# Selecting the Best Spanning Tree in Metro Ethernet Networks using Genetic Algorithm

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3 Chromosome Representation For GA approach, it is important to figure out the adequate chromosome representation of problem. One of the classical theorems in graphical enumeration is Cayleys s theorem [5]. It states that there are N (N-2) distinct labeled trees for a complete graph with N vertices. Prüfer provided a constructive proof of Cayley s theorem establishing a one-to-one correspondence between such trees and the set of all permutation of N-2 digits. This means we can describe our tree with N-2 uniquely digits for N vertices. We name it Prüfer Number. You can see a simple tree and its Prüfer number in figure (1). We use Prüfer number as a chromosome in GA approach. The Prüfer number encoding procedure is as below: Step 1) Let i be the smallest leaf node and node j be incident to node i. Set j be the first digit in the encoding. The encoding is built by appending digits to the right. Step 2) Remove node i and the edge from i to j. Step 3) Repeat above operation until only one edge is left. In a Prüfer number encoding, a tree is encoding as a Prüfer vector and a set of its eligible nodes (the set of all node not included in P). The decoding procedure is as below: Step 1) Let node i be the smallest eligible node of and node j be the leftmost element of P. If i j, add the edge (i,j) into the tree T. If i is no longer eligible, then remove node i from. Delete j from P. If j does not occur anywhere in the remaining part of P, then put it into. Repeat the process until P is empty. Step 2) For the remaining last two nodes u and v of, add the edge (u,v) into the tree T. 3.2 Crossover and Mutation The genetic algorithm uses the individuals in the current generation to create the children that make up the next generation. Besides elite offspring, which correspond to the individuals in the current generation with the best fitness values, the algorithm creates: Crossover offspring by selecting vector entries, or genes, from a pair of individuals in the current generation and combines them to form a child. Mutation offspring by applying random changes to a single individual in the current generation to create a child. Crossover and Mutation are two crucial factors in the biological evolutionary process. There are several types of crossover operator such as: single point, two point, uniform, and etc. In this paper, we use uniform crossover operator. Uniform crossover at first generates a uniformly random crossover mask and then exchanges relative genes between parents according to the mask. Figure (2) illustrates the crossover operation. Mutation operator forces random changes in various chromosomes. It guarantees that we will not be placed in local minimum. There are several types of mutation operation, such as: uniform, Gaussian, and etc. In this paper, we use uniform mutation operator. Figure (3) illustrates the mutation operation. Mask Parent 1 Parent Offspring Offspring Fig. 2: Illustration of crossover operation Fig.1: A tree and corresponding Prüfer Number. Genotypes (chromosome values) are uniquely mapped on to the decision variables (phenotypic) domain. Fig. 3: Illustration of mutation operation 3.3 Evaluation and Selection Operator In GA approach, we need to evaluate our population by an evaluation function. We name it f_obj(x), in which X is the Chromosome (In our research is Prüfer number). The

4 109 main idea is to minimize the f_obj(x). It is clear that for the maximization, we need to minimize the g_obj(x)=f_obj(x). In GA approach, the evaluation function is used to select the best chromosomes from the population. There are several ways to select the selection operator, such as: Stochastic Uniform, Uniform, Roulette, and etc. In this paper, we use Roulette wheel. Each slice in Roulette wheel is proportional to its fitness value. Figure(4) illustrates an example of Roulette wheel Fig. 4: An example of Roulette Wheel. 2 1 to generate two offspring. All offspring constitute a set denoted by S. Step 3) (Mutation) Selection the parents for mutation from set S with probability P m. For each chosen parent, the proposed mutation operator is applied to it to generate a new offspring. These new offspring constitute a set denoted by S 1. Step 4) (Selection) Select the best N individuals among the set P(t) S 1 as the next generation population P(t+1), let t=t+1; Step 5) (Termination) If termination condition hold, then stop, and keep the best solution obtained as the approximate global optimal solution of the problem; otherwise, go to step 2. We used 500 generation as our termination condition. For GA approach, we need chromosome and evaluation function. We define the chromosome with three genes as defined in Figure (5). The chromosome length depends on the number of nodes we will use in our network. To define the evaluation function, we first define the normalized costs as: (6) The total cost that must be minimized is: (9) We use the total cost as our evaluation function. So, the GA approach, will select the spanning tree with the minimum total cost as the best. The overall GA procedure is outlined as follows: Step 1) (Initialization) Choose the population size N, proper crossover probability P c and mutation probability P m, respectively. Generate initial population P(0). Let the generation number t=0; Step 2) (Crossover) Choose the parents from P(t) with probability P c for crossover. If the number of parents chosen is odd, then randomly choose additional one from P(t). Afterwards, randomly match every two parents as a pair and use the proposed crossover operator to each pair (7) (8) Fig. 5: Chromosome with three genes After finding the best spanning tree, we can force STP to select it by assigning proper values to link costs and switch IDs. 4. Simulation Results In this section, we show the computational complexity efficiency of proposed GA approach compared with our previous work [3]. We simulated our algorithm with MATLAB R2007b [6]. Figure (6) shows a simple network graph with 5 switches and 8 links. Figure 6. A simple network graph

5 110 Assume the same links with bandwidth equal to 1Gbps and the same switches with switching capacity equal to 4Gbps. There is 50 Mbps traffic demands from node 5 to node 2 and 100 Mbps from node 5 to all other nodes. Now, consider the link load balancing (LLB) criterion 1, 0, 0. First, we examine the default spanning tree created by standard STP showed in figure (7). For simulation of standard STP, we used OPNET simulator tool [7]. Figure7. Spanning tree created by standard STP The best spanning tree selected by our GA approach based on LLB is shown in figure (8): Figure 8. Spanning tree created by GA approach Now, we can see the effectiveness of our approach in terms of link load balancing. As mentioned before, variance of the normalized link loads ( ) is a good metric for comparison. This metric is for STP and for our approach. In our previous work [3], we compute the total cost for each spanning tree and then we select the best one with minimum total cost. This algorithm is very time consuming, especially when our network is large. In this paper, we proposed a new approach using Genetic Algorithm. It reduces the computational complexity by selecting the best spanning tree in a stochastic manner. Prüfer number can be encoded and decoded in times that are O(NlogN)[4]. This part of calculation is common between our previous algorithm and our new GA approach. In our previous work, there was N (N-2) spanning trees that must be evaluated, and for each spanning tree, we must calculate its costs that is O(N). Therefore, for each spanning tree, we have a computational complexity of O(N 2 logn) so, the computational complexity order of our previous method is : Where A is a constant. In Table (1), you can see the complexity values of our previous algorithm calculated according to formula (10) for different values of nodes (N). It is difficult to determine the computational complexity of GA approach, but we can estimate it by heuristic methods. One of the proposed heuristic methods in literature is evaluating the complexity by O(x K ) [8], where x is complexity order of coding and decoding in GA approach and K is an integer number. As mentioned before, the coding and decoding complexity is O(NlogN). To estimate the complexity in our GA approach, we simulated four different cases and the real complexity is measured for each case. Results can be seen in Table (2). By considering these results and above mentioned notes, the complexity of our GA approach can be described by: log (11) where B is a constant coefficient. By replacing different values of K, (K=2,3,4) in (11) and comparing it with values indicated in Table (2), we found that K=3 is the best estimation. Therefore, the computational complexity of our GA approach can be estimated as: log (12) The comparison between measured (real) complexity values and estimated values is indicated in Table (3). GA is a stochastic process, so the measured complexity values may vary in different simulation tries for the same network. Therefore, although the estimated values are not exactly the same as measured ones, they are good estimations for the complexity order of our GA algorithm. Figures (9) and (10) show the complexity of our previous approach and GA approach, respectively. As you can see, the complexity of our previous approach is very high for medium and large networks. In such cases, GA approach is helpful. Nodes=5 Nodes=10 Nodes=20 Nodes=50 Table 1: Complexity of our previous approach Traditional Algorithm Complexity e e e e+083 log (10)

6 111 Order of Complexity Table 2: Measured Complexity of GA approach Nodes=5 330 Nodes=10 Nodes=20 Nodes=50 GA Complexity e e e+006 Table 3: Comparison between measured complexity values and estimated values for GA approach GA Complexity (Real) GA Complexity (Estimated) Nodes= Nodes= e e+004 Nodes= e e+005 Nodes= e e N Fig. 9: Complexity Our previous algorithm 5. Conclusion In this paper, we introduced a new Genetic Algorithm approach for finding the best spanning tree in a Metro Ethernet network. We select the best tree not only based on shortest path selection but also based on load balancing on links and switches. We defined three coefficients corresponding to above criterions respectively. This allows us to weight the importance of each criterion based on our goal. After finding the best spanning tree, we can force STP to select it by assigning proper values to link costs and switch IDs. In our previous algorithm, we search in all possible spanning trees whereas in this paper we showed that GA approach can reduce the computational complexity. References [1] M. Huynh, and P. Mohapatra, Metropolitan Ethernet Network: A move from LAN to MAN, Computer Networks 17, Elsevier, pp , [2] P.M.V. Nair, Quality of service in Metro Ethernet, Ph.D. thesis, Southern Methodist University, [3] G. Mirjalily, H. Karimi, F. Adibnia and S. Rajai, A new approach to select best spanning tree in Metro Ethernet Networks, accepted in 8 th IEEE CIT conf., Sydney, Australia, July [4] L. Hanr, and Y. Wang, A novel Genetic Algorithm for Degree-Constrained Minimum Spanning Tree Problem, IJCSNS, vol. 6, no. 7A, pp , [5] A. Cayley, A theorem on trees, Quarterly Journal of mathematics 23, pp , [6] A. Chipperfield, P. Fleming, H. Pohlheim, and C. Fonseca, Genetic Algorithm TOOLBOX for use MATLAB, MATLAB user s guide, V. 1.2, [7] OPNET Technologies, [8] J. A. Foster, Genetic Algorithm hardness and Approximation Complexity, Technical report No. LAL 9504, University of Idaho, Order of Complexity N Fig. 10: Estimated Complexity of GA algorithm Farhad Faghani was born in Iran in He received the B.S. degree in Electrical Engineering (Electronic) from Isfahan University of Technology (IUT) in 2004 and M.S. degree in Electrical Engineering (Communication Systems) from Yazd University in 2007 with honors. Now, he is Ph.D. student in Electrical Engineering of Yazd University and Instructor of Islamic Azad University (Najaf-Abad Branch). He has published three books in Electrical Engineering filed for undergraduate students. He is working on Data Communication Networks, specially on IP QoS, Traffic Engineering and Metro Ethernet Networks.

7 112 Mirjalily was born in Iran in He received the B.Eng. degree from Mashhad University, Iran, in 1991 and the MS. degree from K. N. Toosi University of Technology, Iran, in 1994, both in electrical engineering and with honors. In 2000 he completed the Ph.D. program at Tarbiat Modarres University, Iran. He has been a visiting researcher at the Communications Research Laboratory, McMaster University, Canada in Since Fall 2001, he has been with Yazd University, Iran as an assistant professor. His current research interests include IP QoS, Traffic Engineering and Metro Ethernet Networks. Dr. Mirjalily is a member of IAEE, IAENG and IEEE. He has published over 30 papers in national and international journals and conference proceedings. He has a grant from Iranian Telecommunication Research Center (ITRC) to design Metro Ethernet Network for Yazd state.

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