# Generalized Multiobjective Multitree model solution using MOEA

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2 GMM-model using a MOEA, section 4 presents experimental results. Final conclusions and future wor are left for section 5. 2 Generalized multiobjective multitree model In previous wor we have proposed a Generalized Multiobjective Multitree model (GMM-model) [13] that considers simultaneously for the first time, multicast flow, multitree, and splitting. The proposed GMM-model considers a networ represented as a graph G(N, E), with N denoting the set of nodes and E the set of lins. The set of flows is denoted as F. Each flow f F can be split into K f subflows that after normalization can be denoted as f ; = 1, K f. In this case, f indicates the fraction of K f f F it transports, and f = 1. For each flow f F we = 1 have a source s f N and a set of destination or egress nodes T f N. Let t be an egress node, i.e. t T f. Let f t X denote the fraction of subflow f to egress node t ij f t assigned to lin (i,j) E, i.e. 0 X ij 1. GMM-model considers 11 objective functions: Maximal lin utilization (φ 1 ), Total hop count (φ 2 ), Hop count average (φ 3 ), Maximal hop count(φ 4 ), Maximal hop count variation for a flow (φ 5 ), Total delay (φ 6 ), Average delay (φ 7 ), Maximal delay (φ 8 ), Maximal delay variation for a flow (φ 9 ), Total bandwidth consumption (φ 10 ), and number of subflows (φ 11 ). Moreover, considers 7 constraints: Flow conservation constraints for every source node, for every destination and for every other node; a subflow uniformity constraint, to ensure that a subflow f always transports the same information; lin capacity constraint; constraint on the maximum number of subflows. At this point, it is important to point out that the mathematical solution of the proposed GMM-model is a complete set X* of Pareto optimal solutions x* X*, i.e. any solution x outside the Pareto set (x X*) is outperformed by at least one solution x* of the Pareto set ( x* f x ); therefore, x can not outperform x* even if not all the objective functions are considered. Consequently, under the same set of constraints, any previous model or algorithm, that only considers a subset of the proposed objective functions, either as a SOP or MOP, can find one or more solutions calculated with the GMM-model or dominated by solutions x* X* of this model. In conclusion, by using the GMM-model it is possible to calculate the whole set of optimal Pareto solutions. This includes any solution that has been previously found using most of the already published alternatives that consider any subset of the proposed objective functions. Now it may be clear why we call this model generalized. 3 GMM resolution using a MultiObjective Evolutionary Algorithm To solve the GMM-model, a Multiobjective Evolutionary Algorithm (MOEA) approach has been selected because of its well-recognized advantages when solving MOPs in general and TE load balancing in particular [2, 3, 7, 8]. A MOEA, as a genetic algorithm, is inspired by the mechanics of natural evolution (based on the survival of the fittest species). At the beginning, an initial population of P max feasible solutions (nown as individuals) is created as a starting point for the search. In the next stages (or generations), a performance metric, nown as fitness, is calculated for each individual. In general, a modern MOEA calculates fitness considering the dominance properties of a solution with respect to a population. Based on this fitness, a selection mechanism chooses good solutions (nown as parents) for generating a new population of candidate solutions, using genetic operators lie crossover and mutation. The process continues iteratively, replacing old populations with new ones, typically saving the best found solutions (which is nown as elitism), until a stop condition is reached. In this paper, an algorithm based on the Strength Pareto Evolutionary Algorithm (SPEA) [12] is proposed. It holds an evolutionary population P and an external set P nd with the best Pareto solutions found. Starting with a random population P, the individuals of P evolve to optimal solutions that are included in P nd. Old dominated solutions of P nd are pruned each time a new solution from P enters P nd and dominates old ones. 3.1 Encoding Encoding is the process of mapping a decision variable x into a chromosome (the computational representation of a candidate solution). This is one of the most important steps towards solving a TE load balancing problem using evolutionary algorithms. Fortunately, it has been sufficiently studied in current literature. However, it should be mentioned that to our best nowledge, this paper is the first one that proposes an encoding process, as shown in Fig. 1, that allows the representation of several flows (unicast and / or multicast) with as many splitting subflows as needed to optimize a given set of objective functions.

3 In this proposal, each chromosome consists of F flows. Each flow f, denoted as (Flow f), contains K f subflows that have resulted from splitting and which flow in several subflows (multitree, for load balancing). Inside a flow f, every subflow (f,), denoted as (Subflow f,), uses two fields. The first one represents a tree (Tree f,) used to send information about flow f to the set of destinations T f, while the second field represents the fraction f of the total information of flow f being transmitted. Moreover, every tree (Tree f,) consists of T f different paths (Path f,,t), one for each destination t T f. Finally, each path (Path f,,t) consists of a set of nodes N l between the source node s f and destination t T f (including s f and t). For optimality reasons, it is possible to define (Path f,,t) as not valid if it repeats any of the nodes, because in this case it contains a loop that may be easily removed from the given path to mae it feasible. Moreover, in the above representation, a node may receive the same (redundant) information by different paths of the same subflow; therefore, a correction algorithm was implemented to choose only one of these redundant path segments, maing sure that any subflow satisfies the optimality criteria. CHROMOSOME (Subflow f,) (Tree f,) f (Tree f,) (Path f,,t) (Flow 1) (Flow f) (Flow f) (Subflow f,1) (Subflow f,) (Subflow f, K f ) (Path f,,t 1 ) (Path f,,t) (Path f,,t Tf ) N sf N n Fig 1. Chromosome representation. 3.2 Initial Population To generate an initial population P of valid chromosomes we have considered each chromosome at a time, building each (Flow f) of that chromosome at a time. For each (Flow f) we first generate a large enough set of different valid paths from source s f to each destination t T f (see line 1 of Fig. 6). Then, an initial K f is chosen as a reasonable random number that satisfies constraints on the maximum number of subflows. To build each of the K f subflows, we randomly generate a tree with its root in s f and leaves in T f by randomly selecting a path at a time for each destination, from the previously generated set of paths. Trees are conformed by a path-set, which can contain redundant segments; i.e. two paths belonging to a tree with different destinations can meet themselves in more than one node causing redundant subflow information transmission between those pair of nodes. To correct this anomaly, a repair redundant segments process is N t defined: the shortest path of this tree should be taen as a pattern and then for each of the remaining paths in the tree, find its shortest segment starting at the latest node (branching node) in the pattern. The resulting segment will be a pattern segment starting at its source to the branching node joint with the old segment starting at branching node to the destination. Later, an information fraction of f =1/ K f is initially set. In this initialization procedure (see lines 2-3 of Fig 6), chromosomes are randomly generated one at a time. A built chromosome is valid (and accepted as part of the initial population) if it also satisfies lin capacity; otherwise, it is rejected and another chromosome is generated until the initial population P has the desired size P max. 3.3 Selection Good chromosomes of an evolutionary population are selected for reproduction with probabilities that are proportional to their fitness. Therefore, a fitness function describes the quality of a solution (or individual). An individual with good performance (lie the ones in P nd ) has a high fitness level while an individual with bad performance has a low fitness. In this proposal, fitness is computed for each individual, using the well-nown SPEA procedure [12]. In this case, the fitness for every member of P nd is a function of the number of chromosomes it dominates inside the evolutionary set P, while a lower fitness for every member of P is calculated according to the chromosomes in P nd that dominate the individual considered. A roulette selection operator is applied to the union set of P nd and P each time a chromosome needs to be selected. 3.4 Crossover We propose two different crossover operators: flow crossover and tree crossover. With flow crossover operator (line 10.a in Fig 6), F different chromosomes are randomly selected to generate one offspring chromosome that is built using one different flow from each father chromosome, as shown in Fig 2. A father may be chosen more than once, contributing with several flows to an offspring chromosome. The tree crossover operator is based on a two-point crossover operator, which is applied to each selected pair of parent chromosomes (line 10.b in Fig 6). In this case, the crossover is applied by doing tree exchanges between two equivalent flows of a pair of randomly selected parent-chromosomes, as shown in Fig. 3.

4 CHROMOSOMES 1 (Flow 1) (Flow 2) 2 (Flow 1) (Flow 2) F (Flow 1) (Flow 2) is selected as a Mutation Point. The segment mutation phase consists in finding a new segment to connect the selected Mutation Point to destination t (see Fig 4), followed by the already explained (see B - Initial Population) repair redundant segment process, to achieve better chromosome quality. Path f,,t Mutation Point N sf N j oldest segment N t OFFSPRING (Flow 1) (Flow 2) Fig 2. Flow crossover operator. CHROMOSOMES Two-point crossover 1 (flow f) 2 (flow f) OFFSPRING 1 (Sub f,1) (Sub f,) (Sub f,1) (Sub f,) (Sub f,1) (Sub f,) N sf N j new segment N t Fig 4. Segment mutation. Finally, the subflow fraction mutation phase is applied to (Subflow f,) by incrementing (or decrementing) flow fraction f in δ (see Fig 5), followed by the normalization process that has already been explained. Flow f Mutation Point 2 (Sub f,1) (Sub f,) Normalization process (Tree f,1) f * 1 (Tree f,) f * (Tree f, K f ) f K f * Fig 3. Tree crossover operator. Tree crossover without normalization of the information fraction f usually generates infeasible chromosomes. f * = f Therefore, a normalization process used as a last step of a tree crossover operator. K f = Mutation This operator could improve the performance of an evolutionary algorithm, given its ability to continue the search of global optimal (or near optimal) solutions even after local optimal solutions have been found, not allowing the algorithm to be easily trapped in local suboptimal solutions. Each time that an offspring chromosome is generated, a (generally low) mutation probability p m is used to decide if the mutation operator should be applied to this chromosome (line 11 in Fig 6). To apply a mutation operator, we first randomly choose a (Flow f) of the new offspring, in order to later select (also randomly) a (Subflow f,) on which the mutation will actually apply; therefore, what we implement is a subflow mutation operator. For this wor, we propose a subflow mutation operator with two phases: segment mutation and subflow fraction mutation. For segment mutation phase, a (Path f,,t) of (Tree f,) is randomly chosen. At this point, a node N j of (Path f,,t) f,is (Tree f,1) f 1 (Tree f,) f (Tree f, K f ) f Kf Incrementation / decrementation subflow fraction (Tree f,1) f 1 (Tree f,) f ±δ (Tree f, K f ) f Kf Normalization process (Tree f,1) f * 1 (Tree f,) f ±δ * (Tree f, K f ) f * Kf Fig 5. Subflow fraction mutation following by normalization process. 1 Obtain initial set of valid paths 2 Generate the initial population P of size P max Normalize fractions and remove redundant 3 segments for every chromosome in P 4 Initialize set P nd as empty DO WHILE A FINISHING CRITERIOM IS NOT SATISFIED 5 { 6 Add non-dominated solutions of P into P nd 7 Remove dominated solutions in P nd 8 Calculate fitness of individuals in P and P nd 9 REPEAT P max times { Generate new chromosomes-set C using 10 a Tree crossover with Selection (in P P nd) and normalization process b Flow crossover with Selection (in P P nd ) With probability p m mutate set C and 11 normalization process and remove redundant 12 segments Add to P valid chromosomes in C not yet included in P 13 END REPEAT 14 END WHILE Fig 6: Proposed MOEA 4 Experimental Results Although the aim of this wor is to present a general model and not to discuss the best way to solve it, this section presents a simple problem and the corresponding

5 experimental results using the proposed MOEA, as an illustration of what has been previously stated. 4.1 Networ Topology The chosen topology is the well-nown 14-node NSF (National Science Foundation) networ ( N =14). The costs on the lins represent the delays (d ij ) and all lins are assumed to have 1.5 Mbps of bandwidth capacity (c ij = 1.5 Mbps (i,j) E). Two flows with the same source, s f =N 0, are considered. The egress subsets are T 1 ={N 5, N 9 } and T 2 ={N 4, N 9, N 12 }. The transmission rates are b 1 =256 Kbps for the first flow and b 2 =512 Kbps for the second flow. 4.2 Resolution of the Test Problem A complete set of found non-dominated solutions (best calculated approximation to the optimal Pareto set X* in a run) had 748 chromosomes. This large number is due to the large number of objective functions and the fact that the same set of subtrees with different fractions f of the flows are considered as different solutions because each one represents another compromise between conflicting objective functions. When needed, a clustering technique, that is included in the implemented SPEA, may be used to reduce the number of calculated non-dominated solutions to a maximum desired number [12]. Since there is no space to present all 748 nondominated solutions, Table 1 shows some of the best calculated solutions considering one objective function at a time. Each row (identified by an ID given in the first column) represents a non-dominated solution whose chromosome is omitted to save space. The following 11 columns represent the different objective functions. The cells in bold emphasize an optimal objective value. To better exemplify a solution, certain chromosomes were omitted from Table 1. Some solutions presented in Table 1 are clearly nondominated because they are the best ones in at least one objective, lie the ones with ID=1 with the minimum value of φ 1 or ID=10 with the minimum value of φ 2 and φ 3 to just name a few. However, most solutions are nondominated because they are different compromise solutions. As an example, solutions from ID=1 to ID=15 are all optimal considering φ 4, but each one represents a different compromise between conflicting objective functions. In this example when a flow is not split into subflows, we potentially need the least amount of LSPs (φ 11 =2) and there is no hop count or delay variations between subflows (φ 5 =0, φ 9 =0), as shown in solutions with ID=2, 10, 18 and 20. However, it is possible to have a delay variation (φ 9 >0) even when there is no hop count variation (φ 5 =0) if at least one flow is split (φ 11 >2), lie the non-dominated solutions with ID=16 and 17. Table 1 Some calculated Pareto Front solutions ID φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 10 φ ,7% 26 3, , ,3% 16 3, , ,0% 26 3, , ,3% 25 3, , ,3% 26 3, , ,3% 25 3, , ,3% 27 3, , ,3% 26 3, , ,0% 25 3, , ,3% 15 3, , ,3% 26 3, , ,3% 26 3, , ,0% 28 3, , ,3% 25 3, , ,3% 26 3, , ,0% 25 3, , ,0% 28 4, , ,0% 19 3, , ,3% 153 4, , Correlation Analysis A correlation analysis between each pair of objective functions was also performed to get an idea of the real necessity of using (or not) that large a number of objective functions. A very large correlation clearly means that if one objective function is optimized another one with a high correlation is also indirectly optimized. Table 2 presents these correlation values between the 11 objective functions, considering the whole experimental set of 748 non-dominated solutions. Table 2 Correlations between objective functions φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 10 φ 11 φ 1-0,22-0,16-0,24-0,26-0,25 0,19-0,28-0,19 0,09-0,20 φ 2 0,69 0,77 0,80 1,00-0,80 0,78 0,78 0,56 0,98 φ 3 0,78 0,48 0,66-0,35 0,62 0,41 0,52 0,59 φ 4 0,82 0,76-0,61 0,90 0,73 0,47 0,72 φ 5 0,81-0,81 0,85 0,93 0,34 0,80 φ 6-0,81 0,79 0,78 0,53 0,98 φ 7-0,69-0,78-0,36-0,85 φ 8 0,83 0,40 0,75 φ 9 0,33 0,77 φ 10 0,53 As shown in Table 2 there are very large correlations between some objectives, such as: the total hop count (φ 2 ) and total delay (φ 6 ), with a correlation of almost 1, this is easy to understand given that a longer path usually implies more delay; the total hop count (φ 2 ) and number of subflows (φ 11 ), with a value of 0.98, given that the use of splitting implies the use of multiple-routes and therefore, more lins;

6 the maximal hop count (φ 4 ) and maximal delay (φ 8 ), with a correlation of 0,90, because the longest path in the hop count normally has the longest delay. Since the same reasoning applies for the minimal path, it is also easy to understand the high correlation of 0,93 for the hop count variation (φ 5 ) and delay variation (φ 9 ). Finally, the total delay (φ 6 ) and the number of subflows (φ 11 ) with 0.98, as a logical consequence of the high correlation of both objective functions with the total hop count (φ 2 ). More experimental results are needed to mae a final conclusion, but it is clear that all objective functions are not really needed at the same time. We have considered them for sae of completeness, just to mae sure that an optimal solution from a previous wor that consider a given objective will also be a solution of the GMMmodel. 5 Conclusions Any optimal single objective solution, lie the ones proposed in several previous papers, would be a solution of the GMM-model or dominated by a Pareto solution of it when all analysed objective functions are simultaneously considered. To solve the presented GMM-model, a Multi-Objective Evolutionary Algorithm (MOEA) inspired by the Strength Pareto Evolutionary Algorithm (SPEA) has been implemented, proposing new encoding process to represent multitree-multicast solutions using splitting. This MOEA found a set of 748 non-dominated solutions for a very simple multicast test problem based on the well-nown NSF networ. A correlation analysis of this set of non-dominated solutions was also included, emphasizing that several objective functions are highly correlated and therefore, not really needed for some practical applications. For future wor, we plan to improve MOEA solving more complex problems, considering different topologies and including some objective functions that still haven t been considered, such as Pacet Loss. [3] J. Crichigno B. Barán. Multiobjective Multicast Routing Algorithm for Traffic Engineering. IEEE ICCCN [4] Y. Donoso, R. Fabregat, J. Marzo. Multi- Objective Optimization Algorithm for Multicast Routing with Traffic Engineering. IEEE ICN [5] R. Fabregat, Y. Donoso, J.L. Marzo, A. Ariza. A Multi-Objective Multipath Routing Algorithm for Multicast Flows. SPECTS [6] J.C. Chen, and S.H. Chan. "Multipath Routing for Video Unicast over Bandwidth-Limited Networs". GLOBECOM [7] X. Cui, C. Lin, Y. Wei. A Multiobjective Model for QoS Multicast Routing Based on Genetic Algorithm. ICCNMC [8] A. Roy, S.K. Das. QM2RP: A QoS-based Mobile Multicast Routing Protocol using Multi- Objective Genetic Algorithm. The Journal of Mobile Communication, Computation and Information, Wireless Networs, May [9] Y. Cui, K. Xu, J. Wu. Precomputation for Multiconstrained QoS Routing in High-speed Networs. INFOCOM [10] D. A. van Veldhuizen. Multiobjective Evolutionary Algorithms: Classifications, Analyses and New Innovations. Ph.D. thesis. Air Force Institute of Technology, [11] C. Kim; Y. Choi; Y. Seo. Y. Lee. A Constrained Multipath Traffic Engineering Scheme for MPLS Networs. ICC [12] E. Zitzler and L. Thiele. Multiobjective Evolutionary Algorithm: A comparative case study and the Strength Pareto Approach. IEEE Trans. Evolutionary Computation. Vol. 3, No. 4, [13] B. Barán, R. Fabregat, Y. Donoso, F. Solano, JL. Marzo. Generalized Multiobjective Multitree model. Proceedings of IIiA Research Report, ref. IIiA RR, Institut d'informàtica i Aplicacions, University of Girona. September References [1] Y. Seo, Y. Lee, Y. Choi, C. Kim. Explicit Multicast Routing Algorithms for Constrained Traffic Engineering. IEEE ISCC [2] J. Crichigno, B. Barán. Multiobjective Multicast Routing Algorithm. IEEE ICT 2004.

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