Design of a Heuristic Topology Generation Algorithm in Multi-Domain Optical Networks

Designing an excellent original topology not only improves the accuracy of routing, but also improves the restoring rate of failure. In this paper, we propose a new heuristic topology generation algorithm—GA-PODCC (Genetic Algorithm based on the Pareoto Optimality of Delay, Configuration and Consumption), which utilizes a genetic algorithm to optimize the link delay and resource configuration/consumption. The novelty lies in designing the two stages of genetic operation: The first stage is to pick the best population by means of the crossover, mutation, and selection operation; The second stage is to select an excellent individual from the best population. The simulation results show that, using the same number of nodes, GA-PODCC algorithm improves the balance of all the three optimization objectives, maintaining a low level of distortion in topology aggregation.


Introduction
The scale of optical networks is rapidly expanding with the development of optical communication technology [1], the structure of optical networks has also changed greatly.To adapt to the new structure, optical networks are already moving towards multi domain in recent years.The excessive network information, meanwhile, causes bottlenecks in the scalability of optical networks and increases the necessity to provide a better scalability.The generation of original topology is one of the impetuses to solve the problems of scalability in large-scale multi-domain optical networks [2].Meanwhile, original topology also influences the performance of all upper-layer topology-related strategies [3].
The generation of original topology is a NP (Non-deterministic Polynomial) complete problem.Since the network topology is needed in urgent, a heuristic algorithm is proposed quickly.The available resources are taken as the constraint to determine the network topology.The network topology design problem is formulated as a K-Maximum Spanning Tree Problem with degree bound and has been proven NP-Hard [4].Owing to the 3-Dimensional integration has the higher packing density and the shorter wire length, they design a novel 3D torus topology along with simplified inter-layer and vertical optical routers, in order to achieve a better performance [5].The other paper also addresses an NP-hard problem, referred to as network topology design with minimum cost subject to a reliability constraint (NTD-CR), to design a minimal-cost communication network topology that satisfies a pre-defined reliability constraint [6].For the complex topology network, someone introduces an algorithm to construct a plant topology from analyzing correlations in operations data and iteratively combining pieces of information to the final result [7].Considering the design of survivable virtual network mapping in multi-domain optical networks, the researchers propose a heuristic algorithm for survivable virtual network mapping using a partition and contraction mechanism (PCM) and based on cut set graph theory with the objective of minimizing the total network link cost [8].Most topology generation algorithms aim at optimizing the overall topological performance at the expense of other topological performance when concurrently optimizing some objectives.Therefore, there is a demand to improve the balance among all optimized objectives.

Topology Generation Model
Genetic algorithm utilizes group search to obtain optimal solution, especially for complicated nonlinear problems [9].The model of multi-objective optimization is given before designing the topology generation algorithm.The objective functions are as follows: ( ) ( ) ( ) x represents the situation of link configuration between nodes i and j, where max λ represents the maximum amount of business requests that a link can carry, so the first constraint represents the sum of business requests of link ( ) , i j is less than max λ .i b is the number of optical transmitters owned by node i; The second constraint limits the number of links from node i to other nodes less than i b , representing an out-degree constraint; Similarly, the third constraint is an in-degree constraint; The last constraint limits the value of ij x to 0 or 1.

GA-PODCC Algorithm
Based on the above generation model, we propose GA-PODCC algorithm, which optimizes the link delay, resource allocation and resource consumption simultaneously.In order to continuously approach the optimal population and obtain a solution that optimizes the three objectives, the politropism and full search of solutions are achieved through selection and competition of population between parents and offspring.

Gene, Individual and Population
GA-PODCC algorithm needs to initialize a population which consists of many individuals (also known as chromosome).Figure 1 shows the chromosome's structure, in which each element on the chromosome is a gene of the chromosome and each gene represents the situation of link configuration between two nodes.
Gathering a number of individuals forms a population, the structure of a population is shown in Figure 2.

Crossover Operation
The crossover times of the proposed operation is half of the individual number in parent population.Figure 3 shows a schematic diagram of the crossover operation.The crossover type we adopted is uniform crossover, in which two parent individuals are randomly selected from the parent population, and all the genes on the chromosome are exchanged with a crossover probability c p .Two new individuals are produced after the exchange.Then repeat the above crossover operation until n new individuals are produced.Equation ( 5) provides the crossover probability of the i th gene on the chromosome.

Mutation Operation
Mutation operation is applied after the crossover operation.Combined with the chromosome's structure, the so-called genetic mutation is that gene on chromosome with a certain probability changes from the original 0 to 1, or from 1 to 0.
Figure 4 shows the mutation operation.
The main purpose of introducing mutation operation is to provide the genetic algorithm has a characteristic of being easy to fall into the local optimum.Genetic mutation with a certain probability can make the algorithm jump out of the local optimum.When selecting mutation probability m p , adjustability should be considered.Equation (6) gives the mutation probability of the ith gene on the chromosome.
We set , respectively, providing a low mutation probability and an excellent global searching ability in the initial stage of the algorithm execution.In the middle and later stages of the algorithm execution, the mutation probability becomes higher in order to maintain the diversity of the population and prevent it from falling into the local optimum.Similarly, the mutation type adopted in this paper is uniform mutation.

Selection Operation
Through crossover and mutation operations, n parent individuals in the initial population can produce n new offspring individuals.Here, the selection operation selects n individuals from parent and offspring individuals as the parent individuals of the next generation population 1 P , and the remaining n individuals are eliminated.In this way, while producing more outstanding individuals, the outstanding individuals of the original parent population are preserved.Figure 5 shows the selection operation.( ) ( ) ( )

Genetic Operation
Genetic operation is the last step of GA-PODCC algorithm.The genetic operation is divided into two stages.In the first stage, the crossover, mutation, and selection operation are performed for internal individuals of the population, producing the next generation population.The first stage of genetic operation is repeatedly executed until the genetic algebra equals _ Max Gen .Figure 6 depicts the first stage of the genetic operation.
Subsequently, three sets of the optimal solutions corresponding to different optimization objectives are obtained.There are n optimal solutions for each optimization objective and the amount of total optimal solutions equals to 3n.Each set of optimal solution is sorted from high to low according to the value of fitness function.
The second stage of the genetic operation is to select one optimal solution that optimizes the three objectives from the 3n solutions.The introduction of a new elimination mechanism is required, the average computing time complexity of algorithm is approximately to n 3 .
Figure 7 shows the second stage of the genetic operation.First, the optimal solutions of the link delay are tested in fitness function Equation (8).If the test   8), corresponding optimal solutions of the link delay are remained, or eliminated.In the same way, the optimal solutions of the link delay are tested in fitness function Equation (9).After the two tests, there are a (a < n) optimal solutions of the link delay.Similarly, some optimal solutions of the static resource configuration and consumption are eliminated.The number of the remaining optimal solutions is b (b < n) and c (c < n), respectively.
The second stage of the genetic operation is repeated until the single optimal solution is obtained.The second stage of the genetic operation takes the performance of three optimization objectives into account.All optimal solutions need to be tested by the other two fitness functions, so the last remaining optimal solution must be the best.After executing two stages of the genetic operation, an excellent network topology can be obtained.

Simulation
In addition to the above three performance indexes, a new performance index called generational distance (GD) is added to the simulation experiment.Here, GD is a convergence evaluation index [10] and is used to evaluate the approximation quality to global non-inferior optimal region.It can be acquired by the formula: n is the number of individuals in the non-inferior solution set.i d is the Euclidean distance between the individual and the global non-inferior optimal region.A smaller GD value can lead to a close distance between the population individuals and the global non-inferior optimal region, thus achieve a better performance.

Experimental Scheme 1
Under the same experimental conditions, the algorithm performance indexes of the GA-PODCC, GA-POCDC, and DC-GALD are compared in the experimental Scheme 1.
The simulation parameters of the network model are set as follows: The number of individuals is 50 and the value of _ Max Gen is 5000.When genetic alge- bra equals to 10, 100, 500, 1000, 1500, 2000, 2500, 3000, 3500, 4000, 4500 or 5000, the corresponding performance indexes are calculated respectively for the experimental analysis.Finally, it shows all the topologies generated in the last generation and chooses the best network topology.

Comparison of the Link Delay
The following conclusions can be drawn from the data shown in Figure 8: In Communications and Network terms of the link delay, the solutions found by the GA-PODCC algorithm are significantly better than that of the other two algorithms at the beginning.That is to say, GA-PODCC algorithm has a superior searching ability in the link delay.
When genetic algebra larger than 1000, the optimal solutions in the link delay of the GA-PODCC algorithm basically remains stable.While the link delay of the GA-POCDC and the DC-GALD algorithms is basically stable only when the genetic algebra is larger than 3000 generations.This is because the GA-PODCC algorithm puts the parent individuals and the offspring individuals together to produce individuals of the next population.Since the space of the selection operation is enlarged, it also guarantees that the elite parent individuals can enter into the next population when offspring individuals are produced.

Comparison of the Static Resource Consumption
The following conclusions can be drawn from the data shown in Figure 9: The static resource consumptions of GA-POCDC and GA-PODCC algorithm are almost identical at the beginning, but the GA-PODCC algorithm decreases faster with the increase of genetic algebra and thus converges to the optimal solution earlier.After the genetic algebra is larger than to 200, the static resource consumption of the GA-PODCC algorithm basically remains stable.However, the GA-POCDC algorithm does not stabilize the static resource consumption until

Comparison of the GD Value
The following conclusions can be drawn from the data shown in Figure 10: The GA-PODCC algorithm has a smaller GD value than the other two algorithms.
Because the GA-PODCC algorithm has a fair fitness allocation mechanism, while the other two algorithms adopt an equal probability to the randomly assign fitness value, the GA-PODCC algorithm has better performance in the convergence evaluation.
From the analysis of the above three performance indexes, the GA-PODCC algorithm has less link delay and static resource consumption, obtaining a more satisfactory convergence evaluation index and a better foundation for routing and survivability strategy of optical network.

Experimental Scheme 2
Experimental scheme 2 is designed to verify the impact of the topology generation mechanism on subsequent topology aggregation algorithms.We used ML-S algorithm [11]

Figure 3 .
Figure 3. Schematic diagram of the crossover operation.

_
Max Gen is maximum genetic algebra.After a large number of experimental simulation tests, the three values are set as purpose of crossover operation is to produce new individuals, enabling the genetic algorithm to enhance global searching ability.To eliminate the bias about locations and orders of genes on chromosomes, uniform crossover is chosen.Therefore, the search of non-inferior solutions can be improved.L. Wang et al.DOI: 10.4236/cn.2018.10300669 Communications and Network The key of the selection operation is to determine the selection mechanism, which means that to figureoutthe design of the fitness function.The design principle of the fitness function follows that the value of individual fitness function is directly proportional to the adaptability.The design of the fitness function needs to consider three objective functions.The first objective function represents the total link delay of the network, which is a minimum objective function.Therefore, the design of the fitness function is to invert the minimum objective function to the maximum objective function, and use the inverted function as the first fitness function Equation (7).Similarly, the other two fitness functions follow the same design.

Figure 4 .
Figure 4. Schematic diagram of the mutation operation.

Figure 5 .
Figure 5. Schematic diagram of the selection operation.

Figure 6 .
Figure 6.Flow chart of the first stage in the genetic operation.

Figure 7 .
Figure 7. Second stage of the genetic operation.

Figure 8 .
Figure 8.Comparison of the link delay of three algorithms.

Figure 9 .
Figure 9.Comparison of the static resource consumption of three algorithms.

11 )Figure 10 .
Figure 10.Comparison of the GD value of three algorithms.

Figure 11 .
Figure 11.Comparison of the aggregation performance indexes under two different topologies.