PPX. Precedence-based Crossover. A subset of precedence relations of the parents genes are preserved in the offspring. A binary random mask is generated,.
An Experimental Comparison of Two Different Encoding Schemes for the Location of Base Stations in Cellular Networks Carlos A. Brizuela and Everardo Guti´errez Computer Science Department, CICESE Research Center Km 107 Carr. Tijuana-Ensenada, Ensenada, B.C., M´exico {cbrizuel, egutierr}@cicese.mx, +52–646–1750500
Abstract. This paper presents preliminary results on the comparison of binary and integer based representations for the Base Stations Location (BSL) problem. The simplest model of this problem, which is already NP-complete, is dealt with to compare also different crossover operators. Experimental results support the hypothesis that the integer based representation with a specific crossover operator outperforms the more traditional binary one for a very specific set of instances.
1
Introduction
The planning process of cellular radio networks provides a vast class of combinatorial problems. All these problems have a great importance from the application point of view. Most of these problems are of extreme complexity. One of the most studied problem has been the frequency assignment problem (FAP), for a review see [14]. Other problems are the assignment of cells to switches (ACS) in order to reduce the handoff and wiring costs [12], optimal location of base stations (BSL) [11, 8, 5, 4, 10], among others. These problems (FAP, ACS, and BSL) are equivalent to the following traditional combinatorial optimization problems: graph coloring, bin packing, and minimum dominating set, respectively. There have been many proposed methodologies to find good approximation algorithms for this kind of problem. Some of these methodologies try to find provably good approximations [2], but with very conservative results. Another successful approach is to use meta-heuristics like Genetic Algorithms (GA), Tabu Search (TS), Simulated Annealing (SA), and others (see [1]). The main drawback (by definition of heuristics) of these approaches as opposed to approximation algorithms [2, 9], is that they cannot guarantee a given performance for any instance of the problem. In this paper we deal with the Base Station Location (BSL) problem which is a variant of the Minimum Dominating Set (MDS) problem. A problem whose decision version belongs to the NP-complete class. Previous works using GA’s to deal with this problem considers a binary encoding [4, 10]. Here, we propose and conjecture that an integer based representation could be more appropriate for this problem.
The remainder of this paper is organized as follows. Section 2 states the problem we are trying to solve. Section 3 proposes the methodology. Section 4 presents the experimental setup and results. Finally, section 5 gives the conclusions and ideas for future research.
2
Problem Statement
The BSL problem in Cellular Radio Networks consist of a set of demand nodes DN and a set of possible base station locations B. The problem is to decide where to build the BS’s such that the maximum possible number of demand nodes are covered, subject to some restrictions related to interference, limited budget, maximum number of built BS’s in a given area, among others. A model for this problem is proposed in [5] and one of the requirements is: (R2) Every possible base station location i ∈ B has a maximal broadcast range rmax (i). The actual broadcast range of the built BS has to be determined and is denoted as r(i) ≤ rmax (i). The sets Ni of those demand nodes that are supplied by i are then given by Ni = {j|dist(i, j) ≤ r(i)}. Instead we propose that the range for a given base station i ∈ B is defined by a function fi (d, θ, h), where d is the distance from the base station i to a point in the space, θ the angle with respect to a given reference line, and h is the relative height between the BS and the point in the space. With this function we let open the possibility to include all restrictions that may appear in real-world problems. If we make θ and h constant then we have the same requirement as in [5]. Once this function is defined we can represent the problem as a graph G = (DN, E) where DN, the set of vertices, represents the demand nodes and E, the set of edges, represents the set of demand nodes and their corresponding covering BS’s. That is, the presence of the edge (u, v) ∈ E means that both vertices (v, u) are demand nodes, and one of them v/u is a possible base station location covering the demand node u/v, respectively. In real-world applications we need to know the minimum number of base stations needed to provide a given coverage. This problem is a variant of what is called Minimum Dominating Set (MDS) problem (see [11]). In order to state the variant let us first introduce the definition of the MDS problem itself. Definition 1. Minimum Dominating Set Problem [11]. Given a graph G = (V, E) where V is the set of vertices and E the set of edges, a subset V0 ⊆ V is said to be dominating if for every vertex v ∈ V − V 0 there is an edge connecting to a vertex u ∈ V0 such that (u, v) ∈ E. The MDS problem consists of finding such a V’ of minimum cardinality |V0 |. Then the variant can be stated as follows: given a number q, find the MDS of a subset S⊆ V such that q = |S|. If the cardinality of V is N then we need N MDS problems in the worst case. to solve q The MDS problem was proven to be NP-complete ([6], page 190). Furthermore, non-approximability results show that it cannot be approximated within
(1 − )ln|V| for any > 0 ([2], page 370). This result implies that for the worst case scenario even approximations to this problem are difficult to solve. This situation motivates the design of efficient meta-heuristics for specific instances.
3
Methodology
Approximation algorithms in the concept of Hochbaum [9] and Ausiello et al. [2] and meta-heuristics have been the two main approaches to tackle this problem and its variants. The method we propose here is based on Genetic Algorithms and falls in the second category. Genetic Algorithms to solve this problem were previously proposed by Cal´egari [4], Krishnamachari and Wicker [10], and Meunier et al. [13]. 3.1
Encoding
Previous works based on GA’s [4, 10] proposed a binary representation for each solution. In this representation, the locus indicates the BS number, a value of 1, at this locus, indicates that the location is selected, and a value of 0 indicates that it is not. The length of the chromosome is given by the number of possible locations. Cal´egari [4] proposes a parallel GA and its main contribution is in the parallel model he uses. Krishnamachari and Wicker [10] compare different meta-heuristics (GA, Tabu Search (TS), Simulated Annealing) and Random Walk for this problem. They conclude that the GA along with TS provide the most consistent results. Maunier et al. [13] presents a GA to solve the problem and considers three different objective functions: minimize the number of selected sites, maximize the throughput, and minimize the interference with some additional constraints. They apparently use the binary encoding for the location of BS, although it is not explicitly indicated in the document [13]. We propose an integer based representation where each integer value indicates the number of the selected base station. The motivation behind the use of an integer based representation is that we can explicitly fix the maximum number of desired base stations by means of the chromosome length. This is not easily done with the binary encoding since we need a mechanism to control the maximum number of one’s in the chromosome. The mechanism implies an extra computational cost. In our proposed representation each gene in the chromosome, e.g. C = [1375], indicates the number of selected locations, i.e. locations 1,3,5, and 7 are selected. This representation may contain zeros and its maximal length will be the number of possible locations. By looking at this representation and the binary representation we may see no difference at all. However, in the integer representation, as it was mentioned before, we can fix the length of the chromosome, and this will not be a direct thing to do in the binary representation. Another important point in using this representation is that it is possible to design specific crossover operators that are not easily implemented with the binary representation.
The conjecture is that the neighborhood generated by this representation and its crossover operators efficiently exploit the combinatorial structure of the problem. 3.2
Algorithm
Here we present the pseudocode for the implemented algorithm. 1. for i = 1 to P op Size 2. do Generate Individual(i) 3. for i = 1 to P op Size 4. do Evaluate Objective Function(i) 5. for i = 1 to P op Size 6. do Select an Individual by RWS and save it in Temporal Population 7. Replace Population with Temporal Population 8. for i = 1 to N um Of Iter 9. do Crossover with probability P c 10. Apply Generational Replacement 11. Apply Mutation with probability P m 12. for j = 1 to P op Size 13. do Evaluate Objective Function(j) 14. for j = 1 to P op Size 15. do Select by RWS and save it in Temporal Population 16. Replace Population with Temporal Population In this algorithm P op size is the algorithm parameter indicating the population size. The function Generate Individual(i) randomly generates an individual. Evaluate Objective Function(i) computes the fitness for each individual i, two different functions are implemented, minimization of fA = k RNβ [10] and maxγ imization of fB = RN [4]. Here, N is the number of selected locations, R is the covered area rate, i.e the number of covered nodes over the demand nodes, k, β, γ are constants used to give more or less importance to the minimization/maximization of the number of selected locations or the covered area rate. The selection mechanism is the Roulette Wheel Selection (RWS). The number of iterations is given by N um Of Iter. The crossover rate and the mutation rate are given by P c and P m, respectively. Crossover All crossover operators we use here are standard in principle. The modification occurs when we are trying to copy a gene that has been already copied into the offspring. In this case a zero is copied in the offspring. PMX. Partially-Mapped Crossover. In this crossover two random positions are selected. Figure 1 illustrates how this operator works, genes from loci 1 to s1 − 1 of parent 1 are copied to loci 1 to s1 − 1 in the offspring. Loci s1 to s2 of parent 2 are copied to loci s1 to s2 in the offspring, and finally loci s2 + 1 to n of parent 1 are copied to loci s2 + 1 to n in the offspring. In every situation if we are trying to copy a gene that have been already copied in the offspring
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Fig. 2. Mutation operator OP1.
then a zero is copied instead. This situation generates a conflict as it is shown in Fig. 1. This is a modified version of the PMX explained in the book by Gen and Cheng ([7], page 5). PPX. Precedence-based Crossover. A subset of precedence relations of the parents genes are preserved in the offspring. A binary random mask is generated, the 1’s indicate that genes from parent 1 are to be copied and the 0’s indicate that genes from parent 2 are to be copied, in the order they appear from left to right. The same conflict resolution as in PMX is applied. This is a modified version of the PPX presented in [3]. TPX. The position of some genes corresponding to parents are preserved in the offspring in the order they appear. As in PPX a random mask is generated, however, for this operator the 1’s indicate the selected loci to copy from parent 1, and the 0’s indicate the loci to be copied from parent 2. Again the conflict resolution of PMX is used. This is a modified version of the standard two point crossover that can be found elsewhere ([7], page 408). Mutation After a set of trials with different mutation operators we have decided to use a single mutation operator which is described in the following. OP1. The mutation locus is randomly selected, then the gene is replaced by a randomly generated integer in the range 1 to n. If the new value already exists in the chromosome, then the gene is replaced by a zero. Figure 2 illustrates this operator.
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Fig. 3. Grid and cell structure for the artificial problem. M xM grids grouped in P xP cells of QxQ grids
4
Experimental Setup and Results
The instance data are generated in two different ways. The first one is based on the work by Cal´egari [4] and the second one on the work by Krishnamachari and Wicker [10]. These methods are explained in the following subsections. 4.1
Uniform Generation
This artificial model considers L possible locations distributed in a grid structure of M xM . The grid model is divided into P xP sub-grids, each of size QxQ grids, this structure is shown in Fig. 3. The first P 2 possible locations are fixed in the center of each cell such that these locations have a coverage of 100%, representing the optimal solution. Another L − P 2 possible locations are set randomly and uniformly over the whole M xM grid structure. Three different instances were generated, the parameters for each instance are shown in Table 1.
Table 1. Three instances for the artificial problem Instance Area (M xM ) Cell Size (QxQ) Optimal Number of Cells I1 287 x 287 41 x 41 49 I2 217 x 217 31 x 31 49 I3 147 x 147 21 x 21 49
Two different experiments are performed over these instances. For the first experiment the length of the chromosome (integer encoding) is fixed to a maximum length of 49, which is the optimal number of needed BS’s. The second experiment considers a chromosome length equal to the number of possible locations, i.e. equals the number of bits in a binary encoding. We study three different crossover operators for the integer based representation and a single operator (one point crossover) for the binary representation, as it is used in [4].
Table 2. Mean objective value for instance I1. P c = 0.6, P m = 0.6, N um Of Iter = 1000, N um runs = 50, P op Size = 30, and γ = 4 Chromosome Length Crossover Mean Objective Value (f¯B ) Standard Deviation (%) ppx 0.0204033 0.16 tp 0.0204001 0.27 K pmx 0.0202978 1.19 ppx 0.0164076 4.23 tp 0.0167186 3.70 KN pmx 0.0176148 4.86 Binary 0.0168669 5.87
Table 3. Mean number of base stations for instance I1. P c = 0.6, P m = 0.6, N um Of Iter = 1000, N um runs = 50, P op Size = 30, and γ = 4 ¯ ) Standard Deviation (%) Chromosome Length Crossover Mean Number of BS (N ppx 49.0 0.00 tp 49.0 0.00 K pmx 49.0 0.00 ppx 56.8 3.50 tp 56.0 3.52 KN pmx 53.1 3.08 Binary 53.7 3.53
The mean objective function, the mean covered area, and the mean number of selected base stations over the whole set of runs is computed. This is done for each proposed algorithm. The parameters are fixed according to Cal´egari [4] where the crossover rate P c = 0.6, the mutation rate P m = 0.6, the maximum number of iterations is set to 1000, the population size is fixed to 30, and γ = 4. Table 2 shows the mean objective function for both experiments. K indicates chromosome length of 49, and KN a chromosome of full length. For the KN case, we see that the integer encoding with PMX operator outperforms the binary encoding in terms of the objective function mean and standard deviation (maximization problem). Table 3 shows the mean number of BS, in this case also PMX outperforms the binary representation (minimization). Table 4 shows the results for mean coverage, again PMX outperforms the binary encoding (maximization). For all cases the algorithm performance gets worse when changing from K to KN . This could be due to higher population diversity for KN (longer chromosome), and the required number of iterations to reach convergence.
Table 4. Mean Coverage for instance I1. P c = 0.6, P m = 0.6, N um Of Iter = 1000, N um runs = 50, P op Size = 30, and γ = 4 ¯ Standard Deviation (%) Chromosome Length Crossover % Mean Coverage (R) ppx 99.99 0.04 tp 99.99 0.07 K pmx 99.86 0.30 ppx 98.20 0.77 tp 98.35 0.53 KN pmx 98.29 0.68 Binary 97.50 0.78
4.2
Shadow Fading Model
This approach, for instance generation, was proposed by Krishnamachari and Wicker [10]. The idea here is that the covered area is determined by what is called uncorrelated log-normal shadow fading model. In this model, the power loss in dB at a distance d from the base station is given by the equation [10] Ploss = A+Blog(d)+G, where G is a zero-mean Gaussian random variable with variance σ 2 , A and B are constants, as in [10] A = 50, B = 40, and σ 2 = 10. The Ploss is computed for every base station and every point in the grid, if Ploss < 100 dB at a given node then the node is covered by base station i. Notice that the demand nodes are considered to be the whole grid space. This is done without loss of generality since, the specific set of demand nodes can be included at any moment. In this case the number of possible locations is randomly generated over a whole grid of size M xM . Fig. 4 a) shows a randomly generated pattern for 51 possible locations, Fig. 4 b) shows the covered area for one of the 51 base stations (gray node at the center), and Fig. 4 c) shows the covered area when the 51 locations are selected.
Fig. 4. Possible Base Station Location and Coverage. a) Random Generation of 51 BS. b) Coverage of a single BS. c) Coverage when all 51 BS’s are selected
Fig. 5. Mean objective function values. P c = 0.7, P m = 0.5, N um Of Iter = 1000, N um Of runs = 50, β = 3, k = 10000
Fig. 5 presents the mean values for the objective function over 50 runs, for different population sizes, and different crossover operators. The parameters values used in each case are indicated in the figure caption. In this case we see a clear superiority of the PMX crossover operator over the others and over the binary representation in terms of objective function optimality (minimization problem). Fig. 6 a) shows the mean coverage, here, the binary shows poorer results than the other operators (maximization). In Fig. 6 b) we see results for the mean number of selected base stations, in this case, the binary encoding presents the best result for all population sizes (minimization). Remember that what the algorithm should optimize is the objective function, nor the covered area neither the number of base stations. Fig. 7 shows a sub-optimal solution of 13 Base Stations and a coverage of 92.33% obtained with the PMX operator.
Fig. 6. Mean number of selected base stations, and coverage for different population sizes. a) Mean Coverage and b) Mean Number of BS’s. P c = 0.7, P m = 0.5, N um Of Iter = 1000, N um Of runs = 50, β = 3, k = 10000
Fig. 7. Sub-optimal solution with 13 selected base stations. Coverage 92.33%. PMX crossover. P c = 0.7, P m = 0.5, N um Of Iter = 1000, N um Of runs = 50, β = 3, k = 1000
5
Conclusions
An integer based representation for the base station location problem has been proposed. The representation along with a particular crossover (PMX) outperforms a binary representation that uses standard operators, over a specific set of instances. In this representation the maximum number of desired base stations can be directly specified by the chromosome length helping to fix the maximum number of base stations to be used. The representations are compared over two different set of instances. A set of easy instances where the optimal number of base stations is know and easy to compute. The other set, a difficult one, where we do not know the optimal number of base stations and it is not easy to compute. For this case (the difficult one) a clear superiority of the integer representation is observed. Two main paths are in our interest for future research. The first one is related to the study of how the generated neighborhoods exploit the combinatorial structure of the problem to get better results. The second one is related to the average case performance analysis of the approximation algorithms proposed by Glaβer [8] and their comparison with our proposed procedure.
Acknowledgments The authors would like to the thank the anonymous referees for their useful comments and interesting ideas for future research.
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