Algorithms for Intermediate Waveband Switching in Optical WDM Mesh Networks

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1 Algorithm for Intermeite Wven Swithing in Optil WDM Meh Network Ajy Toiml 1 n Byrv Rmmurthy 1 OIT-Mi-Atlnti Croro, Univerity of Mryln-College Prk College Prk MD 070 U.S.A jyt@mxgigpop.net Deprtment of Computer Siene n Engineering Univerity of Nerk-Linoln Linoln NE U.S.A. yrv@e.unl.eu Atrt Wven withing i tehnique tht llow multiple wvelength to e withe together ingle unit. Wven withing tehnique h een proven to reue the with ize onierly in lrge network. Aggregtion of wvelength into wven n i-ggregtion of wven k to wvelength n e one t en-noe or intermeitenoe. Mot of the reerh on wven withing h oniere oure-to-en withing. In intermeite wven withing, ggregtion n/or i-ggregtion n e one t n intermeite noe. In the ontext of intermeite wven withing, the prolem of grooming wvelength uh tht the numer of port ve i mximize i non-trivil. Reent reerh whih oniere intermeite wven withing foue on routing n wvelength ignment prolem uh tht the port ving i mximize. In thi work we fou on the prolem of intermeite wven withing oniering tti trffi n uming tht routing n wvelength ignment i known. We efine two intermeite wven grooming polie, intermeite-to-etintion wven withing (ITD- WBS) n oth-en-to-intermeite wven withing (BETI-WBS), epening on where long the pth ggregtion/iggregtion of wven i one. We preent greey lgorithm to ompute wven for the two intermeite wven grooming polie n nlyze their omputtionl omplexitie. I. INTRODUCTION Wven withing i tehnique tht llow the grouping of multiple wvelength into ingle wven n withing them ingle unit. Wven withing reue the numer of port in the withe. Multiple lightpth hving ommon egment n e groupe n withe ingle unit. Depening upon where (long the pth) the ggregtion of wvelength into wven n the iggregtion of wven into wvelength i one, everl vrition re poile. The oure-to-en wven withing (ETE-WBS) i the implet form of wven withing where multiple onnetion etween oure n etintion noe re groupe into wven. The intermeite wven withing n e further lifie into everl type epening on ggregtion n/or i-ggregtion t intermeite noe. In intermeite-to-etintion wven withing (ITD- WBS) the grooming of wvelength into wven i one t n intermeite noe n iggregtion into iniviul wvelength i one t the etintion noe. The intermeite-to-etintion wven withing n e pplie to multiple onnetion hving the me etintion. In oure-to-intermeite wven withing (STI-WBS) the grooming i one t the oure noe n the iggregtion i one t n intermeite noe. The eon wven grooming poliy i othen-to-intermeite (BETI) wven withing where oth STI n ITD wven withing re llowe t the me time. Wven grnulrity i efine the numer of wvelength tht n e groupe or ggregte into wven. An optil with i i to upport uniform wven withing if the grnulrity of ll the wven i n ritrry ontnt g. In ontrt, for n optil with tht upport non-uniform wven, wven grnulritie vry in et i.e., {g 1,g,...,g k }. Intermeite wven withing w firt tuie in [] where it i hown tht it reue the with ize y ftor of two or more. The tuy of wven withing in pper [] howe tht it further reue the numer of port in the withing noe y t let n orer of mgnitue. In pper [], the uthor tuy the prolem of intermeite wvelength n wven withing in network tht upport ynmi trffi requet.the uthor of pper [] ree the prolem of wven withing fouing motly on the routing n wvelength ignment prolem for mximizing the numer of port ve uing wven withing /07/$ IEEE. 1

2 They preent heuriti, it performne nlyi n n integer liner progrmming formultion to the prolem. In thi work we re the prolem of wven grooming uming tht the routing n wvelength ignment i lrey known. The ret of the pper i orgnize follow. In Setion II we provie the prolem efinition n preent our nottion. In Setion III we preent the lgorithm for olving the uniform/nonuniform wven withing whih llow only intermeite-to-etintion grooming. In Setion IV we preent the lgorithm for olving the uniform/nonuniform wven withing with oth intermeiteto-etintion n oure-to-intermeite grooming. In Setion V we preent the onluion. II. PROBLEM STATEMENT AND NOTATIONS A. Prolem Sttement In thi work we onier the following prolem. We re given routing n wvelength ignment of et of tti onnetion requet in grph G = (V,E). The prolem i to groom wvelength into wven uh tht the numer of port ve i mximize. Two vrition of the prolem exit epening upon the type of wven withing upporte in the network i.e., uniform or non-uniform wven withing. The prolem lo h vrition epening upon the wven ggregtion n i-ggregtion llowe in the network. In thi work we onier ITD-WBS n STI-WBS. We formlize eh of the vrition of the wven withing prolem in thi etion. Intermeite-to-etintion uniform wven withing: (ITD-UWBS) Intne: Given grph G =(V,E) repreenting the topology of the network, et of κ route emn P = {p 1,p,...,p m } ll hving the me etintion, numer of lightpth i orreponing to eh emn i i.e., C = { 1,,..., m } n n integer g repreenting wven grnulrity. Tk: Compute n ggregtion of wvelength into wven uing ITD wven withing poliy uh tht totl numer of port ve in the entire network i mximize. In the ITD-WBS prolem, n intne i efine to onit only of onnetion tht hve the me etintion. Thi i eue in network tht only llow intermeite-to-etintion wven withing (ITD-WBS) poliy, only onnetion tht re etine to the me etintion n e groupe into wven. Therefore the omplete et of pth n e prtitione into u-et e on their etintion noe n the wven grooming only our mong pth within p 1 : ; 1 = p : ; = () p 1 : ; 1 = p 1 : ; 1 = p : ; = Fig. 1. Illutrtion of trnformtion of pitte grph into tree uing Initilize lgorithm. (, ) (, ) (, 18) (, ) (, ) (, 1) (, ) (0, ) (1, ) Fig.. The etintion-roote pitte tree ompute y the Algorithm 1 for n intne of the ITD-UWBS prolem where the tuple (h u,r u) orreponing to eh noe repreent the noe height n reiul pity. prtition n not ro prtition. B. Nottion A wven B of grnulrity g i enote y (Q,,, g) where Q = {(p 1, 1 ), (p, ),...(p m, m )} i et of tuple (p i, i ), where p i i route-emn etween pir of noe n i i the numer if unit (lightpth) of the route-emn. Noe n re ggregting n iggregting noe. The um of i of ll the route-emn in the wven B of grnulrity g i t mot the wven grnulrity g i.e., 1 i m i g. The numer of unit of emn of the route-emn p, whih i yet to e route, fter ggregting unit into wven B i lle the reiul emn r where = + r. The numer of wvelength port ue y wven of length L n grnulrity g i g +(L +1). The numer of port require for routing g wvelength level onnetion of length L i g(l +1). Therefore the numer of port ve y wven route of length L n grnulrity g i (L + 1)(g 1) g. III. ALGORITHM FOR ITD-UWBS PROBLEM In thi etion we preent lgorithm for the ITD- UWBS prolem. ()

3 A. Initiliztion In thi etion we preent n lgorithm to trnform n intne of the ITD-UWBS prolem into etintion-roote pitte tree T. The input to the lgorithm onit of et of κ pth P = {p 1,p,...,p m } in the grph G ll hving the me etintion n their orreponing pth pitie C = { 1,,..., m }. The lgorithm for initiliztion i outline y Algorithm 1. The lgorithm ompute u-grph T uing pth in P. Let the ege of the u-grph T e irete from etintion noe to orreponing oure noe lef noe. The grph T i now trnforme into tree y eleting yle in T n moifying the pth oringly. Let u illutrte the trnformtion into tree uing n exmple intne hown in Figure 1. Eh noe u in the tree T i oite with vrile R u tht repreent the um of the reiul pitie r j of ll the pth p j tht hve ommon egment from noe u to etintion noe. The Algorithm 1 ompute height h i of eh noe in the tree T where the root noe h height 0. The Algorithm 1 now initilize the reiul pity of eh lef noe j of the tree T with the pity of the pth p i where j i the oure noe of the pth p i. The lgorithm then ompute the reiul pity of eh intermeite noe in the tree the um of the reiul pitie of it hil noe. Fig. how the tree ompute n initilize y the Algorithm 1 for n intne of the ITD- UWBS prolem. Algorithm 1 1: Input: (G, P,C) : Output: Detintion-roote pitte tree T : ompute grph T uing pth in the et P : trnform T into tree y eleting yle in T n moifying pth oringly : ompute the height h i for eh noe i in the tree T : initilize the reiul pity R j of the lef noe j to n i where j the oure noe of the pth p i 7: ompute the reiul pity R i of eh intermeite noe i the um of the reiul pitie of it hil noe B. Algorithm to olve the ITD-UWBS Prolem The lgorithm for olving the ITD-UWBS prolem i greey lgorithm n i outline y Algorithm. The input to the lgorithm onit of et of κ pth P = {p 1,p,...,p m } ll hving the me etintion n their orreponing pth pitie C = { 1,,..., m } n n integer g repreenting wven grnulrity. The lgorithm output et of wven B = {B 1,B,...}. The Algorithm ontrut etintion-roote pitte tree T uing the Algorithm 1 n input P n C. The Algorithm iterte with ereing tree height. In eh itertion with height i the lgorithm ompute ll poile wven of length i. To ompute wven, it elet noe u of height i n reiul grnulrity t let g. Now, if ((S u = (i+1)(g 1) g) > 0) i.e., the numer of port ve y forming wven B =(Q, u,, g) from noe u to noe of grnulrity g i non-negtive then we form wven B n it to the et of wven B. We upte reiul pitie of ll the noe long eh of the pth p Q. Let u illutrte the working of the Algorithm on the intne hown in Fig.. Fig. () how the wven B 1 =(Q 1,,,) where Q 1 = {(p 1, )}, ompute y the Algorithm in the firt itertion. The reiul pitie of ll the noe long the wven B 1 re upte oringly n re hown in Fig. (). In the eon itertion, the Algorithm ompute the wven B =(Q,,,) where Q = {(p, )}. The Algorithm iterte until there exit no noe in the tree with reiul grnulrity greter thn n non-negtive S vlue. The Algorithm ompute five wven hown in Fig.. Algorithm 1: Input:(G, P,C,g) : Output: Wven et B : run Algorithm on input (G, P,C) : for i = h; i ; i o : for eh u where h u = i, n R u g o : if ((S u =(i + 1)(g 1) g) > 0) then 7: form wven B =(Q, u,, g) from noe u to root noe n to B 8: upte the reiul pitie R u of ll the noe long the pth inlue in the wven : en if 10: en for 11: en for Let u iu the omplexity of the Algorithm. The time tken to ompute the tree T y Algorithm 1 uming tht the grph ompute y the Algorithm 1 in Step oe not hve yle i O(nm) where n i the numer of noe n m i the numer of pth in the et P. Conier the following imple moifition to Algorithm where inte of forming ingle wven in Step 7 from noe u to noe, we form R u /g numer of wven. With thi moifition eh noe in the tree i exmine t mot one. After forming the wven, the lgorithm nee

4 () (e) () () () (f) Fig.. The illutrtion of omputtion of uniform wven of grnulrity y Algorithm on the intne hown in Fig.. The numer jent to eh noe repreent the urrent reiul pity of the noe. 8 1 h f e t i j k l p g t 1 t o t n m t 1 t t t t h i p g f e j k l o n m Fig.. The etintion-roote n oure-roote tree ompute y the Initilize Algorithm for n intne of the BETI prolem. to upte the reiul pitie of the noe long the pth in the wven. Therefore for eh noe exmine we nee to upte reiul pitie of t mot O(m log n) noe. Therefore the omplexity of the Algorithm i O(mn log n). In the next etion we preent lgorithm for wven withing where oth intermeite-to-etintion n oure-to-intermeite grooming i llowe. Algorithm The Initiliztion Algorithm for the BETI prolem. 1: Input: (G, P, C) : ompute grph T t n T uing pth in the et P : uper etintion noe n uper oure noe to tree T t n T repetively : ege from noe to ll the etintion noe in tree T t : ege from noe to ll the oure noe in tree T : trnform T t n T into tree y eleting yle in T t n t n moifying pth oringly 7: ompute the height h i for eh noe i in the tree T 8: initilize the reiul pity R j of the lef noe j of the tree T t to n i where j the oure noe of the pth p i : initilize the reiul pity R j of the lef noe t j of the tree T to n i where t j the oure noe of the pth p i 10: ompute the reiul pity R i of eh intermeite noe of the tree T t n T the um of the reiul pitie of it hil noe A. Algorithm to olve the BETI Prolem IV. ALGORITHM FOR BETI WAVEBAND SWITCHING In thi etion we preent n lgorithm for vrition of intermeite wven withing where ggregtion n i-ggregtion n e one t oth the en noe i.e., oure n etintion noe. We onier uniform wven. Uing the Algorithm, we trnform n intne of the BETI prolem into etintion-roote pitte tree T t n oure-roote pitte tree T. All the ege in the grph T t (orreponingly, T ) re irete from etintion to oure noe (orreponingly, from oure to etintion noe) of the pth in P. The outline of the BETI Algorithm i given y Algorithm. The input to the lgorithm i n intne of the BETI Prolem n it output et of wven B = {B 1,B,...}. Let u iu the working of BETI Algorithm in etil. The lgorithm ontrut etintionroote n oure-roote pitte tree T t n T repetively uing the Initiliztion lgorithm with input P n C. Leth n h e the height of the tree T n T repetively. Let h e the mximum of the height of oth the tree. The Algorithm iterte with ereing height n ompute wven in eh itertion. In n itertion with height i, the lgorithm elet noe u of

5 Algorithm The BETI Algorithm for omputing the wven. 1: Input:(G, P, C) : Output: Wven et B : run Algorithm on input (G, P, C) to ompute tree T t n T : let h t n h e the height of the tree : let h e the mximum of the height h n h t : for i = h; i ; i o 7: for eh u in T t n T where h u = i in the orreponing tree, n R u g o 8: if ((S u =(i + 1)(g 1) g) > 0) then : form wven B =(Q, u,, g) from noe u to root noe / orreponing to tree T t /T n to B 10: upte the reiul pitie R u of ll the noe long the pth in inlue in the wven in oth the tree T t n T 11: en if 1: en for 1: en for height i from either of the tree with reiul pity t let g. Note tht the me noe my hve ifferent height in oth the tree. We form the wven if n only if ((S u = (i + 1)(g 1) g) > 0) hol i.e., numer of port ve i non-negtive. Now, if we elet noe u in the tree T then we form wven B =(Q, u,, g) from noe u to noe of grnulrity g. But if we elet noe u in tree T then we form wven B = (Q,, u, g) from noe to noe u of grnulrity g. We the wven to the et of wven B. We upte reiul pitie of ll the noe long eh of the pth p Q in oth the tree T n T. The lgorithm ontinue to form wven of length i until no noe u exit with R u g n ((S u = (i + 1)(g 1) g) > 0) fter whih it ontinue with the next itertion erementing the height. etintion-to-intermeite n oure-to-intermeite grooming (oure-to-intermeite grooming) i llowe. The wven withing where intermeiteto-intermeite grooming i llowe i hr prolem. In future we pln to tuy intermeite-to-intermeite wven grooming n it performne in omprion to oure-to-intermeite grooming. REFERENCES [1] L. W. Chen, P. Senguomlert n E. Moino, Uniform v. Non-uniform Bn Swithing in WDM Network, IEEE BroNet - Optil Networking Sympoium, Boton, MA, Otoer 00. [] X. Co, V. Ann n C. Qio, Multi-Lyer veru Single- Lyer Optil Cro-onnet Arhiteture for Wven Swithing, IEEE INFOCOM 0, Hong Kong, April 00 [] L. Noirie, M. Vigoureux, n E. Dotro, Impt of intermeite grooming on the imenioning of multi-grnulrity optil network, in Proeeing - OFC 001, p. TuG, Anheim, CA, Mrh 001. [] R. Lingmplli n P. Venglm, Effet of wvelength n wven grooming on ll-optil network with ingle lyer photoni withing, in Proeeing - OFC 00, p. TuG, Anheim, Cliforni, Mrh 00. [] M. Li n B. Rmmurthy, Integrte Intermeite Wven n Wvelength Swithing for Optil WDM Meh Network, IEEE INFOCOM 0, Brelon, Spin, April 00. [] X. Co, V. Ann, Y. Xiong n C. Qio, Performne Evlution of Wvelength Bn Swithing in Multi-fier All- Optil Network, IEEE INFOCOM 0, Sn Frnio, CA, Mrh 00. V. CONCLUSIONS In thi pper we tuie the prolem of intermeite wven withing uming tht routing n wvelength ignment re lrey known. Three type of grooming poliie exit when oniering intermeite wven withing, nmely, etintion-to-intermeite, oure-to-intermeite n intermeite-to-intermeite grooming. We preente lgorithm for uniform n non-uniform wven withing where either etintion-tointermeite or oure-to-intermeite grooming i llowe. We lo preente lgorithm for uniform n non-uniform wven withing where oth