A Multistart Randomized Greedy Algorithm for Traffic Grooming on Mesh Logical Topologies

Transcription

1 Tehnial Report Università di Trento, Diembre A Multistart Randomized Greedy Algorithm for Traffi Grooming on Mesh Logial Topologies Mauro Brunato Roberto Battiti Deember 11, 001 Abstrat In this paper we onsider a logial topology design problem on DWDM optial networks where the traffi is quantized at sub-wavelength resolution and the ritial fator to determine the fitness of a solution is the number of lightpaths required, that is proportional to the number of hardware modules to handle the traffi. The problem has been shown to be NP-hard. We review some of the previous work in the field, examine a number of regular but unsatisfatory solutions and desribe a greedy-based iterated heuristi belonging to the GRASP family to minimize the number of lightpaths. At the end we present some experimental results that ompare our GRASP heuristi with other greedy-based methods and with regular topologies. Keywords Logial topology design, Traffi grooming, Optimization algorithms, GRASP 1 Introdution Signifiant advanes in optial ommuniations tehnology are leading to the reation of optial Wide Area Networks (WANs) with large link apaity. With the introdution of the Dense Wavelength Division Multiplexing (DWDM) tehnology, the whole bandwidth of every fiber is potentially available for transmission, as many independent signals an be aommodated on the same physial link by using densely paked adjaent wavelengths. The link between two nodes in a WAN is omposed of a thik bundle of fibers, usually in the hundreds; ommerial systems allowing a few hundreds of DWDM wavelength arriers per fiber have already beome available, and eah arrier has a data transmission speed of various Gigabits per seond. On the other hand, every single wavelength needs fairly expensive hardware equipment to be exploited (a tunable laser, a photodetetor, fast eletronis). Università di Trento Dipartimento di Matematia, via Sommarive 14 I Pantè di Povo (TN) ITALY brunato battiti@siene.unitn.it

2 b a 5 a b Photodiode Laser 4 a b a b 1 Node Eletroni Loal add/drop data router 3 a b Loal network / omputer Figure 1: A network node However, many low-bandwidth appliations do not require a whole wavelength arrier; to allow a better traffi optimization link traffi should be quantized below the wavelength level by using Time Division Multiplexing (TDM). Paking many traffi units into less wavelengths is an effetive way to redue the number of hardware omponents, hene to ut off the overall ost. As a onsequene, tehnologial onerns shift from improving the amount of bandwidth to the redution of hardware ost through areful resoure optimization. Another advantage of the DWDM tehnology is that it allows a limited amount of stati routing without eletroni onversion. Upon arriving into a node, the signal travelling through a fiber is split into its omposing wavelengths (see Fig. 1). Every wavelength an be treated in a different way: some of them are onverted into eletroni format for loal treatment or for traditional routing, some other wavelengths are relayed through an internal fiber diretly to an output fiber for retransmission. It is possible to set up a so-alled lightpath, i.e. a diret onnetion between nonadjaent nodes ating as a logial one-hop link, where all intermediate nodes are transparent to it. Transpareny of intermediate nodes is ruial in order to abstrat the atual data flow pattern, alled the logial or virtual topology and depending on lightpaths, from the physial fiber onnetion layer [6]. Traffi handling tehniques inluding multiplexing and onversions of signals are olletively known as grooming tehniques [1]. In partiular, in the following setions the term grooming will refer to the at of merging many low-bandwidth traffi onnetions into a single wavelength by way of TDM. Most studies about grooming tehniques are foused onto the urrent optial infrastruture based on SONET/SDH rings. Many problems are addressed in this ontext. Among them, the above ited ost redution problem has been introdued in [9, 8, 10] and a relevant researh effort is still being spent on the subjet with the proposal and

3 evaluation of heuristi methods [14, 18, 3, 5,, 19, 4] under various assumptions about tehnologial devie availability, ring flexibility and traffi requirements. Grooming methods for general topologies have been disussed in [11], where the problem is treated as an Integer Linear Program and solved as a multiommodity flow problem for limited dimensions of the instanes. The fous of this work is to study the effiay of greedy and iterated greedy algorithms in a simplified ontext. Therefore we shall only onsider logial topology design, with no onern about its implementation on the physial layer. In other words, we assume that mapping a logial topology onto the physial layer is always feasible as in [11]. In a pratial appliation this an be the ase if abundant fibers have been installed, whih is the ase of many wide and metropolitan area networks. In a future extension of this work we will present problem formulations where the atual realization of the logial topology on a given physial topology is also onsidered. The problem of logial topology design originates from the introdution of ATM networks and was reently boosted by introdution of an optial layer in digital paketswithed networks [7, 15] and, later, by the eventual introdution of all-optial systems [13, 1] and by the strong need of a speifi IP-aware optial infrastruture [17]. The proposed optimization tehnique, is based on GRASP (Greedy Randomized Adaptive Searh Proedure). GRASP is a metaheuristi for ombinatorial optimization. GRASP usually is implemented as a multistart proedure, where eah iteration is made up of a onstrution phase, where a randomized greedy solution is onstruted, and a loal searh phase whih starts at the onstruted solution and applies iterative improvement until a loally optimal solution is found [16]. In Setion we state the problem that we to onsider; in Setion 3 we provide a few simple theoretial results suh as lower bounds and values for regular topologies. Next, in Setion 4 we introdue and disuss some optimization heuristis and we analyze some experimental results in Setion 5. Traffi grooming in logial topologies Suppose we have a network with N nodes. We are not onerned with the physial topology of the network, although we assume that it is onneted, i.e. it is always possible to establish a ommuniation path between any two nodes. We are also given a stati traffi pattern in the form of a matrix (T ij ) whose entry T ij ontains the number of traffi units required for ommuniations from node i to node j. A traffi unit is the unit of bandwidth onsidered, and all ommuniation requirements are given as multiples of it. We want to set up a logial topology, i.e. a set of lightpaths between nodes, in order to satisfy all traffi requirements, onsidering that every lightpath an arry at most time-multiplexed traffi units. Given a lightpath P from node i to node j, we define node i as its soure, i = sr(p ), and the node j as its destination, j = dest(p ). Many different lightpaths may have the same soure and destination, to fulfill traffi requirements. A hain of lightpaths from node i to node j is a simple walk in the lightpath graph, i.e. a sequene of lightpaths P = (P 1,..., P n ) suh that: 3

4 a 1 b e d 3 4 Figure : A logial topology over 4 nodes the first lightpath originates from node i: sr(p 1 ) = i, the last lightpath ends into node j: dest(p n ) = j, and two subsequent lightpaths join at the same node: k = 1,..., n 1 dest(p k ) = sr(p k+1 ). A traffi unit from node i to node j an be routed diretly through a lightpath established from i to j, or it an be routed through a onatenation of lightpaths from i to j. For example, if N = 4 we ould design the topology in Fig., where two lightpaths are established through nodes 1 and. Traffi between nodes 1 and 4 may be routed through a onatenation of lightpaths, while traffi from to 4 may be routed in many different ways. Note that we are not onerned with the atual implementation of the lightpaths, i.e. the mapping of the logial topology onto the physial links. For example, the atual onfiguration ould be that shown in Fig. 3; however, we suppose that the physial link apaity is enough to arry any reasonable logial topology and that the intermediate nodes traversed by a lightpath in the physial layer are ompletely transparent to the lightpath. Given nodes i and j, the k-th traffi unit from i to j (k = 1,..., T ij ) shall be routed through a hain of lightpaths P ijk, made of one or more lightpaths, originating from i and ending in j. A routing assignment (P ijk ) i,j=1,...,n;k=1,...,tij is suitable if no lightpath appears more than times. Our objetive is to minimize the number of lightpaths needed by the logial topology: this is equivalent to minimizing the number of ritial eletroni omponents, whih is our main goal. The problem an be stated as follows. LOGICAL GROOMING PROBLEM given a set of N nodes, a traffi pattern (T ij ) and a lightpath apaity, find a suitable logial topology and a 4

5 3 1 a e d b 4 Figure 3: A mapping of the logial topology onto the physial layer routing assignment for eah traffi unit suh that the number of lightpaths is minimum. 3 Exat results In the following setion we derive a lower bound for the number of lightpaths in any problem instane. We also onsider some regular topologies and derive exat results for the number of lightpaths required in these ases. However, none of these solutions is near-optimal, and in Se. 4 we shall introdue heuristis to obtain more satisfatory topologies. 3.1 Lower bound The overall traffi arried by the network is, trivially, N i=1 j=1 N T ij. If every lightpath ould be ompletely filled with traffi units, and no traffi needed to be routed through more than 1 lightpath, then the number of lightpaths needed would be N N i=1 j=1 LB = T ij. This number represents a lower bound for the number of lightpaths. In fat, if less than LB lightpaths were used then at least one traffi unit ould not fit. In ase of uniform traffi T ij = T, i j, this means that the least number of wavelengths is T N(N 1) LB uniform =, 5

6 whih results in a lower bound of 1 lightpaths for T = 3, N = 8 and = Simple ases 3..1 Complete topology One lightpath per ouple of nodes, hop distane is 1 for every onnetion. Number of lightpaths for onnetion (i, j) is Tij L ij =. The total number of lightpaths is N N i=1 j=1 Tij For instane, if T ij = 3 for i j, = 8 and N = 8, the total number of lightpaths is 7 8 = Star topology All lightpaths having a speial node (e.g. node 1) as an endpoint. Edge (i, 1), i =,..., N, arries all traffi outgoing from node i, and thus requires N j=1 T ij lightpaths. Inoming traffi to node i is arried by edge (1, i), requiring N j=1 T ji lightpaths. In the ase of uniform traffi T, edge (i, 1) arries traffi T (N 1), thus requiring L i1 = T (N 1)/ outgoing lightpaths. For the same reason, eah of the N 1 peripheral nodes requires L 1i = L i1 inoming lightpaths, so the overall number of lightpaths is T (N 1) (N 1), whih leads to 4 lightpaths for 8 nodes and traffi equal to 3 units per node pair Unidiretional ring topology In this ase node i, i = 1,..., N 1, is onneted to node i + 1, while node N is onneted to node 1. Let us label the nodes with their soure edge labels; the overall traffi arried by edge i, i = 1,..., N, is C i = h=1 k=i+1. ( i N ) i 1 T hk + T hk + k=1 6 h 1 h=i+ k=i+1 T hk.

7 1. funtion assignpath (node i, node j) returns lightpathchain. find shortest existing lightpath hain LPC 3. from i to j with non-null spare load 4. if not found LPC 5. P new lightpath from i to j 6. LPC = single hop hain (P) 7. return LPC Figure 4: the basi Greedy move So the number of required lightpaths is Ci if j = (i mod N) + 1 L ij = 0 otherwise. In the ase with uniform traffi T ij = T, i j, every node (i, i + 1) is rossed by a path of length 1 (the ommuniations from node i to node i + 1), two paths of length and so on, up to N 1 paths of length N 1. So the total load of an edge is T ( (N 1)) = T N(N 1). Thus the omplete number of lightpaths required is T N(N 1) if j = (i mod N) + 1 L ij = 0 otherwise. There are exatly N edges on the ring, so the total number of lightpaths is T N(N 1) N, aounting for 88 lightpaths if T = 3, N = 8 and = 8. 4 Greedy and Iterated Greedy shemes In the following, every lightpath is assoiated to a load, an integer value reporting the number of traffi unit routed to that lightpath. The spare load of a lightpath is the apaity minus its urrent load, and aounts for the number of traffi units that an be added to the lightpath before saturating it. The spare load of a lightpath hain is the smallest spare load of its omponent lightpaths. The basi omponent of our greedy sheme is funtion assignpath, shown in Fig. 4, whih tries to alloate a hain of lightpaths to a traffi unit going from node i to node j. 7

8 1. funtion setgreedytopology. erase all lightpaths and all routing information 3. for eah ouple of nodes (i, j) in random order 4. for eah traffi unit t {1,..., T[i][j]} 5. LPC assignpath (i, j) 6. route traffi unit t from node i to node j through LPC 7. for eah lightpath P in LPC 8. inrease load in P by 1 Figure 5: the Greedy topology algorithm The funtion exeutes a breadth-first searh from node i through the logial topology graph that is under onstrution until it gets to node j. Only lightpaths having spare apaity of at least 1 are onsidered (lines -3). If no lightpath hain an be determined (line 4), a new lightpath is reated from node i to node j (line 5) and the lightpath hain is instantiated to that single path (line 6). The overall greedy algorithm, whih is shown in Fig. 5, begins by resetting all load and routing information (line ) in order to start the assignment from srath. Then for eah ouple of nodes and eah traffi unit it invokes funtion assignpath (line 5) to retrieve a lightpath hain suitable for routing that traffi unit. Then it sets all routing information needed to use the lightpath hain (line 6) and it inreases the load of every lightpath in it by 1 (lines 7-8). The basi idea of our greedy sheme is that, whenever we need to alloate a route for a traffi unit, we want to use residual traffi apaity if possible (so that no new lightpaths must be reated), otherwise we prefer a new lightpath with many spare units of traffi in the hope that future traffi alloations will use it. Of ourse this heuristi is not optimal. This an be seen experimentally (Se. 5) and theoretially, as the NP-hard multiommodity flow problem an be redued to this problem [11]. In partiular, earlier assignments annot take advantage of lightpaths that have been reated only later (this is the main problem with all greedy shemes), so some lightpaths are eventually underused. To allow older assignments to take advantage of all lightpaths that have been reated, we an repeatedly apply the same greedy sheme without restarting from srath eah time; as we an see in Fig. 6, for eah ouple of nodes we forget everything about their routing assignment while keeping all other lightpaths (lines 4-9), and rebuild it just as we did in the greedy sheme (lines 11-14). This time, however, all other ouples of nodes mintain their urrent assignment, so many lightpaths are available when a suitable route mst be found from i to j. By implementing the greedy sheme, followed by repeated appliation of the GRASPIteration funtion, we obtain an iterated optimization sheme similar to the GRASP tehnique [16]. In our ase the proedure starts in a feasible state obtained by a greedy funtion. This funtion s randomness lies in the order the nodes are onsidered. Also the iteration sheme has some amount of randomness beause of node ordering. 8

9 1. funtion GRASPIteration. for eah ouple of nodes (i, j) in random order 3. for eah traffi unit t {1,..., T[i][j]} 4. LPC routing of traffi unit t from node i to node j 5. for eah lightpath P in LPC 6. derease load of P by 1 7. if load of P is null 8. delete lightpath P 9. forget routing of traffi unit t from node i to node j 10. for eah traffi unit t {1,..., T[i][j]} 11. LPC assignpath (i, j) 1. route traffi unit t from node i to node j through LPC 13. for eah lightpath P in LPC 14. inrease load in P by 1 5 Experimental results Figure 6: the GRASP iteration algorithm We have implemented the greedy and GRASP algorithms as disussed in Se. 4 in C++, and we applied them to various types of traffi. In partiular, we hose to onsider a uniform pattern where traffi between eah pair of nodes is onstantly equal to 3 or 5, and a server-type traffi where all node pairs have a unit traffi requirement unless the soure node is one of three designated servers,in whih ase the required traffi is 10 units. In the first ase the traffi matrix is uniform; in the seond ase, all entries are equal to 1 with the exeption of three rows, equal to 10. In both ases the diagonal is null T uniform =.... T server = As a possible andidate for an alternate startup funtion we also implemented a random assignment funtion suh that for every ouple of nodes and every traffi unit a random lightpath hain was determined. In Figures 7 and 8 we show the omparison among the greedy and random setup shemes, the two orresponding GRASP tehniques (random- and greedy-initiated) and the regular star topology disussed in Se. 3. Eah value shown is the average of five runs; the iterated shemes were run times. The x axis represents the size of the network, while the y axis indiates the number of lightpaths that were established by the algorithm. In this ase, the algorithms had another onstraint: as traffi is symmetri, we also keep a symmetri routing sheme by iterating through node ouples (i, j) suh 9

10 Random Greedy Star Random-fed GRASP Greedy-fed GRASP Lower Bound Number of lightpaths Nodes Figure 7: Uniform traffi, 3 traffi units per node pair, symmetri routing that i < j. In fat, in the ase of symmetri traffi we ould use full-duplex optial links [11]. One run of GRASP iterations on a problem instane with uniform traffi equal to 5 and 0 nodes took 11 seonds on a Pentium III Windows mahine with a GNU ompiler and the Cygwin Unix API emulation libraries. However, we have no evidene that suh a large number of iterations is neessary: in fat, the minimum was always reahed in the first 100 iterations. Further investigation will allow us to identify a suitable number of iterations for eah ase. In both ases (traffi 3 and 5), the greedy algorithm performs better than the random sheme. However, when the GRASP sheme is used the initial assignment is not important and the two urves overlap almost perfetly. In partiular, when the traffi is equal to 5 the greedy-initiated GRASP sheme improves the solution by 10% in the 5-node ase, up to 3% in larger graphs. Even though the star topology allows a lower number of lightpaths when ompared to the greedy and random shemes, we have verified that, when used to initialize the GRASP algorithm, it does not lead to better results. Moreover, it has no randomness in it, so it is not suitable as a starting point for our searh proedure. In the ase of server traffi, the symmetry onstraint was not applied, and the results are shown in Fig. 9. In this ase, too, the greedy sheme outperforms random assignment, but this is ininfluent when the GRASP algorithm is applied. 10

11 10000 Random Greedy Star Random-fed GRASP Greedy-fed GRASP Lower Bound Number of lightpaths Nodes Figure 8: Uniform traffi, 5 traffi units per node pair, symmetri routing Conlusions We have onsidered a logial topology design problem where the lassial routing problem is omplemented with sub-wavelength resolution of traffi. As far as we know, this problem was onsidered for mesh networks only in [11], while the two separate problems have been vastly analyzed in the literature (see Se. 1). We have analyzed a few regular topologies and we have defined an iterated greedy sheme. Experimental analysis has shown a greater improvement over simple greedy, random and regular topologies when applied to regular traffi shemes and simple server traffi models. At present, the omplete network is revised at every step of our GRASP approah. Future work will be dediated to experimenting Loal Searh tehniques to allow a smoother transition of the network logial topology towards the minimum, and possibly the ability to serve dynamially haning traffi patterns while limiting the amount of lightpath reorganization required at every step. Referenes [1] Rihard Barr and Raymond A. Patterson. Grooming teleommuniations networks. Optial Networks Magazine, (3), 001. [] Roberto Battiti and Mauro Brunato. Reative searh for traffi grooming in WDM networks. In S. Palazzo, editor, Proeedings of IWDC001, Leture Notes in Computer Siene. Springer-Verlag, September

12 1000 Random Greedy Random-fed GRASP Greedy-fed GRASP Number of lightpaths Nodes Figure 9: Server traffi [3] Randall Berry and Eytan H. Modiano. Reduing eletroni multiplexing osts in SONET/WDM rings with dynamially hanging traffi. IEEE Journal on seleted areas in ommuniations, 18(10): , Otober 000. [4] Gruia Călinesu and Peng-Jun Wan. Traffi partition in WDM/SONET rings to minimize SONET ADMs. In IWST 001, 001, to appear. [5] Angela L. Chiu and Eytan H. Modiano. Traffi grooming algorithms for reduing eletroni multiplexing osts in WDM ring networks. IEEE Journal of Lightwave Tehnology, 18(1): 1, January 000. [6] I. Chlamta, A. Ganz, and G. Karmi. Lightpath ommuniations: A novel approah to high bandwidth optial WANs. IEEE Transations on Communiations, 40(7): , 199. [7] Aura Ganz and Xudong Wang. Effiient algorithm for virtual topology design in multihop lightwave networks. IEEE/ACM Transations on Networking, (3):17 5, June [8] Gerstel, Lin, and Sasaki. Wavelength assignment in a WDM ring to minimize the ost of embedded SONET rings. In INFOCOM1998, [9] Gerstel and Ramaswami. Cost effetive grooming in wdm rings. In INFO- COM1998, [10] Ornan Gerstel, Rajiv Ramaswami, and Galen H. Sasaki. Cost-effetive traffi grooming in WDM rings. IEEE/ACM Transations on Networking, 8(5): , Otober

13 [11] V. R. Konda and T. Y. Chow. Algorithm for traffi grooming in optial networks to minimize the number of transeivers. In IEEE HPSR001, 001. [1] Rajesh M. Krishnaswami and Kumar N Sivarajan. Design of logial topologies: A linear formulation for wavelength routers with no wavelength hangers. IEEE/ACM Transations on Networking, 9(): , April 001. [13] Emilio Leonardi, Maro Mellia, and Maro Ajmone Marsan. Algortihms for the topology design in WDM all-optial networks. Optial Networks Magazine, 1(1):35 46, January 000. [14] Liwu Liu, Xiangyang Li, Peng-Jun Wan, and Ophir Frieder. Wavelength assignment in WDM rings to minimize SONET ADMs. In INFOCOM 000, pages , 000. [15] Rajiv Ramaswami and Kumar N. Sivarajan. Design of logial topologies for wavelength-routed optial networks. In INFOCOM1995, [16] Mauriio G. C. Resende and Celso C. Ribeiro. Greedy randomized adaptive searh proedures. In F. Glover and G. Kohenberger, editors, State-of-the-Art Handbook in Metaheuristis. Kluwer Aademi Publisher, 001. To appear. [17] D. A. Shupke and D. Sellier. Lightpath onfiguration of transparent and stati WDM networks for IP traffi. In ICC001, 001. [18] Peng-Jun Wan, Gruia Călinesu, Liwu Liu, and Ophir Frieder. Grooming of arbitrary traffi in SONET/WDM BLSRs. IEEE Journal on Seleted Areas in Communiations, 18(10): , Otober 000. [19] Xijun Zhang and Chunming Qiao. An effetive and omprehensive approah for traffi grooming and wavelength assignment in SONET/WDM rings. IEEE/ACM Transations on Networking, 8(5): , Otober