Design of Logical Topologies in Wavelength-Routed IP Networks

Transcription

1 DESIGN OF LOGICAL TOPOLOGIES... 1 Design of Logical Topologies in Wavelength-Routed IP Networks M.Ajmone Marsan ½ A.Grosso ¾, E.Leonardi ½, M.Mellia ½, A.Nucci ½ ½ Dipartimento di Elettronica - Politecnico di Torino ¾ Dipartimento di Automatica ed Informatica - Politecnico di Torino mellia,nucci,leonardi,ajmone@mail.tlc.polito.it, grosso@athena.polito.it Abstract In this paper we discuss the optimal design of logical topologies in wavelength-routed IP over WDM networks supporting both unicast and multicast transfer of IP datagrams under deterministic and stochastic traffic patterns. The paper brings mainly three original contributions: i) it provides a MILP formalization of the optimal Logical Topology Design (LTD) problem in presence of multicast traffic under a perfectly known (deterministic) traffic pattern and proposes sub-optimal greedy and metaheuristic algorithms for its solution; ii) it derives novel, tight optimistic bounds that allow the assessment of the performance of the proposed algorithms; iii) it investigates the optimal logical topology design problem, when traffic patterns are known with a certain degree of uncertainty, hence characterized with a stochastic description, and it presents approaches for the sub-optimal solution of the problem. Focusing on the current Internet routing algorithms, we explicitly consider the routing of multi-hop flows over the logical topology as an input to the problem, not an optimization target. I. INTRODUCTION AND PREVIOUS WORK Internet, the largest data network ever built, is facing a constant increase in bandwidth demand, due to the growth of both the services available on-line, and the number of connected users. The fact that new users are increasingly attracted by new services, causes a positive feedback, whose consequence is the need for continuous upgrades of the Internet infrastructure. Wavelength-Routed (WR) optical networks, which employ Wavelength Division Multiplexing (WDM), are considered the best candidate for the short-term implementation of a high-capacity IP infrastructure, since they permit the exploitation of the huge fiber capacity, and do not require complex processing functionalities in the optical domain. In WR networks, high-capacity (electronic) routers are connected through semi-permanent optical pipes called lightpaths that may extend over several physical links. Lightpaths, thus, can be seen as chains of physical channels through which packets are moved from a router to another toward their destinations. At intermediate nodes, incoming channels belonging to in-transit lightpaths are transparently coupled to outgoing channels through a passive wavelength router that does not process in-transit information. On the other hand, incoming channels belonging to terminating lightpaths are converted to the electronic domain, so that packets can be extracted and processed, and possibly retransmitted on outgoing lightpaths after electronic IP routing. In a WR network, a logical topology, whose vertices are the IP routers and whose edges are the lightpaths, is overlayed to the physical topology, made of optical fibers and optical cross-connects. In principle, the logical topology configuration is independent from the physical topology; however, a number of constraints exist: i) the number of lightpaths departing and terminating at each node is limited by the number of transmitters/receivers and by the processing capability of electronics; ii) the establishment of each lightpath requires the reservation of physical resources (i.e., a WDM channel on the optical fibers along the path); since the number of wavelengths per optical fiber is limited, only a subset of logical topology configurations is feasible; iii) the maximum allowable length of lightpaths (in terms of both physical span and number of devices crossed) may be limited by the degradation of transmission parameters along lightpaths. In order to best exploit the capacity of a WDM infrastructure, a crucial task thus is the identification of the best feasible logical topology for the transport of a given traffic pattern. In recent years, this problem was extensively studied in the case of purely unicast traffic. It was shown that the identification of the optimal logical topology is computationally This work was partly supported by the MURST Italian Ministry for University and Research through the IPPO project.

2 DESIGN OF LOGICAL TOPOLOGIES... 2 intractable for large size networks [1], [2], and several heuristic approaches were proposed for the identification of suboptimal solutions [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. At the same time, the mentioned growth of Internet services has generated an increasing interest in protocols and architectures for the support of multicast applications: a significant fraction of the Internet traffic is expected to come from native multicast applications, a few years from now. As a consequence, the development of logical topology design procedures that can account for multicast traffic flows is necessary. Since multicast traffic flows are characterized by many destinations, replication (branching) of multicast packets somewhere in the network is necessary. Two different approaches are possible in WR networks: replication in the optical domain; replication in the electronic domain. The first approach was considered in [15], where the concept of lightpath was extended to a Light-Tree, which is a transparent one-to-many pipe in which injected packets are passively replicated in the optical domain and delivered to the light-tree end-points. This approach relies on the availability of multicast capabilities in the physical layer, which seem difficult to obtain. In addition, the performance of light-tree based solutions, with multiple packet flows sharing one light-tree, may be severely limited by undesirable replications of packets. In this paper we investigate solutions in which packet replication is performed in the electronic domain. These solutions impact neither the optical architecture of nodes, nor the IP protocol, and in addition appear more flexible and efficient than optical domain replication approaches. Moreover, we must consider the fact that in the IP over WDM context, the routing strategy is assigned a priori ; it cannot be considered a variable of the problem. This key observation led us to the optimization of the topology design, assuming that the IP routing algorithm is fixed. Finally, in all previous works [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] topology optimization is performed assuming a deterministic, perfectly known and stable, traffic pattern. This assumption, however, is quite far from reality, since node-to-node traffic intensities exhibit significant fluctuations and often an unpredictable behavior. For the first time, this paper provides an extension of the logical topology design methodology aimed at achieving a topology optimization which is robust with respect to traffic uncertainties and fluctuation, adopting a stochastic description of traffic patterns. The rest of this paper is organized as follows: in the next section we precisely formulate the logical topology design problem under unicast and multicast traffic, and provide a Mixed Integer Linear Programming Formulation, proving that the problem is NP-hard; we further extend the formulation to take into account traffic uncertainty obtaining a Mixed Integer Non-Linear Programming Formulation. Section III discusses both multicast and unicast routing over the logical topology, and in section IV we propose some simple heuristic and more complex metaheuristic approaches for the logical topology design with affordable complexity. The complexity of the proposed heuristic and metaheuristic algorithms is discussed in some detail in section V. Novel bounds are derived in section VI, and numerical results are presented in section VII. Finally, Section VIII concludes the paper. II. PROBLEM FORMULATION The problem of the optimal Logical Topology Design (LTD) for the transport of a given deterministic unicast and multicast traffic pattern can be stated as follows: GIVEN: i) an existing physical topology, comprising nodes equipped with a limited number of tunable transmitters and receivers, connected by optical fibers that support a limited number of wavelengths; ii) a multi-hop routing strategy, defined both for unicast and multicast flows; iii) a description of the traffic exchanged by sources and sets of destinations; FIND the logical topology that minimizes a target function (or cost). While different functions can be chosen as optimization target, we selected the maximum congestion level in the network, defined as the maximum traffic flow on lightpaths, since this metrics normally drives important network performance indices, such as loss probability and delay [3]: both the queueing delay and the loss probability, indeed,

3 DESIGN OF LOGICAL TOPOLOGIES... 3 mainly depend on the congestion experienced on the highest-load crossed lightpath. Thus, it is important to tightly control the maximum congestion level in the network in order to guarantee short queueing delays and few losses. However, some of the LTD algorithms proposed in this paper can be easily adapted to different target functions. When traffic patterns cannot be described in a deterministic fashion, the LTD problem formulation can be extended to include traffic uncertaintiesas follows GIVEN: i) and ii) as in the deterministic case iii) the probability density functions pdf µ of random variables describing the traffic intensity on traffic relations from sources to destinations in sets ; FIND the logical topology that minimizes a deterministic target function of the random variables describing the flow intensity on links. As regards the choice of the target function, the same consideration made in the deterministic case applies also to the stochastic problem; in particular, the natural extension of the maximum network congestion level under stochastic traffic patterns, becomes the maximum value among the «-percentiles 1 of the traffic distributions on lightpaths. Thus, we selected the maximum value among the «-percentiles as the logical topology performance measure under stochastic traffic patterns. The constraint on the maximum number of wavelengths that each fiber can carry is not considered during the optimization, since new transmission equipments allow the use of very high numbers of wavelengths (presently up to 128), and, in addition, the use of wavelength converters can be instrumental for the relaxation of this constraint [16], [17]. In the next sections we shall refer to the LTD optimization for purely unicast traffic using the acronym ULTD while the acronym MLTD will be used to refer to the problem with both unicast and multicast traffic; we shall refer to the LTD problem under deterministic traffic patterns using the prefix letter D, while the stochastic case will use the prefix letter S. A. MILP formulation of the D-MLTD problem Several different Mixed Integer Linear Programming (MILP) formulations of the D-ULTD problem appeared in the recent literature [1], [5], [6], [7], [9]. In this section we extend the D-ULTD formulation to the case in which multicast traffic is present (D-MLTD). Note that the D-MLTD formulation is not a straightforward extension of the D-ULTD formulation, because the branching capabilities drastically modify the flow conservation equations. Let Ì be the set of traffic requests, whose elements Ø indicate the traffic associated with connection from source to the destinations in set. If ½ then we have a multicast connection. Let Ð be the Ð-th destination in set. Let the binary variables ¾ ¼ ½ indicate whether a lightpath originating from node and terminating in node (i.e., lightpath ) belongs to the logical topology ( ½ if the lightpath is included in the logical topology, ¼ otherwise). The real variables Ö Ð state the percentage of traffic Ø, flowing on lightpath, that reaches destination. As already stated, multicast packets flowing on lightpath can reach several destinations because of branching, Ð thus È Ð ÖÐ in general is larger than Ö, the percentage of traffic Ø flowing on lightpath. In particular, we can state that Ö ÑÜ Ð Ö Ð. We denote with the total traffic flowing on lightpath. Finally, let ÑÜ ÑÜ µ be the maximum amount of traffic flowing on any lightpath in the logical topology. Given Æ, the number of nodes in the network, and given ÆÇ and Æ Á, that represent, respectively, the numbers of transmitters and receivers available at node (i.e., ÆÁ and Æ Ç are the maximum in/out degrees of node in the physical topology), the LTD problem can be formulated as follows: under the following constraints: ÑÒ ÑÜ (1) ½ A percentile is a value on a scale of one hundred that indicates the percent of a distribution that is equal to or below it.

4 DESIGN OF LOGICAL TOPOLOGIES... 4 Flow conservation at each node Ö Ð Total flow on lightpaths for each connection Flow on links Degree constraints Variables range constraints B. Considerations on the problem formulation Ö Ð ½ ¼ ½ if if Ð otherwise Ð (2) Ö Ö Ð Ð (3) Ö (4) Ö Ø (5) ÑÜ (6) Æ Ç (7) Æ Á (8) ¼ (9) ¼ Ö ½ (10) ¼ Ö Ð ½ Ð (11) ÑÜ ¼ (12) ¾ ¼ ½ (13) The above formulation of the topology design problem rests on two assumptions: the traffic between any two nodes can be arbitrarily partitioned and routed on multiple paths, no more than one lightpath is allowed between any two nodes, The first assumption can be relexed by imposing that Ö Ð and Ö ¼ ½µ. Modifying the problem formulation to allow parallel lightpaths requires more significant changes, since the space of the decision variables must be expanded, replacing the with new variables Ô, Ö Ð with ÖÐÔ, Ö with ÖÔ, and with Ô, being Ô ¼ ½ ÑÒ Æ Á Æ Ç µ. The structure of equations (1),..., (13), instead, remains the same. are binary variables (i.e., ÖÐ ¾ ¼ ½ and Ö ¾ While in the case of D-ULTD it is possible to provide a formulation in which only aggregate flows appear, as in [5], thus reducing complexity, the disaggregate flows Ö Ð must be considered in D-MLTD, in order to apply flow conservation equations (2) at each node: the presence of multicast traffic thus leads to a significant increase of complexity in the formulation. The D-MLTD problem is NP-hard, since it falls in the class of general MILP problems, and thus is numerically intractable, even for networks with a moderate number of nodes. Moreover, the D-MLTD problem represents a generalization of the well-known NP-hard D-ULTD problem, in the sense that it includes D-ULTD as particular instance. In addition, the MILP formulation naturally leads to a combined topology and routing optimization. Therefore, the routing strategy is a result of the optimization procedure, together with the logical topology configuration. Thus, MILP, and heuristics based on MILP, such as those relying on continuous relaxation and rounding, do not permit the advance specification of the routing strategy to be considered in the optimization. In the IP over WDM context, however, the routing strategy is assigned a priori : it is not a variable of the problem, but a constraint (see section III for a discussion about this issue). These facts motivated the development of heuristic approaches that permit the specification of the routing strategy as input of the D-MLTD problem.

5 DESIGN OF LOGICAL TOPOLOGIES... 5 C. Extension to the S-MLTD problem The extension of the problem formulation to the case of stochastic traffic patterns is not trivial. A formulation with acceptable complexity can be given when all the traffic concerning any source-destination pair is routed on only one path (i.e., no traffic partitioning on multiple paths is allowed), under the assumption of independence among traffic flows. In fact, in this case it is possible to derive the traffic pdf on any lightpath, that results from the sum of many independent random variables µ whose density function, pdf µ, describes the -th traffic relation. In particular, starting from the D-MLTD formulation, many of the variable definitions and equations still hold in the S-MLTD formulation. More in detail, the definitions of, Ö Ð, Ö and equations 1, 2, 3, 4, 6, 7, 8 still hold. On the contrary, the definition of changes, since these variables now acquire the new meaning of the «percentile of the traffic distribution on lightpath. Let pdf µ be the pdf of random variable describing the traffic flowing through lightpath, thus Ü «µ Ö Ü pdf µ «(14) ½ Thus equation 5 must be reformulated, becoming clearly non-linear. Note however that «µ depends on the Ö in a deterministic way. Assuming statistical independence among the traffic intensities associated with different traffic relations, the link flow intensity density function pdf µ is, indeed, obtained as the convolution among all the densities pdf µ describing traffic relations routed on link : pdf µ Ç Ö pdf µ (15) Given the pdf of traffic flowing on lightpaths, the «-percentile is obtained from equation( 14). As a conclusion, the S-MLTD can be formulated as a Mixed Integer Non-Linear Problem (MINLP), when the traffic between any source-destination pair is routed on only one path (i.e., no traffic partitioning on multiple paths is allowed). This formulation contains a non-linear, complex relation between and Ö Ð, and cannot be exactly solved, even for networks of quite small size. In addition, also the MINLP formulation naturally leads to a combined topology and routing optimization. The development of heuristic approaches that permit the specification of the routing strategy as input of the S-MLTD problem is thus necessary also in this case. III. ROUTING ON THE LOGICAL TOPOLOGY The results of a LTD optimization are generally dependent on the adopted routing: a logical topology configuration can be optimal under a given routing strategy, and at the same time far from optimality under another routing policy. As a consequence, the specification of the routing strategy represents a very important input for LTD heuristic optimization procedures. By optimizing the routing algorithm, it would be possible to optimize the performance on the obtained logical topology. In the current Internet situation, however, unicast IP routing algorithms have been standardized and are widely accepted [18], [19]. They can be described as minimum cost path and shortest path routing algorithms in which only static administrative costs can be considered. This observation indicates that the LTD optimization should be performed adopting a shortest path routing for unicast flows, i.e., a routing strategy that minimizes the number of hops (lightpaths) traversed by each unicast flow. As regards the multicast problem, from a theoretical point of view, the definition of the optimal multicast routing over a network (i.e., the routes that lead to the minimization of the capacity resources used for the transport of multicast flows), can be easily formalized as a minimal Steiner Tree problem, i.e., the problem of finding the minimal graph that connects the multicast traffic sources to a specified set of destinations [23]. However, the minimal Steiner Tree problem is known to be NP-Hard. Thus, the multicast IP routing algorithms proposed in literature can be cataloged as heuristic procedures that obtain a solution of the minimal Steiner Tree at a limited computational cost. Thus, in IP networks, many algorithms were proposed to route multicast traffic [20], [21], [22], some of which have been standardized, but are not yet widely adopted. They can be roughly grouped into two classes: the first is based on distributed algorithms, where shortest paths are used to build the distribution tree. The second class is instead based on

6 DESIGN OF LOGICAL TOPOLOGIES... 6 centralized algorithms, where a special node is elected, which has global visibility of the multicast tree, and can run a centralized algorithm to compute the best multicast tree. The first class of algorithms (often termed source specific ) is preferentially used when the number of participants in the multicast group is small, while the second one (called core specific ) is suited for both small and large groups. In the context of WDM networks, the traffic scenarios foreseen for IP networks of the coming years envisage a significant amount of multicast traffic resulting from the distribution of music, video, large file caches, multimedia events, etc. The number of users of such services can be large, so that centralized multicast routing algorithms can be advantageous. Moreover, since the WDM technology will be mainly deployed in the high-capacity network backbone, traffic scenarios containing single-rooted multicast trees can be expected to be quasi-static 2. These reasons suggested us to utilize as input of our MLTD optimization procedures a centralized multicast algorithm, but any multicast algorithm can be adopted in the optimization procedure. We selected the Selective Closest Terminal First (ËÌ ) algorithm proposed in [24], that, given the multicast traffic source, the group definition, and the network topology, finds a Steiner Tree with a good trade-off between performance and computational complexity. A brief description of the algorithm follows. ËÌ algorithm Let be the source of multicast traffic, and the set of destinations to be reached. Let Ì be a dynamic structure in which the Steiner Tree is gathered: set Ë ; ; Ì =NULL; while ( ) do select the node ¾ closest to nodes in Ë; select ¼, the node in Ë closest to ; extract from ; select the shortest path route Ö from ¼ to ; add the links of Ö to Ì ; insert all the nodes crossed by Ö in Ë; done The algorithm requires at each step the knowledge of all the shortest paths from nodes in Ë to nodes in. To reduce the algorithm complexity, it is possible to associate a weight Û with each node in Ë, and consider only the Þ nodes with highest weight, when selecting the closest destination to Ë. IV. LOGICAL TOPOLOGY DESIGN HEURISTICS In this section we propose heuristic approaches for the D-MLTD and S-MLTD problems; we initially focus our attention on the D-MLTD problem; we first introduce two greedy algorithms and then propose solutions that exploit metaheuristic optimization algorithms developed in the operational research context. Finally, we provide an extension of the proposed heuristic optimization algorithms to cope with the S-MLTD problem. A. Greedy heuristics Greedy heuristic algorithms for MLTD can be applied either to generate an initial solution for the metaheuristic-based algorithms described later in this paper, or to quickly produce reasonable logical topology configurations, when time constraints do not allow the application of computationally intensive algorithms. A.1 Source Copy Multicast Algorithm (ËÇÅ) The first heuristics is named Source COpy Multicast (ËÇÅ) algorithm; it relies on the assumption that multicast IP packets are routed by sending multiple copies from the multicast traffic source node. In this case, each multicast flow degenerates into a set of unicast flows connecting the source to each multicast destination. Thus, topology optimization is obtained ignoring the possibility of performing multicast replication inside the network (multicast replication is ¾ A group with many sources can be split in many groups with one source each, and the same destination set. The Steiner Tree obtained from the union of all the Steiner Trees departing from each root is a valid solution for the original problem.

7 DESIGN OF LOGICAL TOPOLOGIES... 7 allowed only at the host interface). Any unicast topology design algorithm can be used to obtain the desired logical topology; we selected ÅÄÌ, proposed in [3], that is aimed at the maximization of single-hop flows in the network 3. A.2 Route & Remove (ʲÊ) The second heuristic algorithm initially considers a fully-connected logical topology; an iterative removal from the logical topology of the least-loaded lightpaths is executed, until the degree constraints are satisfied. Note that the lightpath load is evaluated by routing both unicast and multicast flows according to a fixed routing strategy, thus fulfilling the routing constraints. To describe this algorithm, we use a bipartite graph associated with the current logical topology according to the following rules: two vertices Ò and Ó Ò in the bipartite graph correspond to each node Ò in the (logical) topology; in the bipartite graph, an edge exists between and Ó, whose weight is initialized to the traffic flow value between nodes and ; a boolean variable is associated with each edge, which can assume the values Removable or Unremovable. Ê²Ê algorithm Select the fully-connected logical topology and mark all lightpaths as Removable. while (all the in/out-degree constraints are not satisfied) do - Solve both the unicast and the multicast routing problems on the current topology and compute traffic flows on lightpaths. - Assign to each edge of the bipartite graph a weight equal to the flow traversing the associated lightpath. - Find a set of edges that can be removed from the graph by solving a 1-minimal Weight Matching 3 (1-ÑÏ Å) on the bipartite graph. Only the edges that are marked as Removable can be chosen in the matching. - Remove all edges in the 1-minimal Weight Matching, together with the corresponding lightpath in the logical topology, only if the resulting logical topology remains connected. If the removal of a matched lightpath would disconnect the logical topology, mark the lightpath as Unremovable. done The reported greedy heuristics entail an increasing degree of computational complexity. The first one is very simple; it transforms the MLTD problem in a ULTD problem, decomposing each multicast connection into several unicast connections. However, it does not consider the routing problem at all. The second algorithm entails a significantly larger degree of complexity, since the topology optimization considers the real routing. Although we could expect this to lead to better performance than the ËÇÅ algorithm, especially when the multicast traffic percentage is significant, numerical results will show that this is not true in general. Thus the ËÇÅ heuristic remains interesting, because it is very simple, neglecting the multicast traffic at all. A.3 Metaheuristic - Tabu Search (Ì Ë) The heuristics we propose next relies on the application of the Tabu Search Ì Ëµ methodology [26]. Ì Ë is based on a partial exploration of the space of admissible solutions, finalized to the discovery of a good solution. The exploration starts from an initial solution that is generally obtained with a greedy algorithm. For each admissible solution, a class of neighbor solutions is defined. A neighbor solution is defined as a solution that can be obtained from the current solution by applying a perturbation. The set of all the admissible perturbations uniquely defines the neighborhood of each solution. At each iteration of the Ì Ë algorithm, all solutions in the neighborhood of the current one are evaluated, and the best is selected as the new current solution. A special rule, the Tabu list, is introduced in order to prevent the algorithm to The maximization of traffic flows that are routed in a single-hop fashion is equivalent to the well-known Æ-Maximum Weight Matching (Æ- Å Ï Å) problem on a bipartite graph when the in/out-degrees of all nodes are equal (i.e., Æ Á Æ Ç Æ ). The optimal solution of this problem can be found with Ç ÆÆ µ operations [25]. Even if the in/out-degrees of nodes are different, a modification of the Æ-Å Ï Å problem can be used to solve the ULTD problem. The minimal Weight Matching, is similar to Maximum Weight Matching algorithm, but it provides a matching with maximal size and minimal weight

8 DESIGN OF LOGICAL TOPOLOGIES lightpaths cycle found a) Old solution b) New solution Fig. 1. Cycle for the perturbation generation in the Ì Ë and Ë algorithms. Solid lines represent present lightpaths, dotted lines represent the identified cycle. deterministically cycle among already visited solutions. The Tabu list consists in a list recording the last perturbations that were accepted. Until a perturbation is stored in the Tabu list, it cannot be used to generate a new solution. After a given number of iterations, the algorithm returns the best visited solution. Note that Ì Ë algorithms can be seen as an evolution of the classical local optimal solution search algorithms called Steepest Descent [26]; however, thanks to the Tabu list mechanism, Ì Ë can accept worse solutions than the current one, and, thus, it is not subject to local minima entrapments. A.4 Metaheuristic - Simulated Annealing (Ë) Like Tabu Search, also Simulated Annealing (SA) [26] is based on a partial exploration of the space of admissible solutions, finalized to the discovery of good solutions. Exploration starts also in this case from an initial solution that is generally obtained with a greedy algorithm. At each iteration of the algorithm, however, only one solution in the neighborhood of the current solution is visited and evaluated. If the new solution performs better than the current one, it is accepted as the new current solution, otherwise it is accepted with probability Ô, and discarded with probability ½ Ô. The probability of accepting a worse solution than the current one generally depends on the iteration count. Usually, the value of Ô is decreased several times during the exploration process, using a decreasing function that emulates the annealing process. As a consequence, in the initial phases of SA the solution space exploration is dominated by randomness; while by decreasing the value of Ô, i.e., by decreasing the algorithm temperature, an increasing degree of determinism is introduced in the exploration. When Ô becomes negligible, the algorithm tends to behave as the classical local optimum solution search algorithm named First Improvement [26]. A.5 Metaheuristics setup Three fundamental aspects that must be defined in the metaheuristic algorithms just described concern: the choice of an initial solution, the definition of the perturbation that generates the neighborhood, the evaluation of the visited solutions. It could be possible to select as initial solution a generic admissible topology, generated with any greedy algorithm, or drawn at random. However, the former approach can sensibly shorten the time required for the execution of the optimization algorithms. Thus we select the result of the Ê²Ê heuristic as initial topology. The perturbation is defined according to the following algorithm: within the current solution, ¾Ä nodes, Ò ½ Ò ¾ Ò ¾Ä, are selected such that lightpaths Ò ¾ Ò ¾ ½ ½ ¾ Ä exist, and lightpaths Ò ¾Ä Ò ½ Ò ¾ Ò ¾ ½ ½ ¾ Ä ½ do not exist in the perturbed solution, lightpaths in are replaced with lightpaths. This is equivalent to identifying a cycle of ¾Ä lightpaths, Ä of which are present in the topology, while the other Ä are missing; in the identified cycle, a present lightpath is followed by a missing one, as shown in Figure 1. To obtain the neighbor topology, we have to remove the existing lightpaths while adding the new ones. If Ä ¾, the resulting perturbation is equivalent to a well known branch exchange operation.

9 DESIGN OF LOGICAL TOPOLOGIES... 9 This class of perturbation guarantees that degree constraints are not violated, thus generating a valid perturbation. However, it does not guarantee that the new topology is connected, thus requiring an explicit test for connectivity. Finally, the evaluation of new solutions is performed by running the chosen routing algorithms for both unicast and multicast traffic, and computing the maximum flow on lightpaths. B. Extension to the S-MLTD problem The extension of the previous algorithms to cope with traffic relations described in stochastic terms impacts only the evaluation of the congestion on the lightpaths in the considered topology, as explained in section II. The evaluation of solutions, indeed, must be performed by taking into account the degree of uncertainty associated with each traffic relation. This can be obtained by routing all the node-to-node traffic relations on the logical topology defined by the current solution, and evaluating the resulting distributions of the traffic traversing each lightpath, as stated by (15). The «-percentile of the traffic traversing each lightpath is then evaluated using (14), and the maximum obtained value is chosen as the performance index for the current solution. Since the exact evaluation of the distribution of the traffic traversing all lightpaths can be too expensive, we propose to approximate it by applying the Central Limit Theorem. Assuming that the random variables associated with individual traffic relations are statistically independent, the random variable describing the traffic flowing on a given lightpath is the sum of several independent traffic variables, and thus tends to be normally distributed. Let Ø and, respectively, denote the average and the standard deviation of the -th traffic relation, and and denote the average and the standard deviation of the traffic crossing the lightpath from to ; then and Ö Ø Ö µ ¾ where Ö is the binary variable that assumes value 1 if the -th traffic relation crosses the lightpath from to, and 0 otherwise. Thus, equation 14, according to the Central Limit Theorem, is the solution of the following equation: ½ Ü erfc Ô¾ «¾ where erfc(ü) is the complementary error function. This approach can be applied to the Ê²Ê greedy heuristic, as well as both the Ì Ë and the Ë metaheuristics. On the contrary, the extension of the ËÇÅ algorithm is not straightforward, as it does not involve the solution of the routing problem, and thus it does not take into account the aggregate flow on lightpaths. V. COMPLEXITY We now discuss the complexity of the heuristics that were described above. Be Æ the number of nodes in the network, and the number of multicast flows that must be transported in the network. Let ÑÜ Æ Ç Æ Á µ denote the maximum in/out degree of the logical topology. The evaluation of the maximum congestion level of lightpaths requires the information about the unicast and multicast traffic routing. The computation of the unicast routing along shortest paths consumes Ç Æ ¾ ÐÓ Æ µµ operations [27]. Once shortest paths are known, the computation of each Steiner Tree according to algorithm ËÌ requires Ç Æ ¾ Þµ operations in the worst case, since, at each iteration, the algorithm requires at most Ç Æ Þµ operations to choose the closest source/destination pair, choice that must be performed for all the nodes in the group destination set, that is Ç Æ µ. Note that the parameter Þ can be in the order of Æ; thus, it significantly affects the algorithm complexity. In conclusion, the computation of the unicast and multicast routing requires Ç Æ ¾ ÐÓ Æ µ Þµµ operations. We can now evaluate the complexity of the proposed greedy heuristics.

10 DESIGN OF LOGICAL TOPOLOGIES ËÇÅ has complexity Ç Æ µ Ç Æ µ, since Ç Æ µ operations are required to transform the multicast requests into unicast requests, and Ç Æ µ operations are needed to run the ÅÏ Å algorithm for all the iteration necessary to satisfy the degree constraints. Ê²Ê has complexity Ç Æ ÐÓ Æ µ Þµ Æ µ, since at most Ç Æ µ iterations are needed to complete the algorithm, each time routing the traffic and performing a Å Ï Å algorithm. Let us now evaluate the complexity of the algorithms based on Ì Ë. At each iteration, the evaluation of all solutions in the neighborhood is necessary; this requires Ç Ä Æ Ä ¾ ÐÓ Æ µ Þµµ operations 4, since Ç Ä Æ Ä µ neighbors are evaluated, and the evaluation of each solution requires the solution of the routing problem. If the number of iterations is Á, the resulting complexity is Ç Á Ä Æ Ä ¾ ÐÓ Æ µ Þµµ. The complexity of each iteration of algorithms based on Ë is instead Ç Æ ¾ ÐÓ Æ µ Þµµ, since at each iteration only one solution is evaluated. Let Á denote the number iterations also in this case; the resulting complexity is Ç ÁÆ ¾ ÐÓ Æ µ Þµµ. VI. LOWER BOUNDS ON CONGESTION In order to evaluate the quality of the solutions provided by the proposed heuristic optimization algorithms, we compute lower bounds on the maximum congestion level achievable for a given deterministic traffic pattern. The availability of bounds is important also because they can be instrumental for the definition of stopping rules for the Ì Ë and Ë algorithms. We restrict our investigation to the D-MLTD problem, even if some of the presented bounds can be easily extended to the S-MLTD context. The presented bounds represent a generalization and extension of the Minimum Flow Tree (Å Ì ) bound [2], [5] to cases where routing algorithms do not permit partitioning of the same traffic relation on several paths and in presence of multicast traffic. A. Unicast traffic For the sake of simplicity, let us consider first only unicast traffic; in order to obtain a lower bound on the maximum congestion, the following three assertions are helpful. Let us consider a logical topology whose source-destination shortest-path length matrix À stores the number of hops from to. Let Ø be the traffic from node to node. Consider any routing algorithm that does not permit partitioning of the same traffic relation on several paths, then: Proposition 1: A lower bound Ä on the maximum congestion level in the network can be obtained by solving the following bin packing problem: optimally distribute objects into bins, where each object is associated with one source-destination flow, and thus has a volume equal to Ø, and each bin represents a lightpath in the topology under examination. The bin-packing problem solutions must satisfy to the two following constraints: objects associated with flows from to must appear at least on bins; each object associated with a flow from to must appear once on a bin associated with a lightpath outgoing from the source node. Proof: On a given topology, the solution of any routing problem defines an assignment of flows (objects) to lightpaths (bins) that satisfies the above defined bin-packing problem. Each solution, however, also satisfies an additional constraint regarding the contiguity of the lightpaths carrying each flow; indeed, they must form a path from the source to the destination. Thus, the optimal solution of the simplified bin-packing problem provides an optimistic result, hence a lower bound on the maximum congestion achievable by any routing algorithm. We must now evaluate the minimum number of times each object appears in the bins, ÓÔØ. Let us first state the following proposition: Proposition 2: Given two different topologies Ì ¼ and Ì ¼¼, comprising the same number of nodes and lightpaths, whose source-destination path length matrix À ¼ and À ¼¼ are such that one of the two conditions below hold: ¼ ¼¼ Remember that ¾Ä is the number of nodes in the cycle that generates the topology perturbation.

11 DESIGN OF LOGICAL TOPOLOGIES for each pair ½ ½ µ such that ¼ ½ ½ ¼¼ ½ ½ ¼ ¼ ¼, there exists a pair ¾ ¾ µ with ¼¼ ¾ ¾ ¼ ¾ ¾ ¼¼ ¼¼ ¼ and Ø ¾ ¾ Ø ½ ½, i.e., for each source destination pair that has a longer path on the first topology, in the second topology there exists a source-destination pair that has an even longer path, and whose traffic is larger, then Ä ¼ Ä ¼¼. Proof: If the first condition is satisfied the proof is straightforward. If, instead, the second condition is satisfied for pairs ½ ½ µ and ¾ ¾ µ, it is possible to transform the optimal bin-packing solution for the second matrix into one of bin-packing solution of the first problem by extracting from the bins ¼¼ objects of volume Ø ¾ ¾, and replacing ¼ of them with objects of volume Ø ½ ½. Note that by construction, this procedure leads to a solution of the first bin-packing problem that cannot perform worse than the optimal solution of the second. This procedure can be iterated for all the source destination pairs that satisfy the second condition. In this case we say that matrix À ¼ lower bounds matrix À ¼¼. It is easy to prove that if there exist three matrices À ¼, À ¼¼ and À ¼¼¼ such that À ¼ lower bounds À ¼¼ and À ¼¼ lower bounds À ¼¼¼, then À ¼ lower bounds À ¼¼¼, i.e., the transitive property holds. We are now able to define a path length matrix À ÓÔØ that lower bounds the path length matrix of every feasible topology; for simplicity we assume ÆÇ Æ Á, but it is possible to generalize the following result when the degree constraints are different node by node. Proposition 3: For any topology with Æ nodes, and in/out degree, a path length matrix À ÓÔØ that lower bounds any feasible path length matrix À, can be provided by the infeasible topology Ì ÓÔØ in which each source node is the root of an optimal tree of degree. In this case, sorting the destination nodes in decreasing order of received traffic from, the first destinations (those that receive more traffic from ) are at one hop distance from, the subsequent ¾ are two hops far, the subsequent are at three hops, etc. Proof: Let À be the path length matrix of a feasible topology with in/out degree equal to. Consider source ; at most paths originating from exist whose length is ½ hop, at most ¾ paths exist whose length is larger or equal to ¾, etc. Sort the destinations in such way that ord µ ord µ if. Then build an infeasible topology Ì ÓÔؼ such that ÓÔؼ ½ for ¼, ÓÔؼ ¾ for ¾, etc. By repeating the procedure for every source we build a path length matrix À ÓÔؼ, such that ÓÔؼ for every µ. Thus À ÓÔؼ lower bounds À. Let us compare À ÓÔØ and À ÓÔؼ ; in each row of both matrices we find elements equal to 1, ¾ elements equal to 2, etc. Only the element position differs. À ÓÔØ can be obtained from À ÓÔؼ, by iteratively exchanging elements of À ÓÔؼ according to the following iterative algorithm: Let À ¼µ À ÓÔؼ ; while (there exist two pairs of nodes ½ µ and ¾ µ such that µ ½ µ ¾ and Ø ½ Ø ¾ µ do exchange the elements, i.e., generate À ½µ that differs from À µ in the two elements ½ and ¾ so that: ½µ ½ µ ¾ and ½µ ¾ µ ½ ; done the matrix À Òµ is À ÓÔØ. Note that for proposition 2, at each iteration of the algorithm, À ½µ lower bounds À µ. Thus À ÓÔØ lower bounds À. The three propositions above allow us to conclude that the solution of the following ILP problem provides a lower bound to the maximum congestion level achievable by any topology with Æ nodes and nodal degree. Note that the total number of lightpaths in the topology is Æ ; let Ð be an index that runs on the lightpaths, ¼ Ð Æ ; let Ç be the set of lightpaths outgoing from node. We group the lightpaths in Æ groups of size ; each group contains the lightpaths outgoing from a source node. Let Ð be a binary variable that assumes the values: Ð ½ if traffic is routed on Ð ¼ otherwise

12 DESIGN OF LOGICAL TOPOLOGIES then the target is to minimize the maximum congestion level: ÑÒ Ä with: subject to the following constraints: and Ä ÑÜ Ð Ð Ð Ð Ð Ø Ð ÓÔØ (16) Ð¾Ç Ð ½ (17) Ð ¾ ¼ ½ Where constraint (16) forces traffic relation to flow on at least lightpaths, while constraint (17) impose that the flow outgoing from source must be routed on one of the lightpaths outgoing from. Thus we have proved the following theorem: Theorem 1: The solution of the above ILP problem provides a lower bound to the D-MLTD problem when routing does not permit partitioning traffic on several paths 5. B. Multicast traffic Consider now one of the multicast groups, say the Ø, whose traffic load is Ø, and whose number of destinations is ½. At least lightpaths are involved in the distribution tree, of which at least one is outgoing from the source. Thus, the previous lower bound can be easily extended to the D-MLTD case by adding the variables Ð to the previous formulation for each multicast group, subject to the constraints: Ð and Ð Ð¾Ç Ð ½ The ILP problem that allows the bound to be obtained is an extension of the well known Bin-Packing problem [28], thus it is NP-Hard. In order to obtain a bound that requires a lower computational complexity, several approximations can be applied to the ILP problem. We describe three possible approximations in increasing order of complexity. B.1 Fluidic Bin-Packing ( È ) bound The first possible approximation considers the continuous relaxation of the ILP problem. Relaxing variables Ð Ð, thus automatically allowing the partitioning of the same traffic relation on several paths, we obtain a bound that is similar to Å Ì [2], [5], differing from it because of the constraint that all flows originated at source must be routed on lightpaths outgoing from. B.2 Branch and Bound Bin-Packing (È ) bound To obtain a more accurate bound, a Branch and Bound algorithm can be applied directly to the full ILP problem. This approach however requires more computational power, and therefore may not be suitable to asses the quality of the solutions generated by the Ì Ë or Ë algorithms. The previous description considers that the number of lightpaths outgoing from the source must accommodate all the flows originating in (the bottleneck is at the source). A similar consideration made on the destination constraints allows us to derive the symmetrical bound, where the lightpaths ingoing to a destination must accommodate all the traffic directed to (the bottleneck is at the destination). In the rest of the paper we present results where both bounds have been evaluated and the maximum among them is taken

13 DESIGN OF LOGICAL TOPOLOGIES C. Partial Relaxation Bin-Packing (È ÊÈ ) bound If the È is computationally too expensive, it is possible to apply the relaxation to continuous of a subset of the integer variables. We propose to solve a partial relaxation bin-packing problem for each source, where only Ð such that ÓÔØ ½ are integer variables. D. Single Hop Bin Packing (ËÀÈ ) bound According to another possible approximation, only the traffic relations originated in are distributed among the outgoing lightpaths of. This is equivalent to set all unicast path lengths ½, and the all multicast group sizes ½ in the previously described ILP formulation. The resulting ILP problem is reduced to Æ independent ILP problems: each one still represents a Bin-Packing problem of smaller size, and can possibly be optimally solved by applying the Branch and Bound algorithm. VII. EXPERIMENTAL ANALYSIS In order to compare the solutions of the LTD problem generated by the different heuristics that we proposed in this paper, we considered quite a large number of different network sizes and traffic patterns, under both deterministic and stochastic traffic. Here we report results only for a small subset of the considered network configurations, highlighting our most interesting findings. We selected traffic patterns where, along with unicast traffic, a limited number of one-to-many multicast connections exist. Each (multicast) connection is characterized by its source node, by a (set of) destination node(s), and by the description of its traffic. First, we focus on deterministic traffic patterns. We report results for networks with ½ Æ nodes, and logical topologies with in/out degree (ÆÁ Æ Ç ). We consider three different traffic patterns, each comprising Æ multicast connections, for each of which both the source and the destinations are randomly selected among all nodes in the network. The average number of destinations of each multicast connection is fixed to ¾ of the total number of nodes. The three traffic patterns can be described as follows: (A) Uniform, low variance Each (unicast or multicast) traffic flow requires a bandwidth whose value is randomly extracted from an exponential distribution with mean ½. This traffic pattern exhibits a high degree of uniformity and regularity, since no hot-spots are present. (B) Uniform, high variance Each (unicast or multicast) traffic flow requires a bandwidth whose value is randomly extracted from a hyper-exponential distribution with mean ½, and variance ¾ ½¼. This scenario exhibits a significantly higher degree of variation and irregularity than the previous one, since the bandwidth variance has been significantly increased. (C) High-Low traffic node 20% of the nodes are High-Traffic (HT), while the rest are Low-Traffic (LT). Thus, traffic relations can be classified as: HT to HT, with high traffic (exponentially distributed with ½¼); HT to LT and LT to HT, with medium traffic (exponentially distributed with ); LT to LT, with low traffic (exponentially distributed with ½). Sources of multicast connections are considered as HT nodes. Since traffic patterns are defined in probabilistic terms, they need to be instantiated to produce a traffic configuration, which can be used as input of the optimization algorithms. In order to understand the relative merits of the different algorithms, we always obtained results for a number of traffic configurations and observed both the extreme behaviors and the average behaviors. Here we shall mainly focus on averages. Note that all traffic values are expressed in arbitrary units, which will also be used to label the results graphs. Although we did not perform a fine-tuning of the metaheuristic algorithm settings, a preliminary set of runs was used to decide the values of their parameters. In the Ì Ë algorithm, the Tabu size was set to ½¾, i.e., at most 12 lightpaths are stored in the Tabu list, and thus are excluded from the neighborhood generation. The neighborhood is generated using nodes cycles. The number of iterations was fixed to 100. In the Ë algorithm, the probability of accepting a worse solutions is initially set to Ô ½¼. Every 1000 iterations, Ô is decreased tenfold. The number of iterations was set so that the numbers of topologies visited by SA and TS are similar. Note that the chosen parameters force both Ì Ë and Ë to perform a deep search in the neighborhood of the current solution, since both the Tabu list size and the SA temperature are small. As regards the multicast routing algorithm parameter, we fixed Þ (only the 3 highest-weight closest nodes are considered in the construction of the Steiner tree).