Topological Design Optimization of a Yottabit-Per-Second Lattice Network Jules R

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 9, NOVEMBER 2004 1613 Topological Design Optimization of a Yottabit-Per-Second Lattice Network Jules R. Dégila, Student Member, IEEE, and Brunilde Sansò, Member, IEEE

Abstract—This paper deals with the topological design of a yottabit-per-second @I yotta a 10PRA multidimensional network. The YottaWeb is a recently proposed architecture based upon agile optical cores that provides fully meshed connectivity with direct optical paths between edge nodes that are electronically controlled. In order to arrange the edge nodes around the agile cores (ACs) into a suitable and efﬁcient YottaWeb, one proposal is to create a multidimensional lattice structure of ACs. The problem of designing such a structure is highly combinatorial. In this paper, we present the problem, that we call nodal arrangement problem, and we propose a meta-search procedure based on Tabu and VNS to solve it. The performance of the algorithm is gauged using a set of randomly generated networks with different distribution of trafﬁc.

Index Terms—Agile optical core, lattice structure, meta-search, Fig. 1. Fundamental concept of the AC. next-generation Internet, nodal arrangement problem (NAP), optical internet, PetaWeb, tabu search, variable neighborhood descent, YottaWeb.

I. INTRODUCTION O KEEP UP with the progression of the current In- T ternet, new proposals led by the next-generation Internet (NGI) initiative [1], supported by Defense Advanced Research Projects Agency (DARPA), have been considered. They con- sist of the development of protocols, standards and testbed networks. One such development is an architecture called the PetaWeb [2]–[5]. The PetaWeb scales to a total capacity of several petabits per second (Pb/s), three orders of magnitude higher than the external capacity of the current global Internet. The concept of the PetaWeb is based on the development of an Fig. 2. General optimization model of the YottaWeb. agile optical core (using the wavelength-division multiplexed (WDM) fibers and optical cross-connectors (OXCs) [5]) that sketched in [6] is to use the agile cores (ACs) from the PetaWeb can provide a high-capacity interconnexion between a transport as building blocks for an expanding network. Hence, the network edge nodes. It also allows to overcome the problems involved network design problem implies the need for the with the current Internet by providing a direct high-capacity efficient use of the AC’s enabling technology. One important interconnection between the edge nodes (see Fig. 1). The parameter to reach the required efficiency, is the choice of a PetaWeb architecture is intended to accommodate thousands of good topology which could allow tremendous growth of the such high-capacity edge nodes distributed nationwide. global capacity and a low number of hops between edge nodes. Based on the motivation that a global high-capacity network, Fig. 2 illustrates the optimization challenge involved in the such as the Internet, could contain millions of high-capacity design of the YottaWeb: it consists in deciding which edges edge nodes. Beshai et al. proposed an architecture that could nodes (ENs) should be connected to which AC. Such a solu- reach an external capacity in the order of yottabit-per-section can be embedded into a virtual conceptual frame. onds, called the YottaWeb [6]. The main idea of the YottaWeb One such virtual frame is a lattice structure, proposed by Beshai et al. [6] for the YottaWeb. The structure is built on Manuscript received February 26, 2003; revised January 26, 2004. This work the basis of the number of ACs attached to the same edge nodes was supported in part by a Collaborative Research and Development (CRD) Grant between Nortel Networks and the National Sciences and Engineering Re- and on the number of edge nodes that are connected to the search Council of Canada. same AC. Thus, an important and highly combinatorial problem The authors are with the GERAD and the Department of Electrical Engi- that gives rise in this context is the nodal arrangement problem neering, École Polytechnique de Montréal, Montréal, QC H3C 3A7, Canada (e-mail: [email protected]; [email protected]). (NAP), which defines which edge nodes should be connected Digital Object Identifier 10.1109/JSAC.2004.829642 together in the conceptual lattice structure.

Fig. 3. Structure of PetaWeb.

The object of this paper is to present a powerful metaheuristic for the solution of the NAP. The algorithm was developed after formally exploring the properties of the lattice structure for the YottaWeb, which is presented in this paper for the first time. This paper is organized as follows. In Section II, the basic elements of the YottaWeb Architecture and their relationship with the PetaWeb, the lattice structure, and the NAP are described. Section III is devoted to a literature review on problems that are similar or are related to the NAP. In Section IV,the resolution algorithm, that we have called Lattice Arrangement Meta-Search Procedure (LAMP) for is presented in detail. Numerical results follow in Section V and the conclusions and recommendations Fig. 4. Parallel-planes optical core node in the PetaWeb. for further work are finally presented in Section VI. of clarity, only two ENs are considered. As explained in [5], II. YOTTAWEB TOPOLOGY AND THE an edge node is connected to a core node with one or more NODAL ARRANGEMENT PROBLEM fiber links, each link having several channels. Fig. 4 shows The enabling technology behind the YottaWeb proposal given the parallel space switches composing a core node. Incoming in [6], is the PetaWeb architecture and in particular, its high- optical signals are demultiplexed and associated to different capacity, distributed, edge controlled, optical core. Indeed, the channels. Outgoing channels associated with a given edge node configuration of the AC allows to consider one PetaWeb as a are multiplexed into a fiber link going back to the edge node. subnet of a greater network. The concept allows information delivery at the optical rate, In what follows, we present the basic configuration of the within the network, since the optical cores are bufferless without PetaWeb, and we show how it gives rise to the YottaWeb ar- connections between them. A time-locking mechanism coordi- chitecture and the virtual lattice structure. The section is con- nates the edge nodes, enabling the PetaWeb core to resemble cluded with an explanation of the NAP, its definition, and some one geographically distributed switch. From now on, to explain previous algorithms that attempted at a solution. the YottaWeb in Section II-B, we will use the term “agile core” (AC) to signify the set of high-capacity optical core nodes of a A. PetaWeb PetaWeb. This set of optical core nodes forming a PetaWeb is Fig. 3 (modified form [5]) sketches the structure of the grouped within the dotted circle of Fig. 3 and is symbolized by PetaWeb, which is a composite-star network. The PetaWeb the star at the right which represents an AC. uses a channel-switching core and a distributed control system that dynamically modifies the routing of individual channels as B. Lattice Topology of the YottaWeb the need arises. The access nodes to the core could be electronic The YottaWeb structure defines a way of efficiently con- switches in packet switching or adaptative circuit mode, with necting the edge nodes to the ACs. A schematic view is high capacity of multiple Terabits 10 per second. The portrayed in Fig. 2. In the figure, the edge nodes can be con- optical core nodes, which should be an array of parallel space nected to several ACs. The fundamental question of the design switches, provide fully meshed connectivity with direct optical is to determine which are the best links for the connection; in paths between edge nodes. The connection of the ENs to the other words, to determine which edge node should be connected optical core through the parallel space switches is depicted to which AC. Of course, the term “best” is used in reference to in Fig. 4 (similar to the one proposed in [5]). For the sake the optimization metric chosen by the designer. DÉGILA AND SANSÒ: TOPOLOGICAL DESIGN OPTIMIZATION OF A YOTTABIT-PER-SECOND LATTICE NETWORK 1615

which the edge nodes in the line are connected. From this deﬁ- nition of the lattice topology, we deduce some of its properties in Section II-C.

C. Some Structural Properties of the Lattice Topology The lattice structure, as defined above, corresponds to a regular form. In that form, each pair of lines representing an AC are parallel or perpendicular. In addition, every AC links the same number of nodes. Those assumptions could be different in an irregular form, which is not studied in this article. In addition, let us classify the ACs following the dimensions. An example is given in Fig. 6, where the AC’s 1, 2, 3, and 4 are the ACs of the 2nd dimension as they are parallel to the axis , and the AC’s 5, 6, 7, and 8 are named the ACs of the first dimension. In general, we define by AC of the th dimension, an AC parallel to the axis . The next proposition gives the conditions of the existence of the regular lattice structure. Proposition 1: Let , , and be respectively the number of nodes, the number of ACs, the dimension of the network and the capacity of each AC. For a regular lattice structure representation, the following two equalities are satisfied. Fig. 5. YottaWeb arrangement into lattice structure. 1) . In the case of the YottaWeb, one of the objectives of the de- 2) . sign is that the connections can be carried out as directly as pos- Proof: Both these two equalities come from the necessity sible from the origin to the destination node. Thus, a suitable of proportionality between the components (nodes, ACs, and optimization measure relates to the number of hops between the their capacity) of a regular lattice topology as defined above. origin and the destination. Let be the set Indeed of edge nodes of the Network, and let be 1) Since there are edge nodes per ACs of each dimension, the set of ACs. Let us also assume that there is traffic demand there are on the whole edge nodes in a lattice between the pairs of edge nodes. Then, the problem addressed is structure of dimension two and edge nodes to find the minimum hop path between every origin–destination. in a regular lattice structure of dimension three. Then, in Such a path is constituted only by links from the edge nodes to dimension with , there on the whole the ACs, as no direct link exist between nodes or between ACs. edge nodes. The minimum hop should also be weighted by the demand in 2) Let us consider , the subsets of the order to favor the use of direct paths for pairs of edge nodes ex- ACs of the same dimension, as defined above. The reader changing high-traffic volumes. This will be further classified in could easily see that each class of ACs of dimension Section II-C. covers the whole set of edge nodes, since each edge Bearing in mind that the addressing and the routing must be as node is connected to one and only one AC of each dimen- quick as possible even in an expanding network, one proposal is sion. Then, with nodes per AC, there are on the whole to use the lattice structure to represent the topology. That struc- divided by ACs per dimension. Finally, there are ture is pointed up by the notion of dimension of the network, ACs in the whole which is defined as the number of ACs at which an edge node regular YottaWeb. is connected to at the same time. Fig. 5(a) illustrates a unit-di- The structure of the lattice topology is comparable to the mension YottaWeb, where each of the four edge nodes is con- topology of the generalized hypercube [7] defined for a multi- nected to only one AC, represented by a single line in the lattice processor interconnection architecture. From that point of view, structure. the lattice structure holds many interesting properties from the The difference between the representation by the lattice hypercube, while its advantage remains its simplicity. Indeed, topology and the traditional one is the use of a “tunnel” (line) to the “tunnel” line linking the nodes simplifies the representation, link all the edges within a subnet, instead of arcs linking them. and the manipulation of subnets defined by ACs. Assuming that two lines (ACs) can only intersect at one point In terms of global capacity, the range of the yotta-bits-per- and defining them to be perpendicular to a regular lattice struc- second can be reached by following a simple calculation. Let us ture (each AC is connected to exactly edge nodes, being consider that the nodes are grouped into a lattice structure and the AC’s capacity), one could easily increase the dimension of that the access capacity per node could be equal to 1 terabit/s the YottaWeb to 2 as in Fig. 5(b) and to more than 2 as needed. 10 bit/s . Let us suppose 1000 edge nodes per subnet. In a A more complex example is portrayed in Fig. 6, where there two dimension YottaWeb, the total external capacity within the are two types of lines connecting the edge nodes: the light ones network is computed as 1000 1000 10 bits/s 10 and the shadowed ones. Each type represents a different AC to exa bits/s. This total external capacity reaches 10 1616 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 9, NOVEMBER 2004

Fig. 6. 16-Node YottaWeb arrangement into two dimensions lattice.

zetta bits/s in dimension three and 10 yotta bits/s TABLE I in dimension four. STEPS OF THE “GREEDY” HEURISTIC

D. Nodal Arrangement Problem (NAP) The problem of arranging the nodes into the regular lattice structure, keeping the number of hops as low as possible, is TABLE II highly combinatorial. In fact, the size of possibilities increases STEPS OF THE TETRIS HEURISTIC nonpolynomially for every node added to the network. Formally, the nodal arrangement problem, which we call NAP could be stated as follows. Given: • set of edge nodes; E. Previous Solution Approaches for the NAP • set of ACs; • capacity of each AC (that is equivalent to the maximum Two heuristics have been proposed up to now by the authors number of nodes that can communicate at the same time of the new architectures [3], [5] to solve the NAP. These are through an AC); the “Greedy” and the “Tetris” heuristics based on the notion of • the dimension of the lattice structure; “Friendship” between nodes which represents the “interaction” • , the traffic matrix between or the amount of the traffic between these nodes. Those heuris- the edge nodes. tics are sketched in Tables I and II, where the objective of the Find: the arrangement of the edge nodes into the lattice struc- arrangement is also to minimize the mean hop distance weighted ture to optimize a certain performance metric. by traffic demand. Performance Metric: There are several possible metrics that These two algorithms start by a step by step construction of can be used to optimize the design of the system. For instance, in the lattice structure. The decision of selecting the next node, is the works related to the well-known problem of logical topology based only on “friendship” comparisons between the untreated design in WDM, several authors have used different types of nodes and a few number of the nodes that have been already metrics such as the mean hop distance, the maximal congestion, placed (only one node in the case of the Greedy algorithm and at the queueing delay or the maximum offered load (see [7]–[10]). most nodes in the case of the Tetris algorithm, where is the However, in this paper, we have chosen to concentrate on one dimension of the lattice structure). The bubble sort at the third metric that links the directness of the routing with the impor- point of the Greedy algorithm (Table I) could be time consuming tance of the traffic carried. and not efficient. The authors have concluded that the Tetris The metric, that we consider is the traffic weighted mean hop algorithm is more efficient than the Greedy heuristic, as each node is placed only once. distance between the origin and the destination edge nodes. In Section III, we explore the optical networking literature to This is an interesting metric since minimizing it can lead to other important benefits for the network and imply the optimization of place the problem of nodal arrangement NAP addressed here, others metrics. For instance, as it is mentioned in [9] and [10], in within its context. minimizing the traffic weighted mean hop, one maximizes the one-hop traffic metric. III. LITERATURE REVIEW In what follows, we describe previous algorithms designed The problem most similar to our NAP is what in the litera- specifically for the NAP. ture has been coined as the optimal node assignment problem DÉGILA AND SANSÒ: TOPOLOGICAL DESIGN OPTIMIZATION OF A YOTTABIT-PER-SECOND LATTICE NETWORK 1617

TABLE III STEPS OF THE VND

(ONAP) [10]. In what follows, we emphasize the similarities heuristics: Tabu Search (TS) from [13] and the variable neigh- and the differences between the NAP and the ONAP. borhood descent (VND) from [14]. They both refer to the pro- In the ONAP, the physical topology can be defined as the cedure for searching a solution among the set of feasible so- set of nodes and the set of physical (most of the time, bidi- lutions , following certain rules and using the notion of the rectional) links. The logical topology, also referred as virtual neighborhood structure. The goodness of the solution is evalu- topology or as lightpath topology,oraslightpath network [11] ated by an objective function with being an element of is the higher level in a WDM transport network. The logical the search space . topology consists of the set of nodes and the set of lightpaths. In Sections IV-A–E, we first review some basic properties Each link in the logical topology is a directed lightpath. From of the TS and the VND. Then, we detail the tools of the Meta- these definitions, the ONAP is generally applied to the design Search Algorithm, called lattice arrangement metasearch proce- of the lightpath network, given a physical topology defined by: dure (LAMP) that we propose for the NAP. the number of nodes, the number of fiber links between pairs of nodes and their length, the number of wavelength per fiber, and A. Tabu Search (TS) the number of transmitters at each node. Given the demand ma- Initially proposed by Glover [13], TS is a procedure that ex- trix, a routing algorithm and a regular existing topology ( MSN, plores the solution space beyond local optimality, by avoiding hypercube, shufflenet, multidimensional torus,…) and a perfor- cycling. TS uses a tabu list which is a set of the solutions de- mance metric, the goal of the ONAP is to find different routed termined by historical information from the last iterations of lightpaths to accommodate the demand matrix. As it is shown in the procedure, where is fixed or is a variable that depends on [10], the problem is known to be NP-hard and some heuristics some properties of the problem. At each iteration, given the cur- have been proposed to tackle it. rent solution and its corresponding neighborhood , the We now state the difference between the NAP and the ONAP. procedure moves to the solution in the neighborhood that • The ONAP addresses the design of lightpaths between improves the objective function, while the moves that lead to so- edge nodes, while the NAP considers aggregate communi- lutions on the tabu list are forbidden. Then, the tabu list avoids cation “tunnels” (ACs) between ENs. The number of com- returning to the local optimum from which the procedure has munication links to seek is drastically reduced in the NAP. recently escaped. An additional basic element of the TS is the This is extremely important for a very large high-capaci- aspiration criterion, which determines when a move is admis- tated network such as the future Internet. sible despite being on the tabu list. The termination criterion • Another interesting feature of the NAP is the fact that it is for the TS is often based on a number of iterations without im- defined within a lattice topology. This allows the planner provement. The reader is referred to [13] for additional expla- to use the elegance of such a topology to address simpler nations of the TS and its efficient application to several classical routing schemes. The lattice topology is in fact well used combinatorial optimization problems. We now explain the VND for such properties in different applications [12]. concept. In Section IV, we explained in detail the proposed meta-search procedure for the NAP. B. Variable Neighborhood Descent (VND) Proposed by Hansen and Mladenovic [14], the VND metaheuristic refers to a local improvement with the use of several IV. META-SEARCH PROCEDURE FOR THE neighborhood structures, instead of just one as is usually the NODAL ARRANGEMENT PROBLEM case. The extended neighborhood allows the method to escape local optima with respect to a smaller neighborhood. As previously stated, the difficulty of the YottaWeb lattice Defining by the neighborhood struc- NAP comes from its huge combinatorial nature. An efficient tures such that is the set of the solutions of the th way to improve the quality of the solution is to use a meta- neighborhood of . Table III presents the steps of the VND pro- heuristic, that is, a procedure consisting of a master strategy cedure as it is designed by its author. that guides and modifies others heuristics to produce solutions The loop step (2) of Table III is repeated at most times, beyond those that are normally generated in the quest for local starting at each iteration with a random solution . Then, a local optimality. In this work, we use a combination of two meta- search algorithm is performed to find the best neighbor of 1618 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 9, NOVEMBER 2004

Fig. 8. Illustration of d d -moves.

Fig. 7. Illustration of d-moves. With these notions, the following paragraphs define three cat- egories of neighborhoods for the LAMP. , using neighborhood . If this local optimum is better Neighborhoods Definition: Let be one possible ar- than the incumbent, then set and restart the search with rangement for the lattice structure and let us define as else, the search proceeds with . the th neighborhood of . is a set constituted of ar- Given that VND provides a way to efficiently explore mul- rangements that can be obtained by some -move(s) from ar- tiple neighborhoods, it is particularly well suited for a meta- rangement . heuristic procedure for the NAP. Let us further define as a neighborhood We describe in Sections IV-C–E, the neighborhood struc- of arrangement . An arrangement is said to be an element tures, the evaluation function, and the fitting of TS and VND of the set , when can be obtained by - moves procedures to design our proposed procedure for the NA from arrangement . problem. Finally, let us define by , the AC’s neighborhood of arrangement . An arrangement is said to be an element of C. Neighborhoods for the Lattice Structure the set , when can be obtained by few numbers of ac-moves from arrangement . From now, let us consider as the search space represented by all the possible arrangements (by permutations) of the edge D. Objective Function nodes into the lattice structure. Before explaining the neighborhood used in the search, and using the classification of the ACs As stated before, the objective function that we consider in the given in Section II-C, we define three kind of moves: -moves, optimization process of the NAP is the mean hop value weighted -moves, and -moves. Fig. 7 illustrates the -move by the demand . which performs node swapping within an AC of the th dimen- We explain in this paragraph the evaluation of this function. sion. Thus, edge node and are swapped within the AC To formally formulate that function, let number 1 by a 1-move. Also, a 2-move swaps the edge nodes be the given traffic matrix, and the matrix of and within the AC number 5. the distances (in hops) between the different pairs of location In the same way, a -move represents a swapping of of the arrangement . Using the same notion nodes belonging to an AC of the th dimension with nodes as in [10], we also define the binary-valued decision variables which belong to an AC of the th dimension, respecting the , , . if node is assigned to location order of the nodes. This move is illustrated in Fig. 8 where, in arrangement ; otherwise, . Assuming that only for the sake of clarity, only the concerned ACs are completely the shortest path distance is used between nodes of the lattice portrayed. The other ACs are represented by dashed lines. structure, the distance matrix is symmetric, i.e, . Thus, In the example, the AC1 and the AC7 are swapped. The op- the mean hop value weighted by the traffic is given by eration is called a -move, since AC7 is an AC of the first dimension and AC1 is an AC of the second dimension. The (1) swapping of these two ACs, as it is sketched in the figure, is achieved by performing successively the moves , , , and . In addition, we define an ac-move, depicted in Fig. 8, which can be rearranged as as the swapping of two ACs. This move leaves every node at its place and swaps only the ACs. In the example of Fig. 8, agile core AC1 is permuted with agile core AC2, leaving the nodes , , , , , and at (2) the place they had before the operation. It is important to notice that the ac-move does not influence the number of hops between where represents the total amount of each pair of nodes, since the nodes remain in the position. How- traffic entering the network. The second expression (2) takes ever, the ac-move is intended to vary a measure due to the lo- into account the symmetry of the distance matrix. Such an equa- cation of the ACs, with respect to the edge nodes and could be tion is useful in practice to differentiate the value of the traffic used in a procedure assessing such measure. weighted mean hop. DÉGILA AND SANSÒ: TOPOLOGICAL DESIGN OPTIMIZATION OF A YOTTABIT-PER-SECOND LATTICE NETWORK 1619

TABLE IV STEPS OF THE LAMP

E. LAMP Algorithm TABLE V TEST PROBLEMS CHARACTERISTICS Table IV shows the different steps of the proposed Meta- Search algorithm. This algorithm is based on the VND previously reviewed, in which a TS is incorporated. Globally, TS is performed using a first subset of neighborhoods , while another subset is used to escape local optimum and to diversify the search. All of the neighborhood structures defined in Section IV-C are considered. On the whole, there are neighborhoods structures of type , of type and of type . The initialization, step 1, is generally performed by setting the order in which the neighborhood structures will be examined. The ordering distinguishes two subsets of neighborhood types. The first subset contains the neighborhood of types , with . The neighborhoods of type and those of type are both in the second subset. The ordering of the first set is random. However, the ordering of the neighborhood structures of types and depends on the objective function and any preference between ACs and ENs. The neighborhoods of the second class are used to diversify the search, but most often they are used to perturb, in order to escape the local optima. The subsequent paragraph details the other main points of the algorithm. Initial Solution: Different initial solutions could be chosen at step 1. In this paper, we consider a random arrangement as a good candidate because of its efficiency in time. We also found Moves are accepted when the derivatives of the objective func- it interesting to start the LAMP with a result given by another tion are under a certain very small threshold. arrangement algorithm as the Tetris heuristic, for instance. V. N UMERICAL RESULTS Tabu List: Given that the TS is done considering a neighborhood with which, the moves are done within an AC, the In this section, we present numerical results to assess the per- length of the tabu list does not exceed the capacity of an AC. The formance of the proposed LAMP algorithm. Several scenarios tabu list allows to direct the moves to avoid cycling. Hence, for have been constructed for different test network problems in dif- a specific dimension , each edge node is moved at most ferent dimensions. Comparative node arrangement results have times. been computed using the Random, the Tetris, and the LAMP al- Function Evaluation: The mean hop value weighted by the gorithms. In each case, the computed optimization metric was traffic request is computed by considering the minimum dis- the mean hop traffic. tance between each pair of nodes. This consideration does not In Sections V-A–D, we detail each of the elements involved restrict the scope of the results as the lattice structure offers in the numerical tests such as the test network problems and the many alternate minimum distance paths between every pair of different traffic matrix generators. nodes which are not connected to the same AC. Next, the test results are presented, drawn, and analyzed. The evaluation of the function is done once at the initialization, step 1. Thereafter, only the differentiate of the objec- A. Test Problems tive function is computed. This differentiation is done, by using Table V presents the characteristics of the network problems only the ENs and, or the ACs, which are involved in the moves. chosen for the tests. We used seven problem tests (from to 1620 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 9, NOVEMBER 2004

TABLE VI TWO TYPES OF TRAFFIC PATTERNS FOR AN EIGHT-NODES TEST NETWORK (CENTRALIZED TRAFFICS ARE HIGHLIGHTED)

TABLE VII COMPARATIVE MEAN HOP VALUES AND PERFORMANCE IMPROVEMENTS WITH RVTG TRAFFIC

) of dimension two, seven (from to ) of dimension 1, the generated traffic is volatile, otherwise, a new traffic three, and six problems of dimension four. matrix is generated. Let be the arithmetic mean of the For each problem, Table V gives the numbers of nodes , traffic , , . Then the number of ACs , the dimension of the network, and the capacity of the ACs .

B. Traffic Matrix Generators An illustrative example of a matrix pattern generated in To simulate the behavior of the YottaWeb, which is supposed this fashion is given in Table VI(a), where the trafficis to be as versatile as the Internet, two types of traffic generators given in petabits per second. The outgoing and incoming have been used that are described as follows. ports of each edge node have, respectively, 2 Pb/s as max- • Random volatile traffic generator (RVTG): With this, imal capacity. The total traffic within the network is ap- an exponential distribution traffic pattern is generated ran- proximately 16 Pb/s. domly between nodes. Afterwards, an index of volatility • Random centralized traffic generator (RCTG): Here, is evaluated, to control how different traffics tend to vary about 1% of the edge nodes of the network are considered very often within the network. If such index is higher than to be server edge nodes with high intensity and the rest DÉGILA AND SANSÒ: TOPOLOGICAL DESIGN OPTIMIZATION OF A YOTTABIT-PER-SECOND LATTICE NETWORK 1621

TABLE VIII REPARTITION OF THE TRAFFIC PER THE NUMBER OF HOPS WITH RANDOM TRAFFIC (RVTG)

present low intensity. In Table VI, an example of a central- Table VII is organized in two parts. The computed mean hop ized pattern is given where the highlighted node number values for a random volatile traffic type are presented in the six holds for the server node. first part, while the second part contains the same measure for For each of these traffic generators, a high-pass filter centralized traffic patterns. For each problem and each type of reduces the noises (in significant traffic) in the simulated traffic, we present the computed mean hop value, evaluated in network. number of hops, the consumed CPU time, evaluated in seconds, and the computed , evaluated in percentage. The best com- C. Results puted mean hop value is highlighted for each test problem. We 1) Experimental Conditions: The tests were performed on would mention that the maximum variance of the computed a 2-GHz Pentium 4 computer. The results computed for each mean hop values is 1/1000. algorithm and each test problem are the averages obtained by applying the algorithm hundreds of times to that problem. Each D. Discussion time, a RVTG traffic matrix and a RCTG traffic matrix were generated. Then, Random, Tetris, LAMP (with random arrange- At this point, some major remarks can be made. ment as initial solution), and LAMP (with Tetris arrangement as • In the case of random traffic, the lowest computed mean initial solution) algorithms, were applied to each instance of test hop values occur mostly with the LAMP algorithm with an network , , using each traffic matrix. initial Tetris arrangement. On the other hand, for the case To further assess the performance of the given algorithms, a of centralized traffic, the lowest values are computed most performance improvement (PI) measure has been used for Tetris frequently by the LAMP with a random arrangement as and LAMP algorithms. PI represents the percentage of improve- initial solution. This remark gives some insights of the be- ment of a given algorithm over the random solution. It is com- havior of the different algorithms with respect to the traffic puted as pattern that is next explained when comparing graphically the algorithms performance. (3) • Given that a straightforward lower bound for the problem is one mean hop, one can say that, for the centralized where represents the objective value of the solution given traffic, and in particular for dimension two, optimal or near by the specific algorithm (“algo”), and the objective value optimal solutions are attained. of the solution given by a random arrangement. • Also, since the variances of the computed mean hop 2) Tables Description: Tables VII–IX show detailed results values are insignificant, the use of the average mean of these four arrangement algorithms applied to each instance values to assess the performance of the different algo- of the problem tested. rithms is justified. 1622 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 9, NOVEMBER 2004

TABLE IX REPARTITION OF THE TRAFFIC PER THE NUMBER OF HOPS WITH CENTRALIZED TRAFFIC (RCTG)

Tables VIII and IX show the percentage of the traffic that is delivered in a given number of hops using the four algorithms. One could see the repartition of the traffic given by the LAMP algorithms compared with the repartition given by the Random and the Tetris. Clearly, the use of the LAMP produces a huge improvement over the Random and Tetris solutions. The improvement is significant for the one-hop traffic. For instance, for problem , 82% of the traffic found by the random solution was three-hop and only 2% was one-hop. With the use of the LAMP, we were able to reach a solution where 46% of the traffic is one-hop and just 6% is three-hop, thus drastically reversing Fig. 9. Illustration of -moves. the inefficiencies of the previous solution. We also underline that, once again, the beneficial effect of the LAMP is even more Thus, there is a great variability of the performance im- striking for centralized traffic. A graphical view of the above provement depending on the dimension. mentioned results is summarized in Fig. 13. Comparing the evo- • When the RVTG traffic is used, both the LAMP algo- lution of the repartition of the traffic in each case, one could see rithm with the Tetris arrangement as an initial solution that the LAMP algorithms convert higher multiple hops traffic and the LAMP algorithm with the initial random arrange- given by the Random and Tetris algorithms, to one-hop traffic. ment have a similar performance [see Figs. 10(b), 11(b), Thus, another performance assessment of LAMP algorithm over and 12(b)]. However, with RCTG traffic, the difference, the Tetris and the Random is given by the measure of the per- in dimension two, between the performance of the LAMP centage of one-hop traffic. (with Tetris) and the LAMP (with Random) becomes sig- Figs. 10–12 that portray the mean hop values and the , nificant [see Fig. 10(d)]. Indeed, LAMP, with a random respectively, in dimension two, three and four, show that the arrangement as initial solution, outperforms the LAMP LAMP algorithm is more efficient in any case, than the Tetris. combined with Tetris. This could be explained by the fact We now point out some additional observations. that, with a centralized traffic, the Tetris algorithm encoun- • If we evaluate the difference between the of the LAMP ters specific local optima from which the LAMP algorithm and the of the Tetris we see that, in dimension two, it is has difficulty escaping thereafter. about 15% [see Fig. 10(b)]. On the other hand, in dimen- • As previously mentioned,the LAMP algorithm could sion four, it can reach more than 45% [see Fig. 12(d)]. reach a quasioptimal solution (with a gap of less than DÉGILA AND SANSÒ: TOPOLOGICAL DESIGN OPTIMIZATION OF A YOTTABIT-PER-SECOND LATTICE NETWORK 1623

Fig. 10. Comparative mean hops and performance improvements results for problems of dimension 2.

Fig. 11. Comparative mean hops and performance improvements results for problems of dimension 3.

10 ), while the Tetris algorithm produces results that are Summarizing, we have evaluated the performance of the always higher than a 1.5 mean hop value. LAMP over the Tetris and Random using several measures. 1624 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 9, NOVEMBER 2004

Fig. 12. Comparative mean hops and performance improvements results for problems of dimension 4.

Fig. 13. Evolution of the repartition of the trafﬁc per the number of hops for random and centralized trafﬁc pattern. DÉGILA AND SANSÒ: TOPOLOGICAL DESIGN OPTIMIZATION OF A YOTTABIT-PER-SECOND LATTICE NETWORK 1625

• performance improvement with respect to the random [6] M. Beshai and A. Yu, “YottaWeb topology,” Pers. Commun., Aug. 1999. grouping; [7] L. N. Bhuyan and D. P. Agrawal, “Generalized hypercube and hyperbus structures for a computer network,” IEEE Trans. Comput., vol. C-33, pp. • mean hop value given over 100 instances of the same test 323–333, 1984. problem; [8] R. Dutta and G. N. Rouskas, “A survey of virtual topology design al- • percentage of one-hop traffic for a given test problem; gorithms for wavelength routed optical networks,” Opt. Networks Mag. “Premiere Issue”, vol. 1, pp. 73–89, Jan. 2000. • CPU time taken by each algorithm. [9] E. Leonardi, M. Mellia, and M. A. Marsan, “Algorithms for the log- In all the cases, except for the CPU time, the LAMP pro- ical topology design in WDM all-optical networks,” Opt. Networks Mag. vides significant improvement with respect to the Tetris and the “Premiere Issue”, pp. 35–46, Jan. 2000. [10] F. Siu and R. K. C. Chang, “Effectiveness of optimal node assignments Random performance. in wavelength division multiplexing networks with fixed regular virtual topologies,” Comput. Networks, vol. 38, pp. 61–74, 2002. WDM Optical Networks: Concepts, VI. CONCLUSION [11] C. S. R. Murthy and M. Gurusamy, Design and Algorithms. Englewood Cliffs, NJ: Prentice-Hall, 2002. The great contribution of the YottaWeb architecture is its ex- [12] A. Kaufmann and G. Boulaye, Théorie des Treillis en vue de ses Appli- cations. Paris: Masson, 1978. pandability and reduction of intermediate nodes between edge [13] F. Glover and M. Laguna, Tabu Search. Norwell, MA: Kluwer, 1997. nodes. The proposed lattice structure for a certain dimension [14] P. Hansen and N. Mladenovic´, “Variable neighborhood search: Princi- network guarantees that the number of hops for any O/D pair ples and applications,” Eur. J. Oper. Res., vol. 130, pp. 449–467, 2001. would be at most . However, the aim of the planner should be to reduce such number of hops to the minimum. This work has shown how to reduce the mean hop values, weighted by the traffic, for communications within the network, and that for net- Jules R. Dégila (S’01) received the B.S. degree in works of more than 4000 nodes in dimension four. In dimen- applied mathematics from the University of Dakar, sions two, three, and four, the LAMP has a superior perfor- Dakar, Senegal, in 1996 and the M.S. degree in operations research from the University of Sherbrooke, mance than the Tetris or the Random grouping. Moreover, we Sherbrooke, QC, Canada. He is currently working found that the performance improvement of the LAMP is even toward the Ph.D. degree in electrical engineering greater when the traffic is centralized. This is particularly impor- at the École Polytechnique de Montréal, Montréal, ON, Canada. His Ph.D. dissertation was conducted tant given that, as the network increases in size, concentration in collaboration with Nortel Networks. of traffic to some major origin-destinations is quite likely. As His research interests include modeling, next-gen- further work, we are currently exploring lower bounding proce- eration network optimization, optical networking, dures for the method and studying update and expansion issues large scale system, and performance analysis. related to the NAP.

ACKNOWLEDGMENT The authors are grateful to M. Beshai and F. Blouin of Nortel Brunilde Sansò (M’92) was born in Rome, Italy, in 1960. She received the E.E. degree from the Univer- Networks for their fruitful comments and for lending us their sidad Simon Bolivar, Caracas, Venezuela, in 1981, traffic generator, to M. Labonté who helped with the implemen- the M.S. degree in reliability and the Ph.D. degree tation of the Tetris algorithm, and to the anonymous referees that in operations research from École Polytechnique de Montréal, Montreal, QC, Canada, in 1985 and 1988, helped improve the final version of the paper. respectively. After Postdoctoral studies at the CRT, University of Montreal, and a Research Fellowship at the REFERENCES GERAD, she joined the faculty of École Polytech- [1] NGI—Publications. NGI Concept Paper (1997). Electronic refer- nique de Montréal in 1992, where she has been a ences. Retrieved Mar. 1, 2002. [Online]. Available: http://www.ccic. Full Professor since 1997. She is currently with the Department of Electrical gov/ngi/pubs/concept-Jul97 Engineering, where she is the Director of the LORLAB, a research laboratory [2] M. Beshai and F. J. Blouin, “Courteous routing,” in Proc. Networks 2000, devoted to the performance, reliability, design and optimization of operational Toward Natural Networks 9th Int. Telecommunication Network Plan- planning of broadband networks. She is Co-Editor of the book Telecommuni- ning Symp., Toronto, ON, Canada, 2000. cations Network Planning (Norwell, MA: Kluwer, 1998) and the forthcoming [3] M. Beshai and R. Vickers, “PetaWeb architecture,” in Proc. Networks book Performance and Planning Methods for the Next Generation Internet 2000 Toward Natural Networks 9th Int. Telecommunication Network (Norwell, MA: Kluwer). Planning Symp., Toronto, ON, Canada, 2000. Dr. Sansò is a recipient or co-recipient of several awards and honors, among [4] F. J. Blouin, S. Yazid, and B. Bou-Diab, “Emulation of a vast adapta- them, the 2003 DRCN Best Paper Award, the Second Prize in the 2003 CORS tive network,” in Proc. Networks 2000, Toward Natural Networks 9th Practice Competition, the 1995 IEEE/ASME JRC Best Paper Award, the 1992 Int. Telecommunication Network Planning Symp., Toronto, ON, Canada, NSERC Women Faculty Award, and the 1992 FCAR Young Researcher Award. 2000. She is an Associate Editor of Telecommunication Systems and has been a referee [5] F. J. Blouin, W. A. Lee, A. J. Lee, and M. Beshai, “Comparison of two and technical committee member for major journals and scientific conferences, optical-core networks,” J. Opt. Networking, Opt. Soc. Amer., vol. 1, no. reviewer for government agencies, and industry consultant. She was the Pro- 1, pp. 56–65, Jan. 2002. gram Co-Chair of the Fifth INFORMS Telecommunications Conference.