A Fast Parallel Routing Algorithm for Strictly Nonblocking Switching Networks

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1 A Fst rllel Routing Algorithm for tritly Nonloking withing Networks Enyue Lu y,. Q. Zheng z, nd Bing Yng Π y Dept. of Mthemtis nd Computer iene, lisury niversity, lisury, MD 8 z Dept. of Computer iene, niversity of exs t Dlls, Rihrdson, X 8 Π Ciso ystems, In., Rihrdson, X 8 y elu@slisury.edu, z sizheng@utdlls.edu, Π inyng@iso.om Astrt - A lss of stritly nonloking (NB) networks B(N; p; ff) n e onstruted from vertil stking of multiple plnes of Bnyn networks. Fst routing lgorithms re needed for finding ville onnetion pths in B(N; p; ff) networks. In this pper, y modeling the swithing routing prolem in NB networks s strong edge oloring prolem, we propose simple nd fst prllel routing lgorithm for routing onnetions in NB B(N; p; ff) networks. he proposed lgorithm n route onnetions in NB B(N; p; ff) networks in O(p N ) time using ompletely onneted multiproessor system of N proessing elements. Our lgorithm n e trnslted into lgorithms with n O(lg N lg lg N ) slowdown ftor for the lss of N-proessor hyperui networks, whose strutures re no more omplex thn single plne of B(N;p;ff) networks. Keywords: Bnyn networks, rosstlk, stritly nonloking networks, grph oloring, prllel lgorithm. Introdution A swithing network usully omprises numer of swithing elements (Es), grouped into severl stges interonneted y set of links. In n eletril swithing network, links re wires nd Es re rossr swithes. In n optil swithing network, links re implemented y optil wveguides nd Es n e implemented y eletro-optil Es suh s ommon lithium-niote (LiNO ) Es (e.g. [,, 8]). Without loss of generlity, we ssume tht the size of n E is, i.e. eh E hs inputs nd outputs. In swithing network, if two inputs (resp. outputs) of n E intend to e onneted with the sme output (resp. input), output link onflit (resp. input link onflit) ours. An eletronilly ontrolled optil E n hve swithing speed rnging from hundreds of pioseonds to tens of nnoseonds []. However, due to the nture of optil devies, optil swithes introdue dditionl hllenges. One is rosstlk prolem, whih is used y undesired oupling etween signls with the sme wvelength rried in two wveguides so tht two signl hnnels interfere with eh other within n E. he rosstlk prolem in photoni swithing networks dds new dimension of loking, lled node onflit, whih hppens when more thn one onnetion with the sme wvelength psses through the sme E t the sme time. If n I/O onnetion pth does not hve ny onflit with other onnetion pths, it is lled onflit-free pth. Nonloking swithing networks hve een fvored in swithing systems euse they n e used to set up ny onflit-free one-to-one I/O onnetion pths. here re three types of nonloking networks: stritly nonloking (NB), wide-sense nonloking (WNB) nd rerrngele nonloking (RNB) [, ]. In oth NB nd WNB networks, onnetion n e estlished from ny idle input to ny idle output without disturing existing onnetions. In NB networks ny of ville onflit-free pths for onnetion n e hosen nd in WNB networks, however, rule must e followed to hoose one. RNB networks n estlish onflit-free pth for the onnetion from ny idle input to ny idle output if the rerrngement of existing onnetions is llowed. Reently, lss of multistge nonloking swithing networks hs een proposed. In this lss eh network, denoted y B(N; x; p; ff), hs reltively low hrdwre ost O(N : lg N ) nd short onnetion dimeter O(lg N ) in terms of the numer of Es. A B(N; x; p; ff), ff f; g, is onstruted y horizontlly ontenting x(» n ) extr stges to n N N Bnyn-type network, nd then vertilly stking p opies of the extended Bnyn. B(N;x;p; ) nd B(N; x; p; ) re similr in struture, ut the ltter does not llow ny two onne- In this pper, the rosstlk is referred to the first-order non-filterle E rosstlk [, ]. In this pper, N = n (n =lgn) nd ll logrithms re in se.

2 tions with the sme wvelength pssing through the sme E t the sme time while the former does. B(N;x;p; ) nd B(N;x;p; ) re suitle for eletroni nd optil implementtion, respetively. It hs een shown tht B(N; x; p; ff) n e NB, WNB nd RNB with ertin vlues of x nd p for given N nd ff [8,,,,]. In swithing network, when more thn one input requests to e onneted with the sme output, output ontention ours. Output ontentions n e resolved y swith sheduling. For set of onnetion requests without output ontention, the proess of estlishing onflitfree onnetion pths to stisfy these requests is lled swith routing. A swith routing (or simply, routing) lgorithm is needed to find these pths. One set of onflitfree pths is found, the Es on these pths n e properly set up. Routing lgorithms ply more fundmentl role in WNB nd RNB networks sine the nonlokingness depends on them. For NB networks, routing lgorithms tend to e overlooked sine onflit-free pth is lwys gurnteed for the onnetion from ny idle input to ny idle output without rerouting the existing onnetions. An effiient routing lgorithm, however, is still needed to find suh onflit-free pth for eh onnetion request. Any routing lgorithm requiring more thn liner time would e onsidered too slow. hus, finding effiient lgorithms to speed up routing proess is ruil for high-speed swithing networks. he fous of this pper is studying the ontrol spet of the lss B(N; ; p; ff) networks, simply s B(N; p; ff), in the ontext of eing used s eletril nd optil swithing networks. In prtiulr, our ojetive is to speed up routing proess in NB B(N; p; ff) networks using prllel proessing tehniques. By exmining the onnetion pity of B(N; p; ff), we redue the routing prolems for this lss of networks to strong edge-olorings of iprtite grphs. Bsing on our model, we propose fst routing lgorithm for B(N; p; ff) using prllel proessing tehniques. We show tht the presented prllel routing lgorithm n route ny set of O(N ) onnetions in NB B(N;p;ff) networks in O(p N ) time, whih improves the est known lgorithm with time omplexity O(lg N p N )in []. he reminder of this pper is orgnized s follows. In etion, we disuss the topology of B(N; p; ff). In etion, we model routing in B(N; p; ff) s strong edge oloring prolems of n I/O mpping grph G(N;K;g). In etion, we present fst prllel routing lgorithm for NB B(N;p;ff) networks. We onlude our pper in etion. Nonloking Networks Bsed on Bnyn Networks. Bnyn-type Networks A lss of multistge self-routing networks, Bnyn-type networks, hs reeived onsiderle ttention. A network elonging to this lss stisfies the properties of short onnetion dimeter, unique onnetion pth, uniform modulrity, et. Bnyn-type networks re very ttrtive for onstruting swithing networks. everl well-known networks, suh s Bnyn, Omeg, nd Bseline, elong to this lss. It hs een shown tht these networks re topologilly equivlent [, ]. In this pper, we use Bseline network s the representtive of Bnyn-type networks. An N N Bseline network, denoted y BL(N ), is onstruted reursively. A BL() is E. A BL(N ) onsists of swithing stge of N= Es, nd shuffle onnetion, followed y stk of two BL(N=)s. hus BL(N ) hs n stges leled y ; ;n from left to right, nd eh stge hs N= Es leled y ; ;N= from top to ottom. he upper nd lower outputs of eh E in stge i re onneted with two BL(N= i+ )s. he N links interonneting two djent stges i nd i +re lled output links of stge i nd input links of stge i +. he input (resp. output) links in the first (resp. lst) stge of BL(N ) re onneted with N inputs (resp. outputs) of BL(N ). o filitte our disussions, the lels of stges, links, Es, inputs nd outputs re ll represented y inry numers. An exmple is shown in Fig.. I N p p p AGE Figure : BL(). BL(N ) is self-routing networks. he self-routing in BL(N ) is deided y the destintion, d n d n d,of eh onnetion. If d n i =, the input of the E on the O

3 onnetion pth in stge i is onneted to the E s upper output, nd to the lower output otherwise (i.e., d n i = ). As shown in Fig., onnetion pths,, nd re set up y self-routing in BL(). By this self-routing property, we hve the following simple ft: Lemm Given ny O(N ) one-to-one distint input/output pirs, the unique pths in BL(N ) for these pirs n e omputed in O(lg N ) time using N proessing elements (Es) if eh E is ssigned to O() pirs. I N plnes AGE O. truture of B(N; x; p; ff) Networks Figure : A network B(8; ; ff). If Bseline network is used for photoni swithing, it is loking network sine two onnetions my pss through the sme E, whih uses node onflit. Even if Bseline network is used for eletroni swithing, it is still loking network sine two onnetions my try to pss through the sme input (resp. output) link, whih uses input (resp. output) link onflit. Fig. shows three onnetion pths,, nd. nd hve link nd node onflits in stges nd. nd hve node onflit in stge. Although Bseline network is loking, nonloking network n e uilt y extending it in three wys: horizontl ontention of extr stges to the k of Bseline network, vertil stking of multiple opies of Bseline network, nd the omintion of oth horizontl ontention nd vertil stking [8,,, ]. In the generl pproh, network is onstruted y ontenting the mirror imge of the first x(< n) stges of BL(N ) to the k of BL(N ) to otin BL(N;x), then vertilly mking p opies of BL(N;x), the extended BL(N ) (eh opy is lled plne), nd finlly onneting the inputs (resp. outputs) in the first (resp. lst) stge to N p splitters (resp. p ominers). peifilly, the i-th input (resp. output) of the j-th plne is onneted with the j-th output (resp. input) of the i-th p splitter (resp. p ominer), whih is onneted with the i-th input (resp. output) of this network. We denote network onstruted in this wy y B(N; x; p; ff), where ff is rosstlk ftor: ff =if the network hs no rosstlkfree onstrint (i.e. the network hs only link onflit-free onstrint) nd ff = if the network hs rosstlk-free onstrint (i.e. the network hs node onflit-free onstrint). If x =, B(N; x; p; ff) eomes B(N; p; ff). In this pper, we fous on designing fst routing lgorithm for lss of NB B(N; p; ff) networks. Fig. shows the struture of B(8; ; ff). For B(N; p; ff), let I e set of N inputs, I ; ;I N, nd O e set of N outputs, O ; ;O N. Let g = i,» i» n. he k-th modulo-g input group omprises inputs I (k )g ;I (k )g+ ; ;I kg, nd the k- th modulo-g output group omprises outputs O (k )g ;O (k )g+ ; ;O kg, where» k» N=g. Let ß : I! O e n I=O mpping tht indites onnetions from I to O. If there is onnetion from I i to O j, then set ß(i) = j nd ß (j) = i; otherwise set ß(i) =. Ifj = ß(i) for ny I i, then set ß (j) =. We sy tht n input (resp. output, link, E) is tive if it is on onnetion pth, nd idle otherwise. An I/O mpping from I to O is one-to-one if eh I i is mpped to t most one O j nd ß(i) = ß(j) for ny i = j. In this pper, ll I/O mppings re one-to-one nd ll onnetions elong to one-to-one I/O mpping.. Designing rllel with Routing Algorithms A trivil lower ound on the time for routing K (» K» N ) onnetions sequentilly in B(N;p;ff) is Ω(K lg N ). his lower ound is otined y Lemm nd ssuming tht for ny onnetion it tkes O() time to orretly guess whih plne to use without using onflit. Clerly, when the numer of onnetion requests is lrge, the routing time omplexity is greter thn O(N ). rllel proessing tehniques should e used to meet the stringent timing requirement []. In [], we proposed prllel routing lgorithm with time omplexity (lg N p N ) for B(N; p; ff) on ompletely onneted multiproessor system. In this pper, we try to improve the time omplexity to O(p N ) using grph oloring pproh. We hoose to present our prllel lgorithm on ompletely onneted multiproessor system. A ompletely onneted multiproessor system of size N onsists of N proessing elements (Es), E i,» i» N, onneted in suh wy tht there is onnetion etween every pir

4 of Es. We ssume tht eh E n ommunite with t most one E during ommunition step. he time omplexity of n lgorithm on suh multiproessor system is mesured in terms of the totl numer of prllel omputtion nd ommunition steps required y the lgorithm. uh multiproessor system is y no mens to e prtil, ut used s generl strt model to derive prllel lgorithms. Effiient lgorithms on more relisti models, suh s the lss of hyperui prllel omputers, whose rhiteturl omplexity is the sme s tht of single plne of B(N;p;ff), n e esily otined from our lgorithms. Grph Model. I/O Mpping Grphs Given ny I/O mpping with K onnetions for B(N;p;ff), we onstrut grph G(N;K;g), nmed I/O mpping grph, s follows. he vertex set onsists of two prts, V = fv;v ; ;vn=g g nd V = fv ;v ; ;vn=g g. Eh modulo-g input (resp. output) group is represented y vertex in V (resp. V ). here is n edge etween vertex i=g in V nd vertex j=g in V if j = ß(i). hus, G(N;K;g) is iprtite grph with N=g verties in eh of V nd V nd K edges, where t most g edges re inident t ny vertex. Clerly, the degree of G(N;K;g), the mximum numer of edges inident t vertex, is no lrger thn g. ine there my e more thn one onnetion from modulo-g input group to the sme modulo-g output group, G(N;K;g) my hve prllel edges etween two verties nd it my e multigrph. Fig. () shows n I/O mpping with inputs, of whih re tive. Fig. () shows the I/O mpping grph G(; ; 8) of Fig. (), where V (resp. V )ofg(; ; 8) hs verties nd eh vertex in V (resp. V ) inludes 8 inputs (resp. outputs) elonging to the sme modulo-8 input (resp. output) group.. Grph Coloring nd Nonlokingness We sy tht two onnetions shre modulo-g input (resp. output) group if their soures (resp. destintions) re in the sme modulo-g input (resp. output) group. Lemm For ny onnetion set C of B(N; ;ff), if no two onnetions in C shre ny modulo-g input (resp. output) group, then the onnetion pths for C stisfy the following onditions: (i) they re node onflit-free in the first (resp. lst) lg g stges; (ii) they re input link onflit-free in the first lg g + (resp. lst lg g) stges nd output link onflit-free in the first lg g (resp. lst lg g +) stges. i (i ) V V ( ) ( ) Figure : () An I/O mpping ß; () An I/O mpping grph G(; ; 8). It is esy to verify tht Lemm is true ording to the topology of BL(N ) (refer to [] for forml proof). We sy tht set C of I/O onnetions is fesile for B(N;p; ) (resp. B(N;p; )) if they n e routed without ny link (resp. node) onflit. sing the ove lemm, the following lim n e esily derived from the results of []. Lemm Given onnetion set C of B(N; ;ff), if ny two onnetions in C do not shre ny modulo- n+ff input group nd lso do not shre ny modulo- n+ff output group, then C is fesile for B(N; ;ff). By Lemm, if we ssign the onnetions of B(N; p; ff) with soures (resp. destintions) pssing through the sme modulo-g input (resp. output) group to different plnes, then we n route onnetions in B(N; p; ff) without onflit. hus, in order to route onflit-free onnetions in B(N; p; ff), we only need to determine whih plne to e used for eh onnetion. o hieve this gol, we deompose set of onnetions into disjoint susets, nd route eh suset in one plne of B(N; p; ff) so tht eh suset is fesile for its ssigned plne. By onstruting n I/O mpping grph G(N;K;g) with g = n+ff, we n redue the prolem of routing K onnetions in B(N; p; ff) to the following strong edge grph oloring prolem: trong Edge Coloring rolem (EC prolem): Given n I/O mpping grph G(N;K;g) with K (< K) olored

5 edges, olor K K unolored edges with set of olors suh tht no two edges with the sme olor re inident t the sme vertex of G(N;K;g) without hnging the olors of the K olored edges. If we n find strong edge-oloring of G(N;K;g) using t most different olors, we ll this oloring strong -edge oloring of G(N;K;g). [ ] [ ] [ ] [ ] [ ] [ ] [ ] ( ) ( ) Figure : () A edge-oloring () A strong edge-oloring. If we onsider the olored (resp. unolored) edges in G(N;K;g) s the existing (resp. new) onnetions in B(N;p;ff), solution to the EC prolem is plne ssignment for routing in n NB network sine rerouting existing onnetions is prohiited. In Fig., we show simple exmple. here re three edges leled,,, respetively. Edges nd hve lredy een olored using olors nd, respetively. An edge oloring solution is given in (), nd n EC solution is given in (). Note tht, in (), n dditionl olor is needed for edge euse the olors of existing olored edges nd nnot e hnged. Routing in tritly Nonloking Networks. trit Nonlokingness he following lemm n e esily derived from the results of []. Lemm If ρ n ( p + ff) ; for even n n+ ( + ff) ; for odd n then B(N;p;ff) is stritly nonloking. For n NB network, we n route new onnetions (s long s these onnetions form n I/O mpping from idle inputs to idle outputs) without disturing the existing ones; however, this routing prolem is hrder thn tht in n RNB network when we need to route the new onnetions simultneously. Bsed on the disussions in etion, we know tht the routing prolem for n NB B(N;p;ff) n e solved y finding strong edgeoloring of the I/O mpping grph G(N;K;g). We onsider sulss of NB networks, B(N;p Λ ;ff) with p Λ = n+ff +. By Lemm, we know tht [ ] [ ] [ ] B(N;p Λ ;ff) is n NB network. ine eh plne of B(N;p Λ ;ff) is Bseline network, the routing of onnetions in ny plne n e done y self-routing. hus, the prolem of routing onnetions in B(N;p Λ ;ff) is redued to finding plne for eh new onnetion so tht ll onnetions, inluding existing ones, re onflit-free. Lemm Any multigrph G hs strong ( )-edge oloring, where is the degree of G. By Lemms nd (proved in []), this n e done y finding strong (g )-edge oloring for G(N;K;g) of B(N;p Λ ;ff) with K existing onnetions nd K K new onnetions, where g = n+ff = pλ +. In the next susetion, we present prllel lgorithm to find strong (g )-edge oloring of G(N;K;g) using strong edge oloring pproh.. Algorithm for trong (g )-Edge Coloring of G(N; K; g) Let G(N;K K ;g) denote the grph otined from G(N;K;g) y removing the K olored edges. ine G(N;K;g) is iprtite multigrph, G(N;K K ;g) is lso iprtite multigrph. he edges etween the sme two verties re lled prllel edges. We sy olor is free t vertex v if none of edges djent to v hs olor. If olor is free t two ends of edge e, then is free for e. One edge e is onflit with nother edge f if e nd f re djent to eh other nd they hve the sme olor. Let E i;j = fe i;j = vi v j je i;j G(N;K K ;g)g. hus, E i;j ontins ll unolored prllel edges etween nodes vi nd v j. Clerly, eh unolored edge is in nd only in one of suh E i;j s. Our lgorithm onsists of g itertions. In eh itertion, we try to olor set of non-prllel unolored edges using one of olors in set of g olors, f; ; ; g g, so tht no two edges with the sme olor djent to the sme vertex. hen for eh edge e with olor g, we reolor it y free olor in f; ; ; g g. In order to find set of non-prllel unolored edges in eh itertion, we need preproessing step. For eh vertex v i, we n sort ll prllel edges in E i;j in nonderesing order of is where is re the input lels orresponding to edges. he sorting for eh E i;j n e done in O(lg je i;j j) time using je i;j j Es. hus, the preproessing step n e done in O(lg g) time using N Es sine je i;j j»g nd i;j je i;jj = N. After this preproessing, the opertion of finding unolored non-prllel edges n e done in O() time in eh itertion. he outline of the lgorithm is listed in Algorithm. he orretness of this lgorithm n e derived from the following fts.

6 Algorithm A trong Edge Coloring of n I/O Mpping Grph G(N;K;g) for l =to g do for ll i; j f; ; ;N=gg do i;j := (i + j + l) modg; if there is n unolored edge in E i;j nd olor i;j is free t oth vi nd v j then ssign olor i;j to this edge; updte free olors t vi nd v j nd remove the olored edge from E i;j ; end if end for end for for ll edges with olor g do olor these edges with one of free olors in f; ; ; g g; end for (i) In itertion i, one unolored edge, if ny, in eh E i;j is seleted. Ft (i) is ssured y preproessing step. (ii) In itertion i, if two edges, one in E i;j nd one in E p;q, re ssigned the sme olor, i.e. i;j = p;q, then i = p nd j = q. Ft (ii) n e proved y ontrdition s follows. Assume there re two pirs of (i; j) nd (i; q) with j = q nd i;j = i;q. (For the se tht there re two pirs of (i; j) nd (p; j) with i = p nd i;j = p;j, the proof is similr). hus, there is l so tht i +j +l i +q + l mod g. hen j q =g x where x is nonnegtive iteger. ine j; q f; ; ;N=gg nd g = n+ff, we hve j q < g. hus, x = nd j = q, whih ontrdits the ssumption. (iii) For unolored edges in G(N;K K ;g), ll g possile olors re tried. Ft (iii) is oviously true from the lgorithm. (iv) After g itertions, none of djent edges is ssigned the sme olor g. By Ft (iii), it is ler for ny non-prllel edges. By preproessing, we know tht ny two prllel edges re olored in different itertions. ine there re totl g itertions nd in eh itertion we ssign different olors to the edges in E i;j, ft (iv) is true. (v) he edges with the sme olor g n e reolored onurrently using the olors in f; ; ; g g so tht none of djent edges is ssigned the sme olor. By Lemm, for ny edge e with olor g, we know suh free olor in f; ; ; g g is ville. ine ll edges with originl olor g re not djent to eh other y ft (ii), the reoloring will not result in ny onflit olors. Now, we show tht this lgorithm n e implemented in O(g) time using ompletely onneted multiproessor system of N Es. By the previous disussion, we know tht the preproessing step tkes O(lg g) time using ompletely onneted multiproessor system of N Es. hen we show tht, eh of the g itertions tkes O() time. We ssoite g-it inry rry C v [ :::g ] with eh vertex v of G(N;K;g) suh tht C v [] =if nd only if olor is free t vertex v, nd ssign g= Es to v. hen the opertions of finding out if given olor is free t v nd updting C v [] n e rried out in O() time. Finlly, the reoloring of the edges with olor g n e done in O(lg g) sine the degree of G(N;K;g) is g. In summry, we hve the following result: heorem For ny I/O mpping grph G(N;K;g) with K (<K) olored edges, strong (g )-edge oloring n e found in O(g) time using ompletely onneted multiproessor system of N Es.. erformne Anlysis ine O(g) =O(p N) in G(N;K;g), y Lemm nd heorem, we summrize the overll performne of our routing lgorithm for NB network B(N;p Λ ;ff) y the following theorem. heorem For n NB network B(N; p; ff) with p p Λ = n+ff +, onnetions from ny K K idle inputs to ny K K idle outputs, with K existing onnetions, n e orretly routed in O(p N ) time using ompletely onneted multiproessor system of N Es. By Lemm, we n derive the minimum numer of plnes, p min, in B(N;p;ff) s follows: If there is no rosstlk-free onstrint (i.e., ff = ), then p min = n for even n nd p min = n+ for odd n. If there is rosstlk-free onstrint (i.e., ff = ), then p min = n + for even n nd p min = n+ for odd n. Compred with B(N;p min ;ff), the hrdwre redundny p red = p Λ p min of B(N;p Λ ;ff) is: p red p=if ff =nd n is odd or ff =nd n is even, p red = N= p if ff =nd n is even, nd p red = N= if ff =nd n is odd. he hrdwre ost of B(N;p Λ ;ff), in terms of the numer of Es, is higher thn tht of B(N;p min ;ff) in hlf of the ses, ut oth hve the sme hrdwre omplexity of (N : lg N ). he time for routing O(N ) onnetions, however, is improved from Ω(N lg N ) to suliner O(p N ) in the worst se. Conlusion he mjor ontriution of this pper is the design nd nlysis of prllel routing lgorithms for lss of stritly nonloking swithing networks, B(N; p; ff). Although the ssumed prllel mhine model is ompletely onneted multiproessor system of N Es, the

7 proposed lgorithms n e trnsformed to lgorithms for more relisti prllel omputing models. Let (N ) e the time for sorting N elements on prllel mhine M with N proessors, then our lgorithms n e implemented with slow-down ftor (N ) on M. It is known tht sorting N numers on the lss of hyperui networks tkes O(lg N lg lg N ) time [, ]. his lss of networks inlude hyperue, ue-onneted-yles, utterfly networks, seline networks, reverse seline networks, Omeg networks, flip networks, de Bruijin grphs, shuffle-exhnge networks, nyn networks, delt networks, idelt networks, k-ry Butterflies, nd Benes networks []. Our lgorithms n route onnetions in B(N;p;ff) with slow-down ftor O(lg N lg lg N ) on ll these relisti prllel mhine models, though some hve topologies tht re quite different from others, whose struturl omplexity is no lrger thn one plne of B(N;p;ff). Compred with sequentil lgorithms, we onsider tht our lgorithms on relisti prllel omputers provide signifint speedup, mking them potentilly vlid nd useful for lrge swithes. he pproh of pplying edge-oloring tehniques to investigte the pity nd routility of RNB swithing networks hs een widely used (refer to [,,, ]). We extended this pproh to NB networks y defining strong edge-oloring. For lss of NB nyn-sed swithing networks we proposed unified mthemtil formultion, nmely EC prolems, for designing prllel routing lgorithms using this pproh. Our lgorithm n find the solutions for EC prolem in suliner time. Finding fster prllel lgorithms for the EC prolem, however, remins to e very hllenging. Referenes [] D.. Agrwl, Grph heoretil Anlysis nd Design of Multistge Interonnetion Networks, IEEE rnstions on Computers, vol. C-, no., pp. -8, July 8. [] V.E. Benes, Mthemtil heory of Conneting Networks nd elephone rffi, Ademi ress, New York,. [] J. Crpinelli nd A. Y. Oru, Applitions of Mthing nd Edge-Coloring Algorithms to Routing in Clos Networks, Networks, vol., pp. -, ep.. [] R. Cypher nd G. lxton, Deterministi orting in Nerly Logrithmi ime on the Hyperue nd Relted Computers, roeedings of the nd Annul ACM ymposium on heory of Computing, pp. -,. [] H. Hinton, A Non-Bloking Optil Interonnetion Network sing Diretionl Couplers, roeedings of IEEE Glol eleommunitions Conferene, pp , Nov. 8. [] D.K. Hunter,.J. Legg, nd I. Andonovi, Arhiteture for Lrge Dilted Optil DM withing Networks, IEE roeedings on Optoeletronis, vol., no., pp. -, Ot.. [] F.K. Hwng, he Mthemtil heory of Nonloking withing Networks, World ientifi, 8. [8] C.. Le, Multi-logN Networks nd heir Applitions in High-peed Eletroni nd hotoni withing ystems, IEEE rnstions on Communitions, vol. 8, no., pp. -, Ot.. [] C.. Le nd D.J. hyy, rdeoff of Horizontl Deomposition Versus Vertil tking in Rerrngele Nonloking Networks, IEEE rnstions on Communitions, pp. 8-, vol., no., June. [] F.. Leighton, Introdution to rllel Algorithms nd Arhitetures: Arrys rees Hyperues, Morgn Kufmnn ulishers,. [] G.F. Lev, N. ippenger nd L.G. Vlint, A Fst rllel Algorithm for Routing in ermuttion Networks, IEEE rnstions on Computers, vol., pp. -, Fe. 8. [] E. Lu nd. Q. Zheng, rllel Routing Algorithms for Nonloking Eletroni nd hotoni Multistge withing Networks, Workshop on Advnes in rllel nd Distriuted Computing Models, April,. [] E.Lu, Mei Yng, Bing Yng nd. Q. Zheng, A Clss of elf-routing tritly Nonloking hotoni withing Networks, roeedings of IEEE Glol Communitions Conferene, Nov.-De.,. [] G. Mier, A. ttvin, nd. G. Colomo, Control of Non-filterle Crosstlk in Optil-Cross-Connet Bnyn Arhitetures, roeedings of IEEE Glol eleommunitions Conferene GLOBECOM, vol., pp. 8-, Nov.-De.. [] G. Mier nd A. ttvin, Design of hotoni Rerrngele Networks with Zero First-Order withing- Element-Crosstlk, IEEE rnstions on Communitions, vol., no., pp. 8-, Jul.. [] N. Nssimi nd. hni, rllel Algorithms to et p the Benes ermuttion Network, IEEE rnstions on Computers, vol., no., pp. 8-, Fe. 8. [] R. Rmswmi nd K. ivrjn, Optil Networks: A rtil erspetive, seond edition, Morgn Kufmnn,. [8] G.H. ong nd M. Goodmn, Asymmetrilly-Dilted Cross-Connet withes for Low-Crosstlk WDM Optil Networks, roeedings of IEEE 8th Annul Meeting Conferene on Lsers nd Eletro-Optis oiety Annul Meeting, vol., pp. -, Ot.. [] M. Vez nd C.. Le, Wide-ense Nonloking Bnyn-ype withing ystems Bsed on Diretionl Couplers, IEEE Journl on eleted Ares in Communitions, vol., no., pp. -, ep. 8. [] M. Vez nd C.. Le, tritly Nonloking Diretionl- Coupler-Bsed withing Networks under Crosstlk Constrint, IEEE rnstions on Communitions, vol. 8, no., pp. -, Fe.. [] C.L. Wu nd.y. Feng, On Clss of Multistge Interonnetion Networks, IEEE rnstions on Computers, vol. C-, no. 8, pp. -, Aug. 8.