Fast triangle flip bat algorithm based on curve strategy and rank transformation to improve DV-Hop performance

Abstract

The information of localization is a fundamental requirement in wireless sensor network (WSN). The method of distance vector-hop (DV-Hop), a range-free localization algorithm, can locate the ordinary nodes by utilizing the connectivity and multi-hop transmission. However, the error of the estimated distance between the beacon nodes and ordinary nodes is too large. In order to enhance the positioning precision of DV-Hop, fast triangle flip bat algorithm, which is based on curve strategy and rank transformation (FTBA-TCR) is proposed. The rank is introduced to directly select individuals in the population of each generation, which arranges all individuals according to their merits and a threshold is set to get the better solution. To test the algorithm performance, the CEC2013 test suite is used to check out the algorithm's performance. Meanwhile, there are four other algorithms are compared with the proposed algorithm. The results show that our algorithm is greater than other algorithms. And this algorithm is used to enhance the performance of DV-Hop algorithm. The results show that the proposed algorithm receives the lower average localization error and the best performance by comparing with the other algorithms.

1. Introduction

 Wireless sensor network (WSN) [1] is a system, which mainly compose of three parts: node, gateway and software. It is an important issue [2] for WSN [3] to locate the information of nodes accurately [4]. Such as mine search and rescue [5], the coordinates of the position should be obtained when survivors are found. In view of the cost, most wireless nodes will not install GPS and other localization devices. Therefore, how to use fewer of the beacon nodes (nodes with GSP localization device) to predict the location of other ordinary nodes (nodes without localization device) has become an urgent problem [6] to be solved.

 Due to the location information is crucial to WSN [7], various localization algorithms [8] have been proposed to improve the positioning accuracy of sensor nodes. And the classification of range-free and range-based algorithm is distinguished according to whether the distances need to calculate accurately. The range-based locating algorithm includes time of arrival algorithm (TOA) [9], Ad hoc positioning system (APS) [10] and received signal strength indicator algorithm (RSSI) [11], also the range-free locating algorithm includes the distance vector hop algorithm (DV-Hop) [12], convex position estimation (CPE) [13], and multi-dimensional scaling (MDS) [14]. The DV-Hop [15] positioning processes are implemented by multi-hop information and distance estimation. The distance between the ordinary node and beacon node is obtained by the following process. Firstly, the ordinary node records the minimum hops to the beacon node, and then calculate roughly the average distance per hop. Finally, the required distance is estimated according to the average distance per hop and the minimum number of hops. To get the accurate value, variety optimization algorithms are used to reduce the error of position [16], such as trilateral measuring.

 Intelligent optimization algorithm [17] is a method constructed based on human cognition and learning experience of nature, which used to solve the complex optimization problem [18]. And in recent years, the bio-inspired optimization algorithm [19] has been applied to various fields [20]. Such as bat algorithm (BA) [21], particle swarm optimization (PSO) [22], firefly algorithm (FA) [23], cuckoo search (CS) [24] and pigeon-inspired optimization algorithm (PIO) [25]. Specifically, a new firefly inspired strategy [26] is proposed to spread and disseminate game in online social networks (OSNs) and decrease the acceptance-discontinuance anomaly. A many-objective optimization algorithm to protect the Privacy protection [27]. An improved cuckoo search algorithm to solve the problem of integer program [28]. A firefly algorithm is used to find the fixed point of a nonlinear function [29]. And most optimization algorithms [30] can be applied to practical problems [31], such as support vector machines [32], 0-1 knapsack problem [33] and IP assignment [34]. A new search algorithm based on cuckoo [35] is proposed to enhance DV-Hop performance. And a hybrid PSO with mutation (HPSOM) [36] is used to detect the code smell.

 Bat algorithm [37], as a kind of swarm intelligence algorithm [38], simulates the echolocation prey behavior of bats to achieve search the optimal solution. And the mode of echolocation behavior is that each bat individual is regarded as a solution of the current the feasible region, each solution can be regarded as a fitness value. Each bat individual chooses a global or local search method according to probability. That is, when the given probability is satisfied, the global search mode is adopted; otherwise, search by local optimization. And each bat follows the current optimal bats by adjusting three parameters, including the pulse wave length, volume, and pulse emissivity. In this way, we can get the optimal solution in searching space. In a word, both of the global and the local search mode adopt the method of random transformation, which means the individuals of the local search in each generation adopt the method of random selection to determine, and the proportion of the local search is determined by the pulse transmission frequency.

 Cai [39] proposed the fast bat algorithm which adopted the triangle reversal curve strategy (FTBA-TC), which is combined with the fast triangular flip and curve decline strategy.  In this paper, we continue to improve FTBA-TC so that the algorithm can balance the proportion of global and local searches. The contribution of this paper is as follows: 1) The rank-based transformation strategy is designed which sort the individual by the fitness value. 2) A threshold is set to dynamically adjust the proportion of performing global search and local search throughout the iteration process, which means that the threshold would change with the increase of the iterator. 3) Both experiments of CEC2013 and DV-Hop are used to demonstrate the proposed algorithm has the best performance.

 The other parts of this paper are introduced as follows: Section 2 describes the localization algorithm of dv-hop. Section 3 describes the idea and implementation process of standard BA in detail. In addition, the main idea of rank-based transformation strategy is introduced emphatically. Section 4 tests our algorithm and applies it to wireless sensor node location. It is effective in improving the accuracy of DV-Hop algorithm that the proposed FTBA-TCR. At the end of the article, the conclusion is drawn in section 5.

 

2. DV-Hop Localization Algorithm

 With the quick development of wireless communication [40], it is crucial to WSN [41] that the position information is attracting more and more attention. The algorithm of range-free and range-based as the branched of the WSN are illustrated in Fig. 1.

 Fig. 1. Classification diagram of localization algorithm

 The fundamentals of DV-Hop algorithm [42] are showed as follows: (1) the average distance per hop of beacon node is estimated according to the information of hop count and distance between beacon nodes; (2) the ordinary node is located by the estimation distance. Where \(\left(B_{1}, B_{2}, B_{3} \cdots, B_{m}\right)\)  indicate m beacon nodes, \(\left(U_{1}, U_{2}, U_{3} \cdots, U_{n}\right)\)  represent  n ordinary nodes. The wireless sensor network has m+n  nodes, consider per hop average distance of the beacon node B_i , after permitting all beacon nodes to broadcast in the network, a certain number of hops (the number of nodes transmitted) are passed. The information of other nodes can be obtained by beacon node \(B_{i} \cdot\left(h_{1}, h_{2}, \cdots, h_{i-1}, h_{i+1}, \cdots, h_{m}\right)\) .   represent the number of hops between beacon node B_i  and B_m . Since the coordinates of beacon nodes is known, the average distance of per hop can be obtained as:

\(H o p_{i}=\frac{\sum_{k \neq i} \sqrt{\left(x_{i}-x_{k}\right)^{2}+\left(y_{i}-y_{k}\right)^{2}}}{\sum_{k \neq i} h_{k}}\)       (1)

where (x_k,y_k)  denotes the coordinates of beacon node B_k.

 In the broadcast process,  B_i may receive multiple hops of B_k  with different paths and nodes. At this time, we only keep the minimum hops received.

 Meanwhile, the relationship among of the beacon nodes, the ordinary nodes and the estimated distance can be expressed as follows:

\(\left\{\begin{array}{c} \left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}=d_{1}^{2} \\ \left(x-x_{2}\right)^{2}+\left(y-y_{2}\right)^{2}=d_{2}^{2} \\ \cdots \cdots \\ \left(x-x_{m}\right)^{2}+\left(y-y_{m}\right)^{2}=d_{m}^{2} \end{array}\right.\)       (2)

\(d_{k}=H o p_{k} \cdot T_{k}\)       (3)

where,d_k  is the estimated distance, the following objective function is obtained by this distance:

\(f(x, y)=\sum_{k=1}^{m} \alpha_{k} \cdot\left|d_{k}^{2}-\left(x-x_{k}\right)^{2}-\left(y-y_{k}\right)^{2}\right|\)       (4)

where, the value of a_k is the reciprocal of the number of hops, which means the larger the number of hops, the smaller the value of a_k .

 

3. Fast Triangle Flip Bat Algorithm Based on Curve Strategy and Rank

3.1 Standard Bat Algorithm

 A heuristic intelligent algorithm [43], bat algorithm [44], simulates the principle of echolocation in bat predation and has the advantages of simple structure, few parameters, strong robustness, easy understanding and implementation. Therefore, it has received extensive attention and has become a hot spot in the field of computational intelligence research [45]. Bat algorithm [46] is designed by Yang in 2010.

 For the minimum objective function  , the variable is  . The attributes of each bats individual is defined as follows:

\(\left\{v_{k}(t), x_{k}(t), f_{k}(t), r_{k}(t), A_{k}(t)\right\}\)       (5)

where \(v_{k}(t)\)  and  \(x_{k}(t)\) denotes the velocity and position of the kth  bat in t  generation, respectively. Meanwhile  \(f_{k}(t), r_{k}(t)\)   and \(A_{k}(t)\)  denotes the frequency, the rate of pulse emission and the loudness, respectively.

 Firstly, the population is initialized, which means that the bat flies at position \(x_{k}(t)\)   with velocity \(v_{k}(t)\) . Meanwhile, the velocity and position are updated by adjusting the frequency. The updating equation is:

\(x_{k}(t)=x_{k}(t-1)+v_{k}(t)\)       (6)

\(v_{k}(t)=v_{k}(t-1)+\left(x_{k}(t)-x_{b e s t}(t)\right) \cdot f_{k}(t)\)       (7)

\(f_{k}(t)=f_{\min }+\left(f_{\max }-f_{\min }\right) \cdot \sigma\)       (8)

where \(x_{\text {best}}(t)\) represents the optimal position in t  generation, and \(\sigma \in[0,1]\) .

 And the local search strategy is showed as follows:

\(x_{k}(t+1)=x_{b e s t}(t)+\delta_{k} \cdot \bar{A}_{k}(t), \quad \text { if } \quad \eta>r_{k}(t)\)       (9)

where \(\delta_{k} \in[-1,1], \eta \in[0,1]\)   and \(\bar{A}_{k}(t)\)  denotes the average loudness.

 The new position is updated only when the conditions are met:

\(\boldsymbol{x}_{k}(t)=\left\{\begin{array}{l} \boldsymbol{x}_{k}^{\prime}(t) \text { if } \beta<A_{k}(t) \text { and } f\left(\boldsymbol{x}_{k}^{\prime}(t)\right)<f\left(\boldsymbol{x}_{k}(t)\right) \\ \boldsymbol{x}_{k}(t-1) \quad \text { otherwise } \end{array}\right.\)

where  \(\beta \in[0,1]\) and \(\boldsymbol{x}_{k}^{\prime}(t)\) denotes the new position updated by Eq. (6) and Eq. (9).

 Furthermore, the updating equations of frequency, rate of pulse emission and loudness are showed as follows:

\(A_{k}(t)=\alpha A_{k}(t-1)\)       (11)

\(r_{k}(t+1)=r(0)[1-\exp (-\gamma t)]\)       (12)

where  \(\alpha \text { and } \gamma\) are predefined parameters, a(0)  and r(0)  are two intial values of loudness and pulse emission, respectively.

 The flow chart of the BA is showed in Fig. 2.

Fig. 2. The standard BA

 

3.2 Rank-based Transformation Strategy

 The fast bat algorithm with triangle flip (FTBA) [47] improves global optimization ability validly. In addition, the FTBA-TC is combining the fast triangular flip and curve decline strategy, which modify the local search ability of the algorithm. According to FTBA, the velocity update equation is showed as follows:

\(\boldsymbol{v}_{k}(t)=\left(\boldsymbol{x}_{m}(t-1)-\boldsymbol{x}_{u}(t-1)\right) \cdot f_{k}(t-1)\)       (13)

\(\boldsymbol{v}_{k}(t)=\left(x_{\text {best}}(t-1)-\boldsymbol{x}_{m}(t-1)\right) \cdot f_{k}(t)\)       (14)

where x_m(t-1)  and x_u(t-1)  refer to two randomly positions in the current generation.

 Based on FTBA-TC [39], the curve decline strategy uses a disturbance parameter \(\tau \cdot x_{\max }\)  to replace the average loudness \(\bar{A}(t)\) .

\(\boldsymbol{x}(t)=\boldsymbol{x}_{b e s t}(t-1)+\boldsymbol{\varepsilon}_{k} \tau(t-1) \cdot x_{\max }\)       (15)

\(\tau(t)=\tau_{\max } \cdot\left[1-\left(\frac{\tau_{\max }-\tau_{\min }}{\tau_{\max } \cdot(L G-1)} \cdot(t-1)\right)^{k 1}\right]^{k 2}\)       (16)

where e_k  is a random vector within [-1,1]  that satisfies uniform distribution, \(\tau(t-1) \cdot x_{\max }\)  is the area search radius,  r(t-1)   decreases linearly with the number of evolutionary iterations increases,   denotes the number of evolutionary iterations,   represents the maximum number of evolutionary iterations, the value of k1  and k2  determined the down trend. The algorithm has the best performance when k1=1  and k2=4 through this experiments.

 In addition, in each generation of bat algorithm, all individuals are arranged with descending order according to the merits and demerits of the adaptive values to obtain the ranking of each bat, which is called the rank of the bat, and the rank is a commonly used statistic in statistics. Generally speaking, the global optimal position is less likely to be in the vicinity of individuals with poor fitness values. Therefore, some individuals with poor global optimal position can adopt the local perturbation mode to carry out mining operations. The   is designed for each generation, if the bats ranked below   (the   represents the number of bats in a population), local search was used, otherwise, global search was used.

 In the early stage of the algorithm, a large area of global optimization is required to determine the optimal position and the optimal position will be exploitated in the later stage. Therefore, in the early stage, a large number of bat individuals are required to conduct global optimization, while in the later stage, a large number of individuals are required to conduct local optimization through disturbance. Therefore, the value of   will increase with the increase of the iteration.

\( {Threshold}=T h_{\min }+\left(T h_{\max }-T h_{\min }\right) \times \frac{t-1}{L G-1}\)       (17)

where, Th_min  is the lower bound of  Threshold, Th_max  is the upper bound of Threshold, t is the current generation,  LG is the maximum evolutionary generations. Eq. (17) indicates that   will increase linearly from Th_min to Th_max  with the increase of the evolutionary generations.

 The transformation strategy is adopted in FTBA-TCR, which is showed in Eq. (17). And the pseudo-code of the FTBA-TCR is showed as (FTBA-TCR).

FTBA-TCR

Begin     Initialize \(v_{k}(t), x_{k}(t), f_{k}(t), r_{k}(t) \ { and }\ A_{k}(t)\)     Calculate the fitness value of each bat and select the best solution While (\(t)     If \(t<0.258 \cdot L G\)         Eq. (13) is used to update the velocity for each bat     Else             Eq. (14) is used to update the velocity for each bat         End         Sort the fitness values     If the individual’s fitness value is weakness, the formula is expressed         as \({Rank}\left(f\left(x_{k}(t)\right)\right)< { Threshold } \times { Popsize }\)             The local search strategy in Eq. (15) is used to update the position     Else     Eq. (6) is used to update the position     End     If \(\boldsymbol{x}_{k}(t+1)=\boldsymbol{x}_{k}(t)\)       Update the bat individual position with  \(\boldsymbol{x}_{k}(t+1)=\boldsymbol{x}_{k}(t)\)     End     Update and save the position of the best solution End     Output the best solution End

 

4. Experiments

4.1 FTBA-TCR

 In order to verify the performance of these strategies, these algorithms are tested in the CEC2013 test suite. The upper limit of parameter Threshold   was Th_max , and the lower limit of parameter Threshold  was Th_min . Since Threshold  is a value within [0,1]  , the upper limit Th_max  was tested at 0.5, 0.6, 0.7, 0.8, 0.9, while the lower limit Th_min  was tested at 0.1, 0.2, 0.3, and 0.4. Therefore, there are 20 combinations of Th_max  and Th_min , which are described in Table 1.

Table 1. The combination of different parameters

 Table 2 shows the results of lists 20 different strategies. Table 3 shows the Friedman tests and gives the values of ranking. The best combination is COM8. Therefore, the value of   is  , the performance of FTBA-TCR algorithm is the best.

Table 2. Comparision of different combination parameters

Table 2. Comparision of different combination parameters (continue)

Table 2. Comparision of different combination parameters (continue)

Table 3. Algorithm ranking corresponding to different combination

 

4.2 Comparison of FTBA-TCR with other algorithms

 To further test the performance of FTBA-TCR, the other four algorithms are compared in CEC2013 test set. The involved algothrims are listed as follows:

 (1)        Bat Algorithm (BA) [46]

 (2)        Particle Swarm Optimizer (PSO) [48]

 (3)        Fast Triangle Flip Bat Algorithm (FTBA) [47]

 (4)        Fast Triangle Flip Bat Algorithm with Curve Strategy (FTBA-TC) [39]

 Table 4 presents the values of mean error function which are achieved by the four algorithms on the CEC2013 test suite. And the results show that FTBA-TCR has the best performance. Specifically, FTBA-TCR performs better than BA on 27 functions. For the PSO and FTBA, FTBA-TCR performs better on 19 functions and 21 functions, respecively. While PSO and FTBA outperform FTBA-TCR on 6 functions and 4 functions, respectively. Meanwhile, compared to FTBA-TC, FTBA-TCR obtains better results on 16 functions, and loses in 7 functions.

Table 4. Comparison results of FTBA-TCR with other algorithms (D = 30)

 

4.3 DV-Hop Localization Algorithm with Different Strategies

 We conducted simulation experiments in MATLAB R2013a to test and verify the performance of the proposed algorithm. And the parameters are set as follows in the simulation environment:

Table 5. Parameter values

 Average localization error was used to evalute the localization performance:

\( {Average \ error}=\frac{100}{n \times R} \sum_{i=1}^{n} \sqrt{\left(x'_{i}-x_{i}\right)^{2}+\left(y_{i}^{\prime}-y_{i}\right)^{2}}\)       (22)

where, the number of ordinary nodes is n ,R   is CR, (x'_i,y'_i)  is the estimated location of ordinary nodes, and (x_i,y_i)  is the real location of ordinary nodes.

Fig. 3. Initial diagram of the node's location

 The following algorithms are applied in DV-Hop algorithm to decrease the average localization error.

 a) DV-Hop algorithm, shorthand for DVHop algorithm

 b) DV-Hop algorithm based on PSO which is shorthand for PSO-DVHop

 c) DV-Hop algorithm based on Standard-BA which is shorthand for BA-DVHop

 d) DV-Hop algorithm based on FTBA which is shorthand for FTBA-DVHop

 e) DV-Hop algorithm based on FTBA-TC which is shorthand for FTBA-TC-DVHop

 f) DV-Hop algorithm based on FTBA-TCR which is shorthand for FTBA-TCR-DVHop

 (1) The changes in communication radius (CR)

 Table 6 and Fig. 4 show the influence of communication radius change on algorithm performance. Obviously, with the increase of CR, the error decreases gradually, and FTBA-TCR-DVHop has the best performance.

Table 6. Comparison of average localization errors of different CR

Fig. 4. Comparison of average localization errors of different CR

 (2) The changes in the total number of nodes

 Fig. 5 and Table 7 describe the change of algorithm performance on the condition that the total number of nodes changes while beacon nodes remains unchanged. The smaller the number of nodes, the larger the average distance error per hop, indicating that the error decreases with the increase of the number of nodes. And FTBA-TCR-DVHop algorithm performance is always better than the other algorithms.

Table 7. Comparison of average localization errors of different node numbers

Fig. 5. Comparison of average localization errors of different node numbers

 (3) The changes in the number of beacon nodes

 Fig. 6 and Table 8 shows the impact of beacon nodes changes on algorithm performance. When there are fewer beacon nodes (e.g., 5), the performance of DVHop algorithm based on intelligent optimization algorithm is worse than DVHop algorithm, but with the change of the amount of beacon nodes, the performance of DVHop algorithm based on intelligent optimization algorithm becomes better than DVHop algorithm. And in most case, the FTBA-TCR-DVHop has the minimum position error.

Table 8. Comparison of average localization errors of different beacon nodes

Fig. 6. Comparison of average localization errors of different beacon nodes

 (4) The number of iterations is different

 Fig. 7 and Table 9 describes the change in the number of with the times of iterations will affect the performance of the DV-Hop algorithm. Obviously, with the increase of iterations, the performance of DVHop algorithm based on intelligent optimization algorithm becomes better. What’s more, FTBA-TCR-DVHop converges faster than other algorithms, which has better converges performance than other algorithms.

Table 9. Comparison of average localization errors of different iterations

 

Fig. 7. Comparison of average localization errors of different iterations

 

5. Conclusion

 DV-Hop is a general range-free positioning algorithm for detecting ordinary nodes position. Because of the simple location principle of DV-Hop algorithm, it brings a higher localization error. In this paper, FTBA-TCR is designed to modify the accuracy of DV-Hop. In FTBA-TCR, the local search ability is affected by the optimal location. In each generation, the rank-based transformation strategy is used to select which individuals to conduct a local search. It was tested in the CEC2013 benchmark and applied into the DV-Hop algorithm. The best combination of threshold is adopted by the CEC2013 test suite. Four different algorithms are used to compare with the proposed algorithm. And the DV-Hop algorithms base these intelligent algorithms are used to verify the performance of FTBA-TCR. The simulation results show that the FTBA-TCR strategy is added to the localization of wireless sensor nodes which achieve better localization performance. In the future, we will integrate other operators and coupling strategies to improve the performance of BA. In addition, the proposed algorithm can be applied for solving practical problems, such as: the vehicle routing problem (VRP) [49], integer programming problems [50], the numerical association rule mining problem [51].