, β diff only a posiion i}. Th s of vics {(i, α) / α [2] d is said o fom h i-h lvl of h Bufly [24]. Bufly inconncion nwok modl of dimnsion n has n lvls and N = 2 n inpu and oupu. Inpu / oupu is s in h 2 n columns a labld fom o 2 n -1 (in binay). Each lvl is numbd, 1, 2,..., n fom op o boom [22]. Th Bufly inconncion nwok modl only nod o nod in a nighboing ow. Edg bwn h vics in h sam column is calld a saigh dg and h dg bwn nods in diffn columns a calld coss scions [8]. In h gaph B (d), whn lvl d is placd wih lvl, hn w said a Wap aound Bufly dimnsion d o WB (d) [25]. A Wap aound Bufly (WB) dimnsion n 3 whn w addd a paicula dg on h gaph is calld Enhancd Bufly [4]. Dfiniion: Tous (m, l) = T (m, l) is a gaph ha conains (m, l) msh wih wap aound sids in ows and columns [26]. Dfiniion : Givn wo gaphs G (V1, E1), and H (V2, E2), Casian poduc opaion dfind G and H is dnod by G H is h gaph (, ), wih V and E as follows 1)V {(a,b)/a V1, y V2. 2)Fo any x (a,b) and y (c,d) in V, (x, y) is and dg in E if and only if (a, c) is an dg in E1 and b d o (b, d) is an dg in E2 and a = c [27]. Poposiion 2.1: Suppos ha wo gaph G = (V 1, E 1 ) and H (V2, E2). Th Casian poduc G H is a conncd gaph which has a siz V1 V2, dgs = dgs G + dg H and diam diam G + diam H [13, 12]. Poposiion 2.2: Suppos ha Cay (S, G) and Cay (S ', G') a wo Cayly gaph, hn h Casian poduc Cay (S, G) Cay (S ', G') H is also a Cayly gaph [3]. Poposiion 2.3: Th dg of inconncion nwok modl Enhancd Bufly dimnsion n dnod as EB(n), n 3, is 5 and h diam is n [4]. Poposiion 2.4: Tous Inconncion Modl dnod as T (m, l) has dgs 4 and h diam = max{ m / 2, l / 2 [26]. Fom h abov dfiniions and poposiions w hav following dfiniion fo h nw inconncion nwok namd Tous-Bufly. Dfiniion: If G = (V 1, E 1 ) is h Tous inconncion nwok modl of siz ml and H = (V 2, E 2 ) is h Enhancd Bufly inconncion nwok modl dimnsion n, hn h Tous-Bufly inconncion nwok modl, dnod as (m, l, n), is h Casian poduc of Tous and Enhancd Bufly, wih m and l is h siz of Tous inconncion nwok modl and n is h dimnsion of h Enhancd Bufly inconncion nwok modl. This is u fo n 3, m 2 and l RESULTS AND DISCUSSION 3.1 Nod dg W hav h following Lmma: Lmma 1: Th dg of ach nod in h Tous -Bufly inconncion nwok modl is 9. Poof: By poposiion 2.1 vy nod in h Tous-Bufly inconncion nwok modl = (m, l, n) has dg , hus h dg = Diam and Nwok Cos W hav h following Lmma fo Diam of Tous- Bufly Inconncions modl: Lmma 2: Th diam of h inconncion nwok Tous-Bufly (m, l, n) is = max { m / 2, l / 2 } + n. Poof: by poposiion 2.1, hn h diam Tous-Bufly is diama Tous + Diam Enhancd Bufly = max { m / 2, l / 2 + n. Fom h abov dg and diam of Tous-Bufly inconncions nwok fomula, w hav h following nwok cos: Nwok Cos of Tous-Bufly inconncion nwok modl is 9 (max { m / 2, l / 2 + n). 3.3 Symi and Rgulaiy. Tous inconncion nwok is a Cayly gaph and Enhancd Bufly is also a Cayly gaph, hnc fom poposiion 2. 2 h nw Tous-Bufly inconncion 32

3 Innaional Jounal of Compu Applicaions ( ) Volum 46 No.16, May 212 nwok is Cayly gaph. I follows ha his nw inconncion nwok is symi and gula. Tabl 1 and figu 1 givs h compaison of dg fo vaious sam pocssos of Hyp-Bufly and Tous mbddd Hypcub nwoks along wih Tous-Bufly nwok. Tabl 1. Dg Compaison of h modls of inconncions nwok Nwok Typ (k, n) TH(16,16,k) (m,l,n) pocsso D g No.of Pocssos Fig 1: Compaison of h dg of 3 modls inconncion nwoks [14] 1[5] Tabl 2. Diam compaison of h modls of inconncion nwok Nwok Typ (k, n) TH(16,16,k) (m,l,n) pocsso Rmak: = Hyp-Bufly, TH =Tous Tous Embddd and = Tous Bufly. In abl 1 i is sn ha Tous-Bufly Inconncion modl has a consan dg,so i is gula whas h Hyp-Bufly and Tous-mbddd-Hypcub has lini dg dpnds on h siz of pocsso. Tabl 2 and figu 2 givs h compaison of diam fo vaious sam pocssos of Hyp-Bufly and Tous mbddd Hypcub nwoks along wih Tous-Bufly nwok. Tabl 3 and figu 3 givs h compaison of nwok coss fo vaious sam pocssos of Hyp-Bufly and Tous mbddd Hypcub nwoks along wih Tous-Bufly nwok [14] 22[5] In abl 2 i is sn ha Tous-Bufly Inconncion modl has low diam han Tous-mbddd-Hypcub and has low diam han Hyp-Bufly fo numb of pocsso 512 ill 496, xcp fo h numb of pocsso and D i a m No.of Pocssos fig 2: Compaison of h diam of 3 modls inconncion nwoks 4. CONCLUSION Fom h abov valuaion modl of Tous-Bufly inconncion nwok and h h compaison abl of 33

4 Innaional Jounal of Compu Applicaions ( ) Volum 46 No.16, May 212 dg, diam and visibl nwok cos, Tous Bufly inconncion nwok has b popis han h modl Tous mbddd Hypcub inconncion nwoks, and Hyp Bufly. This nw inconncion nwok also has symy and gula popis. Hnc his Tous-Bufly Inconncion nwok can b usd as an alnaiv modl fo inconncion nwok. 5. ACKNOWLEDGMENTS Ou hanks o h Univsiy of Gunadama and STMIK Jakaa STI&K who hav sponsod his sach. Tabl 3.Nwok cos compaison of h modls of inconncion nwok Nwok Typ (k, n) TH(16,16,k) (m,l,n) pocsso [14] [5] In abl 3 i is sn ha Tous-Bufly Inconncion modl has low nwok cos han Tous-mbddd- Hypcub and Hyp-Bufly fo all h sam numb of pocsso. N w o k C o s No.of Pocssos Fig 3: Compaison of h nwok cos of 3 modls inconncion nwoks 6. REFERENCES [1] Gu, Huaxi, Xi, Qiming, Wang, Kun, Zhang, Ji dan Li, Yunsong. 26. X-ous: A Vaiaion of ous Topology wih Low Diam and Lag Biscion Widh, ICCSA, pp [2] Zhang, Zhn dan Wang, Xiaoming, 29. A nw Family of Cayly Gaph Inconncion Nwoks Basd on Wah Poduc, ISCST, China, 26-28, Dc, pp [3] Zhang, Zhn, Xiao, Wn Jun, Wi, Wn-Hong, 29. Som Popis of Casian Poduc of Cayly Gaphs, Innaioal Confncs on Machin laning and cybnics, Baoding. [4] Guzid, Osman dan Wagh Mghanad D, 27. Enhancd Bufly : A Cayly Gaph wih Nod 5 Nwok, ISCA Innaional Confnc on Paalll and Disibud sysm, viw as hml ml. [5] Kini, N. Gopalakishna, Kuma, M.Sahish, HS.Muhyunja, 29. Pfomanc Mics Analysis of Tous Embddd Hypcub Inconncion Nwok, Jounal on Compu Scinc and Engining Vol 1(2). [6] Liaw, shng, chyang dan Chang, Gad J., Wid Diams of Bufly Nwoks, Taiwans Jounal of Mahmaics, Vol 3, No. 1,pp.83-88, Mach, [7] Cada, Roman, 29. On Hamilonian cycls in sa Gaphs,Univsiy of Ws Bohmia. [8] Guzid, Osman dan Wagh, Mghanad D, 26. Mapping cycls and Ts on Wap Aound Bufly Gaphs, SIAM Jounal Compuaion, vol. 35, No. 3, pp [9] Wang, Hong, Xu, Du dan Li, Lmin, 27. A Novl Globally Adapiv Load-Balancd Rouing Algoihm fo Tous Inconncion Nwoks, ETRI Jounal, Volum 29, Numb 3. [1] Guzid, Osman dan Wagh, Mghanad D, 28. Exndd Bufly Nwoks. [11] Youyao, LIU, Jungang, HAN, Huimin, DU, 28. A Hypcub-basd Scalabl Inconncion Nwok fo Massivly Paalll Compuing, Jounal of Compus, Vol. 3, No 1. [12] Livingson, Mailynn, Sou, Qunin F., Shif- Poduc Nwoks, Mahmaical and Compuaional Modlling. [13] Day, Khald, Al-Ayyoub, Abdl-Elah, Th Coss Poduc of Inconncion Nwoks, IEEE ansacion on Paalll and Disibud Sysms, Vol. 8 No.2. [14] Shi, Wi and Simani, Padip K, Hyp-Bufly Nwok: A scalabl Opimally Faul Tolan Achicu, Univsiy of Coloado. [15] Kini, N. Gopalakishna, Kuma, M.Sahish, HS.Muhyunja, 21. Tous Embddd Hypcub Inconncion Nwok: A compaaiv Sudy, Jounal on Compu Scinc and Engining Vol 1(4). [16] Alam, Jahangi, Kuma Rajsh, 211. STH:A Highly Scalabl and Economical Topology fo Massivly Paalll Sysm, Indian Jounal of Scinc & Tachnology, Vol 4 No.12. [17] Hamano, Suyadi, Basic Gaph Thoy, Gunadama Univsiy. [18] Enasui, 28. Th Nw Inconncion Nwok Topology: Exndd Lucas Cub Topologi, Dissaion, Gunadama Univsiy. [19] Chung, F.R.K, Diams and Eignvalus, Jounal of h Amican Mahmaical sociy, Vol 2. 34

5 Innaional Jounal of Compu Applicaions ( ) Volum 46 No.16, May 212 [2] Iidon, Mihala, Maula, David W., 22. A 6-Rgula Tous Gaph Family wih Applicaions o cllula and Inconncion Nwoks, Jounal of Gaph Algoihms and Applicaions, Vo 6 no 4. [21] Rahman, MM Haosu, Inoguchi, Yasuki, Faisal, Al Faiz, Kundu, Munaz, Kuma, 211. Symic & Foldd Toi Conncd Tous Nwok, Jounal of Nwok. [22] Hou, Xinmin, Xu, Jun-Ming and Xu, Min, 29. Th fowading Indics of Wappd Bufly Nwoks, Nwoks,DOI 1.12/n. [23] Bmon, J-C, Dao, O, Dlmas and Pnns, S, Hamilon Cycl Dcompsiion of h Bufly Nwok, paalll pocssing l, wold scinific Publishing Company. [24] Kalovic, Rasislav, 26. Boadcasing on Bufly Nwok wih dynamic Fauls Baislava. [25] Jyohi, Papandangal Vijaya, Mahaswai Bommiddy and Klka Indani, Dominaion Numb of Bufy Gaphs, Chamchui Jounal of Mahmaics, Volum 1, No. 1, [26] Xiang, yonghong, 28. Inconncion Nwoks fo Paalll and Disibud Compuing, Dpamn of Compu Scincs, Univsiy of Duham, Unid Kingdom. [27] Yousf, Abdou, Casian Poduc Nwoks, Innaional Confnc on Paalll Pocssing. 35