Constructing Multiple Light Multicast Trees in WDM Optical Networks

Transcription

1 Constrcting Mltiple Light Mlticast Trees in WDM Optical Networks Weifa Liang Department of Compter Science Astralian National University Canberra ACT 0200 Astralia Abstract Mlticast roting a fndamental problem in any telecommnication network We address the mlticast se in a WDM optical network withot wavelength conversion capability To realize a mlticast reqest in sch network it reqired to establh mltiple light mlticast trees MLMT) somes de to the fact that the reqested network resorces are being occpied by other roting traffic and therefore it impossible to establh a single light mlticast tree for the reqest In th paper we first propose a mltiple light mlticast tree model We then show the MLMT NP-hard and deve an approximation algorithm for it which takes and delivers an approximation soltion within s of the optimal where and are the nmbers of nodes links and wavelengths in the network the set of destination nodes and constant We finally extend the problem frther with the end-to-end path delay bonded by an integer and we call th latter problem as the mltiple delay-constrained light mlticast tree problem MDCLMT) for which we propose two approximation algorithms with the performance ratios of One of the proposed algorithms takes and the path delay strictly met; and another takes and the path delay no more than where constant with Introdction Wavelength-divion-mltiplexing WDM) emerging as a key technology for next-generation networks by providing nprecedented bandwidth in a medim that free from indctive and capacitive loadings ths relaxing the limitations imposed on the bandwidth-dtance prodct In WDM optical networks the fibre bandwidth partitioned into mltiple data channels which may be transmitted simltaneosly on different wavelengths In th paper we consider circit-switched WDM networks which can be frther classified as either single-hop or mlti-hop networks [0 ] In single-hop networks each message transmitted from the sorce to the destination withot any optical-to-electronic conversion within the network refore single-hop commnication can be realized by sing a single wavelength to establh a connection bt sch connections may in general be difficlt or impossible to find in the presence of other network traffic Alternatively all-optical wavelength converters may be sed to convert from one wavelength to another wavelength within the network However sch converters are likely to prohibitively expensive for most applications in the foreseeable ftre [5] In th paper we therefore only consider the single-hop WDM optical network Mlticast a fndamental problem in telecommnication networks which ares in a wide variety of applications sch as video conferencingentertainment dtribtion teleclassrooms dtribted data processing etc [2] In WDM networks the mlticast problem also referred to one-tomany roting and wavelength assignment problem RWA) which aims at finding a set of links and wavelengths on these links to establh the connection from the sorce to the destination nodes Lots of effort for mlticast in mlti-hop networks has been taken in past years cost of sing the network resorces in sch networks measred as follows A wavelength traversing a link incrs a cost and the wavelength conversion at a node also incrs a cost Ths the mlticast problem to find a mlticast tree rooted at the sorce and spanning the destination nodes in sch that the tree cost minimized [6] where the set of destination nodes When the size of the set of destination nodes the mlticast problem becomes the optimal semilightpath problem [2] for which Chlamtac et al [2] presented an algorithm for it where the nmber of wavelengths and the nmber of nodes in the network Liang and Shen [7] later provided 7 9 an ; improved algorithm which takes and can be implemented in the dtribted environment effi- Proceedings of the 7th International Symposim on Parallel Architectres Algorithms and Networks ISPAN 0) /0 $ IEEE

2 ciently where the nmber of links in the network For Liang and Shen [6] proposed an approximation algorithm for finding a mlticast tree with minimizing the tree cost Sahin and Azizolg [] considered the mlticast problem nder varios fanot polices and Malli et al [9] dealt with th problem nder a sparse splitting model Sahasrabddhe and Mkherjee [3] formlated the problem as a mixed-integer linear programming problem Znati et al [7] dealt with the problem by decopling the delay from the cost model and presented several hertic algorithms for finding a mlticast tree nder the constraint of end-toend delay from the sorce to the destination nodes In addition there have been several other stdies for constrcting constrained mlticast trees in the WDM optical networks For example Libeskind-Hadas [8] proposed a mlti-path roting model in which the mlticast problem to find a set of paths from the sorce to the destination nodes sch that each path contains a sbset of destination nodes the nodes in the set of destination nodes are inclded by these paths and the cost sm of these paths minimized Zhang et al [6] considered the mlticast problem by focsing on the limited splitting power of optical switches and provided for hertic algorithms for the problem De to crrently it prohibitively expensive to employ the wavelength conversion switches in optical networks in th paper we consider a circit-switched single-hop WDM optical network We extend the mlti-path roting concept [8] frther by proposing a mltiple light mlticast tree model on which we deal with the mlticast problem Specifically given a sorce and a set of destination nodes if the wavelength conversion at nodes not allowed then it impossible to realize a mlticast reqest throgh the constrction of a single light mlticast tree rooted at the sorce inclding all the nodes in de to lack of the available resorces that are being sed by other roting traffics Instead to realize a mlticast reqest it necessitates to constrct several light mlticast trees that cover the nodes in herealight mlticast tree referred to a tree in which every link has the same wavelength To minimize the network resorce consmption for a mlticast session the objective to minimize the cost smmation of all the light mlticast trees We ths refer to the mlticast problem on th cost model as the mltiple light mlticast tree problem MLMT for short) When the set of destination nodes incldes all other nodes except the sorce the problem becomes a broadcast problem which called the light broadcast tree problem If the end-to-end delay of a roting path in each of the trees from the root sorce) to a destination node bonded by an integer then the problem referred to the mltiple delay-constrained light mlticast tree problem MDCLMT for short) Under or model we first show the mlicast problem MLMT) NP-Complete even when the set of destination nodes We then provide an approximation algorithm for it which takes and delivers a soltion within s of the optimal where constant We finally deal with the MDCLMT for which two approximation algorithms with performance ratios of are proposed which trade-offs between the rnning and the accracy of the path delay One of the proposed algorithms takes and the path delay strictly met; and another takes which independent of and the path delay no more than where constant with rest of the paper organized as follows In Section 2 notations are introdced and the problems are defined In Section 3 the MLMT shown to be NP-Complete even when the destination set an approximation algorithm for it proposed and the rnning as well the performance ratio of the algorithm analyzed In Section the MDCLMT considered and two approximation algorithms for it with performance ratios of presented 2 Preliminaries are network model: optical network modeled by a directed graph where a set of nodes vertices) representing switches a set of directed links edges) representing the optical fibers and a set of wavelengths in and Let Associated with each link a wavelength set ) given and for each a non-negative weight assigned which the cost of traversing sing wavelength Th cost reflects the commnication bandwidth consmption on Somes the roting congestion factor on the link also incorporated in the cost by assigning different weights to different wavelengths In some cases it also assmed that an integral delay associated with each link Problem formlation: A mlticast reqest an ordered pair where the sorce of the mlticast session and ) the set of destination nodes Assme that mlticast reqests are made and released dynamically To realize a mlticast commnication reqest the ideal case to establh a single light mlticast tree However de to the fact that some specific network resorce eg a specific wavelength occpying by other commnication traffic at th moment and it impossible to establh sch a tree for the crrent mlticast reqest Instead it reqired to establh a collection of light mlticast trees to realize the reqest Specifically we deal with the following two mlticast problems Mltiple Light Mlticast Tree Problem MLMT) to constrct a collection of mlticast trees 2 sch that i) tree a light mlticast tree rooted at 2 Proceedings of the 7th International Symposim on Parallel Architectres Algorithms and Networks ISPAN 0) /0 $ IEEE

3 inclding a sbset of nodes in and wavelength on each link of ; ii) each node in as a leaf node inclded in one of the trees at most once bt it may serve as a relay node in other trees 2 nion of the nodes 2 in the trees satfies that ; iii) minimized where Clearly In some real- QoS applications not only the cost of mlticast trees the main optimization objective bt also the response the path delay) from the sorce to each destination node in a mlticast session paramont For th latter case another extra metric - an integral delay assigned in addition to the cost metric for each link Ths given a sorce an integral delay and a set of destination nodes the Mltiple Delay-Constrained Light Mlticast Tree Problem MDCLMT) 2 to find a collection of light mlticast trees sch that i) tree a light mlticast tree rooted at covering a sbset of and the wavelength on each link of ; ii) each node as a leaf node inclded in one of the trees at most once bt it may appear as a relay node in other trees and the path 2 delay in the tree from to bonded by ; iii) minimized where 3 Approximation Algorithm for the MLMT 3 Light Broadcast Tree Problem NP- Hard When the destination set the MLMT becomes the light broadcast tree problem In contrast to that the broadcast tree problem the minimm spanning tree) in conventional networks polynomially solvable the light broadcast tree problem in WDM optical networks on the mltiple tree model NP-Complete We state it in the following theorem withot proof de to the space limit major techniqe to redce the 3-CNF SAT problem to it orem light broadcast tree problem in WDM networks on the mltiple light mlticast tree model NP- Complete 32 Approximation Algorithm We have shown that the light broadcast tree problem NP hard so the MLMT We ths focs on deving an approximation algorithm for it basic idea to transform the problem to a directed Steiner tree problem in an axiliary directed graph As a reslt a feasible soltion for th latter problem will give a feasible soltion for the concerned problem For each dtinct wavelength a sbgraph " $ of containing indced by the links that the wavelength on them and every other node in the sbgraph reachable from Let " $ be the sbgraph and let node be inclded in " $ then in " $ relabeled as weight of a corresponding link in " $ if and reachable from and then n the axiliary graph constrcted as follows and where a virtal node representing the sorce represents the node the set of wavelengths on link in and the set of wavelengths on link in weight assignment of links in defined as follows For each link its weight in for each link its weight and for each the weight of link axiliary graph has the following properties i) For each node there at least a directed path in from to sing the links in a specific " $ links and in ii) Each " $ a sbgraph of nodes in between " $ and " are not reachable from each other if We then have the following theorem orem 2 mlticast tree in rooted at inclding the nodes in corresponds to an optimal soltion for the MLMT in Proof Let ; be an optimal soltion for the MLMT in consting of the collection of light mlticast trees and ; be the cost sm of the light mlticast trees in ; Following the constrction of it easy to show that there a corresponding mlticast tree in in which each tree in ; corresponds to a sbtree rooted at of and a mlticast tree rooted at inclding the nodes in Let be the weighted sm of the links in n ; Assme that 9 A; a directed Steiner tree in rooted at inclding the nodes in Following the definition of directed Steiner trees the weighted sm 9 A; of the links in 9 A; 9 A; ; theorem follows D It well known that the directed Steiner tree problem NP hard [3] In the following we aim to find an approximate directed Steiner tree in rooted at inclding the nodes in which in trn will give an approximation soltion for the MLMT detailed algorithm Fig ) described as follows orem 3 Given a directed network with a given sorce and a node set there an approximation algorithm for the MLMT with a performance ratio of 3 Proceedings of the 7th International Symposim on Parallel Architectres Algorithms and Networks ISPAN 0) /0 $ IEEE

4 ; " 5 Algorithm App Mlticast Trees Constrct an axiliary graph ; 2 Find an approximate directed Steiner 9 A9 in rooted at inclding the nodes in ; 3 Tree corresponds to a light mlticast tree in with and a node in inclded in the mlticast tree if node inclded in 3 Figre Algorithm for the MLMT complexity of the proposed algorithm where constant with ) 3 Proof Following the constrction of it contains at most 3 nodes and links Ths Step takes obviosly Step 2 to find an approximate directed Steiner 9 A9 in rooted at inclding the nodes in whichtakes sing the approximation algorithm in [] where a constant with ) 3 Step 3 takes becase the nmber of links 9 A9 bonded by cost 9 A9 9 A9 7@ 9 A; ; byorem 2 where and are constants and ) 3 D Approximation Algorithm for the MD- CLMT In th section we start with a rotine for finding a delayconstrained shortest path in a graph bonded by an end-toend integral delay 5 which will be sed in the proposed algorithms We then provide approximation algorithms for the MDCLMT delay-constrained shortest path problem delay-constrained shortest path problem to find a shortest path in from to nder the constraint that the end-to-end delay of the path bonded by 5 If the path delay 5 not an integer the problem NP-hard [3]; otherwe it polynomial solvable and one sch a dynamic programming algorithm de to Kompella et al [5] described as follows Let be the cost of a shortest path from to with the delay exactly and be the cost of the shortest path in with a bonded delay 5 If there are mltiple shortest paths with the same cost then the one with the least delay chosen Ths and are formlated as follows and n the soltion for the problem to find the vale of Use Dijkstra s algorithm it takes to find a soltion for th dynamic programming problem Let be a delay-constrained shortest path in a sbgraph " of from to with the end-toend delay 5 the cost of where the sbgraph " indced by the links in containing wavelength and a link in It easy to see that the dynamic programming algorithm for the delay-constrained shortest path problem a psedo-polynomial algorithm becase its rnning depends on the delay 5 However if the end-to-end delay not met strictly there a strongly polynomial approximation algorithm for th problem which stated as follows Lemma Given a WDM optical network with a pair of nodes and and an integral delay 5 assme that each link has been assigned a set )of wavelengths each wavelength traversing incrs acost and the delay for on an integer and 5 re an approximation algorithm for finding a delay-constrained shortest path from to with the relaxation of the bonded delay which finds a shortest path with the bonded delay 3 5 where constant ) 3 Proof approximation algorithm [] for the delayconstrained shortest path problem ses a scaling techniqe which delivers an optimal soltion in terms of the path cost bt the path delay bonded by 3 5 where constant ) 3 D 2 Approximation algorithm with strictly delay We here propose an approximation algorithm for the MDCLMT sch that the end-to-end delay of the roting paths mst be met strictly idea behind the proposed algorithm as follows Let the approximation soltion const of light mlticast trees 2 where wavelength on each link of 3 3 For each node inclded in if 2 for all Let be the sbset of 2 whose nodes are inclded in then 3 and 3 Ths we have the following important lemma Lemma 2 Let be the cost of an optimal soltion for the MDCLMT n Proof Assme that the optimal soltion for the MDCLMT consts of light mlticast trees where wavelength on every link of 3 and 3 n If Proceedings of the 7th International Symposim on Parallel Architectres Algorithms and Networks ISPAN 0) /0 $ IEEE

5 ths have D inclded in in the optimal soltion then While inclded in " $ we We ths have an approximation algorithm See Fig 2) for the problem Algorithm App Constrained Mlticast Trees for to do 2 for each node do 3 find a delay-constrained shortest path in " $ from to ; endfor; endfor; partition the nodes in into djoint sbsets sch that a node also in if 5 for to do 6 find a delay-constrained mlticast tree in " $ rooted at inclding nodes in endfor; 7 A feasible soltion for the problem 2 since 2 2 Figre 2 Algorithm for the MDCLMT orem Given a directed network with a given sorce and a node set and an integral delay assme that each link has been assigned a set ) of wavelengths each wavelength traversing incrs a cost and the delay ) for traversing an integer re an approximation algorithm for the MDCLMT with a performance ratio of complexity of the proposed algorithm Proof Let be the cost of the optimal soltion for the MDCLMT n following Lemma 2 for each As a reslt the cost of Ths the total cost of the 2 light mlticast trees 2 rnning from Step to Step 9 where 9 the for finding a delay-constrained shortest path from to in a graph with nodes and links with the end-to-end delay Step 6 takes 9 [5] and the soltion delivered s of the optimal So the rnning for Steps 5 and algorithm therefore takes D 3 Approximation algorithm withot strictly delay We now provide a trly polynomial approximation algorithm for the MDCLMT if the end-to-end delay of a path not met strictly basic idea similar to the one in the preceding section except the following modifications At Step 2 an approximation algorithm instead of an exact algorithm for finding a delay-constrained shortest path will be sed set of destination nodes then partitioned into djoint sbsets and For each sbset at Step 6 a new approach called sticking approach instead of the algorithm de to Kompella et al [5] will be employed to find an approximate light mlticast tree in " $ rooted at inclding all nodes in In the following we present the sticking approach Let 9 9 be a shortest path in " from to with a bonded delay no more than fonded by applying the Goel et al [] algorithm on " A node inclded in set if and only if for any other Let and be the seqence of nodes in sorted in increasing order ie sticking approach to constrct a light mlticast tree inclding the nodes in by adding these paths one by one ntil all nodes in are inclded Assme that a light mlticast tree rooted at inclding nodes in has already been constrcted Now consider adding the path 9 9 to to form a new light mlticast tree inclding the nodes in sch that the path delay in from to each node within Initially 9 9 and Consider adding a path 9 9 to to constrct for Let be a segment of path starting at node and finhing at node We traverse from the sorce Let be the th node in appeared in both and except the sorce Ths path can be partitioned into segments if nodes appearin both and Clearly Let be the tree after having considered the th segment of to where the reslting tree after having considered the previos segments of Denote by the niqe path in from to If stick the entire path to to form a new tree Otherwe constrcted by taking into accont the segments of one by one which described as follows Let be the first node in which also in Following the definition we have node-djoint paths in when the segment the two paths one n there are two from to added to Among consting of tree edges 5 Proceedings of the 7th International Symposim on Parallel Architectres Algorithms and Networks ISPAN 0) /0 $ IEEE

6 only; and another of Let be the delay of a path If then we jst ignore of for any frther consideration and the path delay in from to no greater than becase Otherwe we delete the tree edge from where the parent of in n the path delay in from to each leaf node which inclded in the sbtree rooted at no more than becase Assme that the first segments of have been considered and the reslting tree Now we consider adding the th segment of to in order to form a new tree Let be the lowest common ancestor of and in tree n there are two node-djoint paths in from to which can frther be classified into three cases see Fig 3) s w= r i) r w= path consts of tree edges only path consts of a segment of P i Figre 3 Add segment r s ii) r r iii) w=lca s to r r ) r i) In th case the two paths are consting of the tree edges of and If we jst ignore for frther consideration becase the path delay in from to no more than ; otherwe constrcted by adding to and re- moving the tree edge from at the same Now we see that the path delay in from to each which inclded in a sbtree of rooted at no more than becase the path ii) In th case will not be considered and becase the path delay in from to de to the fact that an ancestor of in iii) and re are two node-djoint paths in from to One consting of the tree edges only and another the joint of two paths consting of tree edges only and at node If then and will not be considered becase the delay in from to Otherwe constrcted by adding the path to and removing the edge from it at the same It not difficlt to show that the path delay in from to each node that inclded in the sbtree of rooted at no greater than de to the fact that and When the last segment assming the th segment) of has been considered the reslting tree a directed tree rooted at inclding all nodes in ) and the path delay in from to no more than for all and We therefore have the following lemma Lemma 3 Let be the light mlticast tree in " $ rooted at inclding the nodes in constrcted by the sticking approach and be the cost of Let be the optimal cost of the MDCLMT n Ittakes to constrct Proof Following the similar argment as Lemma 2 it easy to see that 9 9 with the path delay for all Let be the tree after adding the path 9 9 " $ to tree Fol- lowing the constrction of from it obvios that 9 9 Ths We now analyze the complexity of the sticking approach Assme that the path consts of segments To obtain the tree thereare stages At stage the th 6 Proceedings of the 7th International Symposim on Parallel Architectres Algorithms and Networks ISPAN 0) /0 $ IEEE

7 segment of considered and processing th segment 2 involves finding a lowest common ancestor in tree for a pair of nodes 2and determining whether or not to add th segment to whichtakes Ths it takes to constrct When in the worst case it takes to constrct from reare paths in " $ are to be considered As a reslt the constrction of a light mlticast tree with wavelength takes D We ths have the following theorem orem 5 Given a directed network with a given sorce and a set of destination nodes and an integral delay assme that each link of has been assigned a set ) of wavelengths each wavelength traversing incrs a cost and the delay of traversing an integer ) re an approximation algorithm for the MDCLMT with a performance ratio of and the path delay of complexity of the proposed algorithm where constant Proof Following Lemma 2 3 the soltion delivered 2 by the proposed algorithm rnning of the proposed algorithm analyzed as follows To partition the nodes in into djoint sbsets apply the Goel et al [] algorithm for each indced graph " for any Note that the Goel et al algorithm [] takes on each graph " for each Ths the sed for the partition n apply the sticking approach to each sbset to constrct a light mlticast tree As reslts the light mlticast trees are constrcted which form a soltion 2 for the MDCLMT sed for th step following Lemma 3 theorem then follows D 5 Acknowledgment It acknowledged that the work by Weifa Liang was spported by a research grant DP093 fnded by Astralian Research Concil nder its Dcovery Schemes References [] M Charikar C Chekri T-Y Cheng Z Dai A Goel S Gha and M Li Approximation algorithms for directed Steiner problems J Algorithms Vol [2] I Chlamtac A Faragó and T Zhang Lightpath wavelength) roting in large WDM networks IEEE Jornal on Selected Areas in CommnicationsVol [3] M R Garey and D S Johnson Compters and Intractability: A gide to the ory of NP- Completeness WH Freeman and Company NY 979 [] A Goel KG Ramakrhnan D Kataria and D Logothet Efficient comptation of delay-sensitive rotes from one sorce to all destinations Proc of INFOCOM [5] V P Kompella J C Pasqale and G C Polyzos Mlticast roting for mldia commnication IEEE/ACM Trans Networking Vol pp [6] W Liang and H Shen Mlticasting and broadcasting in large WDM networks Proc of 2th Int l Conf on Parallel Processing Symp IEEE Compter Society Press [7] W Liang and X Shen Improved lightpath wavelength) roting in large WDM networks IEEE Trans Commn Vol [8] R Libeskind-Hadas Efficient collective commnication in WDM networks Proc of IC N [9] R Malli X Zhang and C Qiao Benefit of mlticasting in large WDM networks Proc of SPIE Conf on All-Optical Networks Vol [0] B Mkherjee WDM-based local lightwave networks Part I: single-hop systems IEEE Network Vol6 3) [] B Mkherjee WDM-based local lightwave networks Part I: mlti-hop systems IEEE Network Vol6 ) [2] R K Pankaj Wavelength reqirements for mlticasting in all-optical networks IEEE/ACM Trans Networking Vol [3] L H Sahasrabdde and B Mkherjee Light-trees: optical mlticasting for improved performance in wavelength-roted networks IEEE Commnication Magazine Vol [] G Sahin and M Azizogl Mlticast roting and wavelength assignment in wide-area networks Proc of SPIE Conf on All-Optical Networks Vol [5] J Yates M Rmsewicz and J Lacey Wavelength converters in dynamically reconfigrable WDM networks IEEE Commnication Srveys 999 [6] X Zhang J Wei and C Qiao Constrained mlticast roting in WDM networks with sparse light splitting Proc of IEEE INFOCOM [7] T F Znati T Alrabiah and R Melhem Low-cost delay-bonded point-to-point mltipoint commnication to spport mlticasting over WDM networks Compter Networks Vol Proceedings of the 7th International Symposim on Parallel Architectres Algorithms and Networks ISPAN 0) /0 $ IEEE