Yi Wang，Yubo Peng，Li Chen，Yanzhong Duan，Jing Li

School of Information Science and Technology，Northwestern University，Xi’an，Shaanxi 710127，China

*The corresponding author，email: chenli@nwu.edu.cn

Abstract: Wireless sensor networks are widely used in today’s fields，such as scientific research，industry and agriculture.However，due to the influence of its geographical location and the problems of low coverage and waste of resources caused by random placement，it is very important to adopt appropriate strategies to improve its coverage.To this end，an improved GND-DE (Global and Neighborhood Difference Guided DE) algorithm is proposed.This algorithm uses both the global topology structure and the neighborhood topology structure，combined with the evaluation of contemporary optimization results，and selects the results from the two topology structures.The value-dominant individual，the individual to be evolved and the two dominant individuals calculate the difference operator corresponding to the two topological structures; a diversity neighborhood topology is proposed for the creation of the neighborhood topology;at the same time，the algorithm step size factor F is adaptively adjusted and the JADE external archive mutation strategy is introduced to eliminate the possibility of algorithm search stagnation.In order to verify the effectiveness of its improved algorithm，compared with other mainstream improved algorithms on the CEC2017 test set，it shows that its optimization efficiency and convergence are better than other comparison algorithms; finally，GND-DE is applied to WSN node coverage optimization，which proves the feasibility of its optimization strategy.

Keywords:differential evolution;the global topology;the neighborhood topology;diversity;wireless sensor network

I.INTRODUCTION

Wireless sensor network (WSN) is a multi-hop selforganizing network formed by a large number of communicating sensor nodes deployed in the monitoring area [1].With the rapid development of science and technology，WSN has been widely used in many fields by virtue of its excellent performance advantages.Among them，node deployment design is a key issue in WSN.However，in the practical application，due to the limitations of deployment methods，the network coverage is low and resources are wasted.Therefore，designing an appropriate node deployment method can effectively improve the monitoring efficiency of sensors and the quality of data transmission in the network.For example，Literature [2] aimed at link conflicts and excessive link interference in wireless sensor networks，and proposed a joint resource allocation optimization algorithm based on dual-population differential evolution，reducing network energy consumption，maximizing network capacity，improving resource allocation and enhancing the balance of resource distribution.Focused on the problem of limited energy and unbalanced load of infinite sensor network nodes，a clustering routing algorithm POFCA based on particle swarm optimization fuzzy C-means is proposed，which effectively balances network load and reduce network energy consumption for extending network life cycle[3].The two-stage strategy is adopted to optimize the WSN node location，which improves the WSN node location accuracy and increases the network coverage.The first stage uses the Dv-Hop algorithm for rough positioning，and the second stage uses the differential evolution algorithm to optimize the precise positioning [4].In addition，the particle swarm is combined with the differential evolution algorithm，and the advantages of the two algorithms are combined to optimize the location of WSN nodes to improve the network coverage[5].It can be seen that designing a suitable dominance mechanism and optimization strategy in differential evolution，and improving the global and local adaptive balance of the algorithm[6]，is the key to improving the coverage optimization of WSN network.

Differential evolution algorithm(DE)[7]has the advantages of good convergence performance，fewer parameters and strong robustness among many swarm intelligence optimization algorithms，but in some cases there are still weak local convergence capabilities and poor late-stage efficiency of the algorithm，etc.To this end，many scholars at home and abroad have improved the DE algorithm from different angles.For example，in order to enhance the search accuracy of the algorithm，most algorithms adjust the algorithm parameters adaptively and dynamically change the search range [8-10].In addition，the difference between two random individuals is used to generate a new step size factor NF[11]，and the individual distribution of the current population is fed back to the step size factor，which solves the problem of missing the optimal value due to a long step size.Some algorithms have proposed new mutation strategies to improve the performance of the algorithms in response to the shortcomings of the original mutation strategies [12-14].Learning from the idea of self-organizing mapping network，a structure of self-organizing mapping neighborhood is proposed，and variant individuals are generated according to the neighborhood，and information interaction and co-evolution between subpopulations make the DE algorithm search more refined[15，16].Through the master-slave multi-population distributed framework [17]，the population is divided into three populations: exploratory population，development population，and balanced population for coevolution.Different populations will adaptively select appropriate mutation strategies based on the evolutionary state estimation，so as to make full use of the feedback of the individuals and entire population[18].

When the elder brother heard this a great rage filled his heart, and, without saying one word, he drew his sword and slew his brother, and his body rolled in the dust

On the basis of the above theoretical analysis and based on the crossover strategy，this research proposes a WSN node coverage optimization algorithm based on global and neighborhood difference DE ( GNDDE).The main innovations are as follows:

· Propose a method of constructing neighborhood with diversity perturbation，and obtain neighborhood elite and global elite.

· For the intersection link，the global topological structure and the neighborhood topological structure are combined to optimize the result evaluation value to obtain the global elite difference and the neighborhood elite difference，and then guide the difference operation of the intersection link.

· The JADE mutation operator is introduced into this algorithm，and the step size factor is adjusted adaptively.Compared with the mainstream improved DE algorithm in the CEC 2017 test set，it is successfully applied to WSN node coverage optimization.

II.GND-DE ALGORITHM

The mutation strategy in this paper adopts JADE[16]algorithm with external file mutation strategy，the specific formula is as follows:

Figure 1. Schematic diagram of algorithm idea.

2.1 An External File Mutation Strategy

The schematic diagram of this algorithm is shown in Figure.1 and it can be seen that，GND-DE mainly includes 6 parts:Initialization，elite difference generation，step factor update，mutation，crossover and selection operations.The algorithm starts by initializing the population，and the difference between the global elite and the neighborhood elite is generated by the elite difference(The specific process is: to evaluate the fitness of all individuals in the contemporary population to find the global elite，and at the same time for each individual to be evolved，use the neighborhood generation strategy in this article to construct the neighborhood，and combine the fitness value to use the best individual in the neighborhood as the neighborhood elite;finally，the elite difference generation method is used to determine the generation of elite differences among the individuals to be evolved，the global elites and neighborhood elites respectively).Then，update the step size factorFto generate mutant individuals with external archive mutation strategy，and use binomial crossover and elite difference crossover mentioned in this article to select one to generate experimental individuals，the population updated through the selection strategy.If the termination condition is not met，the next iteration will be performed.Binomial crossover retains the advantages of traditional crossover randomness.Elite difference crossover has a learning function，learning from global and neighborhood optimization.The two strategies operate alternately and cooperate with each other to balance the convergence and diversity of the algorithm.

Where:Fis the step factor;xiis the parent individual;xpbestis randomly selected among thetopindividuals in the population;xr1is randomly selected for the current population;xr2is the archives of current populations and individuals who have failed evolution and are selected collectively.

2.2 Parameter Adaptive Adjustment

Fis the step factor of the DE algorithm，which is used to control the degree of scaling of the difference between the two basis vectors.Global exploration is required in the early stage，so a larger step length is needed to explore more areas.In the later stage，most individuals are centered around the optimal solution of the population.Therefore，it is necessary to perform a local fine search to obtain the local optimum，and at this time，the value ofFshould be small.For this reason，this paper proposes a method of adaptively changing the number of iterationsF.The specific formula is as follows:

The day came that Tibley should go to heaven. He wasn t sure if they would let him in because he had only sewed one coat his whole life. But nothing was said at the door because Kooble Loomploy who had sewed 3,000 coats in his lifetime was arguing with the Lord. I don t understand , said Kooble Loomploy, why everyone else gets to enjoy the same amount of sunshine as me, when I sewed so many more coats! Why look at Tibley over there!! His whole 800 years he only sewed 1 coat. What kind of justice is that? The Lord smiled at Kooble Loomploy with sadness in his eyes. My Son , he said, All that I have is yours! Kooble Loomploy was not consoled by this, but no matter, for he rushed off into the sunshine with everyone else. They danced and played and drank in the warmth and light. They were in heaven!!!

2.3 Global Elite Differences

Elite difference is a discrete guide operator.Global elite difference aims to join the influence mechanism of the global optimal individual and promote the development of individuals near the global optimal value，finally using the elite difference value as a guide to carry out vectorized crossover.The specific schematic is shown in Figure.2 (the values of elite individuals and individuals to be evolved in the figure are random values in the iterative process).The following takes the generation of elite differences in the 4-dimensional problem as an example.The figure shows the individual to be evolved，the elite individual，and the difference among the elites.The 4 dimensions of the individual to be evolved are set as: 3，2，17，-8，and the elite individuals represent the best individuals in the current iterative population or the best individuals in the neighborhood，which are respectively set as: 7，-3，12，-8.The difference of the corresponding dimension between the elite individual and the individual to be mutated is calculated.If the difference is greater than 0，the corresponding elite difference dimension is 1.if the difference is less than 0，the corresponding elite difference dimension is-1.if the difference is equal to 0，the corresponding elite difference dimension is 0.

Figure 2. Example of elite difference generation.

Initialize global elite differencesEg={e1，e2，...，eNP}，NPstands for population size.The difference between each elite isei，ei={e1，e2，...，eij}，wherejrepresents the individual dimensions of the population.And the generation method ofEgis as follows:

For solving the difference in dimensionality and calculating the dimensional difference between the individualxiin the population and the global optimal individualx_gbestasCg，the calculation method is as follows:

From Eq.(3)can obtain the difference of each dimension between the individual and the optimal value，and generate the global elite difference according toCg:As Eq.(3)and Eq.(4)，the global elite difference can be calculated.When the difference value of the individual to be evolved in a certain dimension is 1，which means that the global optimal solution is larger than the individual to be evolved in this dimension，and the individual to be evolved will evolve in a positive direction in this dimension.When the difference value is-1，it will evolve in the negative direction;when the difference value is 0，it means that the individual to be evolved does not need to change in this dimension.

2.4 Neighborhood Elite Differences

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Anne Lisbeth returned to her home, but she was no longer the womanshe had been. Her thoughts were like a confused, tangled skein; onlyone thread, only one thought was clear to her, namely that she mustcarry the spectre of the sea-shore to the churchyard, and dig agrave for him there; that by so doing she might win back her soul.

2.4.1 Weighted Euclidean distance

In order to maintain the diversity of the population in the iterative process，the individual dimensional weight factorpis introduced to perturb the distance calculation.When the weighting factor is larger，the individual diversity is greater，and vice versa.represents the average of all individuals in each dimension，andpi={pi1，pi2，...，pij}respectively represents the weight factor of the i-th individual in the j-th dimension.The calculation formula for themeandof the dimensions and the weighting factorpare as follows:

Wherejrepresents the dimension of the individual andirepresents the individual of the population.As shown in formula Eq.(6)，when the difference between a certain dimension of an individual and the mean of the corresponding dimension is greater，the weighting factorpwill be greater.It can be seen that，prepresents the degree to which the individual’s dimensionality deviates from the mean value of the dimensionality.When an individual maintains a large pvalue in all dimensions，it means that the individual is farther away from where most individuals gather，that is，it has greater diversity.

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In this paper，the dimensional weight factorpis applied to the Euclidean distance to calculate the distance among individuals.Defined in Euclidean space，the distance between the N-dimensional space vectorx={x1，...，xn}andy={y1，...，yn}is:

The story as one about sexual maturity17 and acceptance of a marital30 relationship is more explicit31 in the earliest Grimms manuscript of 1810. In the early version, the frog s desire to sleep with the princess is overt32 and not hidden in ornate details (Zipes 1983).Return to place in story.

Applyingpto the Euclidean distance，the formula for forming the diversity weighted Euclidean distance is as follows:

Romance is not something that can be taught or copied. One can only be romantic through another. Patricia, my wife of fourteen years, has instilled4 the romance in me. I am romantic because of her. Patricia has always brought out the best in me. The many aspects of our romance are too numerous to mention. However, there is one special romantic interlude that I began over fifteen years ago.

From Eq.(8)is used to calculate the weighted Euclidean distance between individualxcand individualxiandpi，jcorresponds to the weight factor of the jth dimension of the i-th individual.When thepi，jis large，the distance dp between individuals will become smaller，that is，when finding the distance betweenxcand individualxiand thepvalue corresponding toxiis larger，it means that the individualxiis farther from the population gathering point，and the weighted Euclidean distance between the individualxcandxiwill be smaller.Therefore，the weighted perturbation of the Euclidean distance through individual diversity is realized.

2.4.2 Neighborhood strategy constructed by diversity disturbance

Construct the neighborhood according to Eq.(8)Diversity-weighted Euclidean distance to calculate the distance between the neighborhood center and the population individuals.Perturbed by the weight factor，the diversity of individuals closer to the neighborhood center is stronger，and the distance between the selection and the neighborhood center is less than that of the neighborhood.Individuals within the radius of the domain construct neighborhoods to achieve the purpose of constructing neighborhoods with diverse disturbances.

2.4.3 Neighborhood elite difference generation strategy

In the process of generating the difference between neighborhood elites in this article，firstly，the neighborhood is constructed by using the diversity disturbance neighborhood construction strategy for each individual.Taking individualxias the center of the neighborhood，given the radius of the neighborhood m，construct a neighborhood with a neighborhood center ofxcand a neighborhood radius ofm，denoted asU(xc，m).Sort the fitness value of the individuals in the neighborhoodU(xc，m) from low to high to select the individual with the smallest fitness value in the neighborhood as the neighborhood elitex_lbest，and calculate the differenceClbetweenx_lbestand the population individualxiby Eq.(9)，as well as the neighborhood elite differenceElby Eq.(10).

The value of neighborhood elite difference is calculated by Eq.(9) and formula Eq.(10).The pseudo code of neighborhood elite difference generation strategy is shown in Algorithm 1.

Global elite differences guide individuals to evolve in the direction of the best individual.When all individuals learn from the best individual，the population diversity will gradually decrease，and it is easy to fall into an endless loop of shallow search.Therefore，this paper proposes a combination of neighborhood elite differences and global elite differences to guide individual crossover.It also proposes to construct a neighborhood model based on the perturbation of individual diversity to find the optimal value from the neighborhood where the individual is located，and determine the neighborhood elite difference with the individual to be mutated.Among them，the diversity disturbance constructs the neighborhood through the weight factor weighted Euclidean distance，which effectively maintains the individual dimensional diversity and promotes the algorithm for the breadth search development.

13. Come in and let me clean thee: In the early part of the story, its elements-the two trips to the woods, the white pebbles, the bread crumbs80 eaten by birds-are exactly as occur in Hansel and Gretel. Here, as in that story, The children s successful return home does not solve anything (Bettelheim, p. 160). The family are still poor, Little Thumb has yet to gain his parent s favor, and so they must be brought back into the wood to face their challenges again.Return to place in story.

2.5 Algorithm Description

2.5.1 Elite Difference Crossover Strategy The crossover strategy of elite differences is guided by global elite differences and neighborhood elite differences，selecting the components between the mutated individualxiand the mutated individualvto form the experimental individualu，instead of the binomial crossover，and the crossover rateCRis used to control the crossover.Since the global elite difference has the same structure as the neighborhood elite difference，Eis used to representEgandEl，and the cross formula is shown as:

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It can be seen from the formula that only the genes of the mutated individuals that meet the conditions are selected to form the experimental vector，and the positive and negative values are determined by the difference between the mutated individual and the individual to be mutated.Then compared with the elite difference value，genes can be effectively selected from mutated individuals to help them learn from the elite.

Through the experimental analysis results of the above 6 algorithms on 4 types of functions，it can be seen that the GND-DE algorithm has the best performance in multi-peak，mixed function and composite function，especially in terms of convergence accuracy，which are higher than other comparison algorithms;for the mixed function，it has the best stability.

2.5.2 Algorithm flow

The GND-DE algorithm generates global elite differences and neighborhood elite differences，and performs feedback control on the algorithm crossover phase.In order to balance the classic binomial crossover and elite difference feedback crossover selection ratio，we introduce hyperparametervcontrol to select traditional binomial crossover or elite difference feedback crosses.The hyperparametertis used to balance the global elite difference and the neighborhood elite difference，and then balance the convergence and diversity of the algorithm.The pseudo code of the algorithm is shown in Algorithm 2.

III.EXPERIMENTS AND RESULTS

3.1 Experimental Environment Configuration and Parameter Settings

Algorithm 1. Pseudo-code generation strategy of neighborhood elite difference.Input: Population size pop，Individual dimension d，fitness，Neighborhood radius m 1: for j =1: d do 2: Calculate the mean dj value of each dimension according to Eq.(5)3: end for 4: for i=1: pop do 5: for j =1: d do 6:Calculate the individual dimension weight factor pi，j according to Eq.(6)7: end for 8: end for 9: for i=1: pop do 10: Calculate the weighted Euclidean distance dp between the center of the neighborhood and other individuals according to Eq.(8)11: end for 12: for i=1: pop do 13: For the individual xi as the neighborhood center，the first m individual resume fields of Euclidean distance are screened，and the best individual from the m neighborhood individuals is selected as the neighborhood elite，and the neighborhood elite is calculated by Eq.(9)and Eq.(10)Difference El 14: end for Output: El

Specifically，for the unimodal functionsf1andf2，the convergence speed of GND-DE is better than other algorithms，especially forf2.In the early stages of evolution of each algorithm，the convergence speed of GND-DE is significantly better than that of the comparison algorithm.This is due to the large step factor in the early stage of the algorithm，the wide search range，and the crossover operation will have a certain probability to be guided by the global elite difference，which improves the convergence speed and optimization efficiency will eventually converge to the global optimal solution.

Algorithm 2. Pseudo code of GND-DE algorithm.Input: Population size pop，Maxiter，Upper bound of step factor F0，Step factor lower bound F1，Hyperparameter v，Hyperparameter t，Crossover rate CR 1: Initialization: population individuals 2: for j =1: Maxiter do 3: for i=1: pop do 4:Calculate the fitness value of individual population fitness 5: end for 6: Calculate the global optimal individual of the population x gbest，fitnessbest 7: for i=1: pop do 8:Calculate the global elite difference Eg value by Eq.(3)and Eq.(4)9: end for 10: Calculate the difference between the neighborhood elites El through Algorithm 1 11: Update step factor F by formula Eq.(2)12: Perform mutation operation according to Eq.(1)to generate a mutated individual 13: for i=1: pop do 14:if rand ＜v then 15:Classic binomial crossover strategy 16:else 17:Elite difference crossover strategy 18:if rand ＞t then 19:Eg combined formula Eq.(11) to guide cross operation 20:else 21:El combination Eq.(11) to guide cross operation 22:end if 23:end if 24: end for 25: Choose Action Update population 26: end for Output: fitnessgbest

In order to compare the performance of various algorithms fairly，the common parameter settings of all algorithms are the same.Each function in the CEC2017 test set was independently solved 30 times，meaning the dimensionD=30 and the population sizeNP=150，and the maximum number of iterations isMaxiter=2000.The other parameters of the comparison algorithm are the best settings recommended in the corresponding literature.

3.2 Experimental Results and Analysis

3.2.1 Solution accuracy and stability analysis This time，the average value (Av) and standard deviation (Sd) of the 30 experimental results were taken，and the convergence accuracy of the algorithm was evaluated by the average value，and the stability of the algorithm was evaluated by the standard deviation，as shown in Table 1.The black value represents the optimal value of various algorithms on the corresponding test function，and the last line is the number of optimal average results obtained by the corresponding algorithm on all test functions.

It can be seen from Table 1 that the GND-DE algorithm has achieved 16 optimal average results on the test functions off1～f30，while SEFDE，PLDE，JADE，CODE and SMGBDE are in 4，4，and 6 respectively，11 and 2 test functions to obtain the optimal mean value，indicating that the GND-DE algorithm has better comprehensive optimization capabilities.In order to further analyze the performance of the algorithm，there is the accuracy and stability analysis of the 6 types of algorithms on 30 test functions such as unimodal，multimodal，mixed and composite.

Table 1. Results of 6 algorithms running on 30 test functions.

(1)Unimodal function(f1～f3):It can be seen from the experimental results that CODE and SMGBDE have achieved 2 optimal mean results in the 3 test functions respectively.GND-DE performance is not ideal，and JADE，SEFDE and PL-DE algorithms have achieved 1 optimal mean value respectively.However，GND-DE has achieved results close to the optimal mean on the two unimodal functionsf2andf3.Among them，the functionf2is 2，5，and 9 orders of magnitude higher than SEFDE，PL-DE and JADE，respectively.In terms of the stability embodied by the standard deviation，JADE obtained the optimal stability of 1 function atf1，SMGBDE atf2，and CODE atf3.Although GND-DE has not achieved optimal stability，its stability onf2andf3is higher than that of SEFDE，PL-DE and JADE;specifically，onf2，the performance of GND-DE and CODE is the same.Compared to the standard deviations of SEFDE，PL-DE and JADE，it was increased by 4，7 and 10 orders of magnitude respectively.Onf3，GND-DE is 2 orders of magnitude worse than SMGBDE，but it is 3，3，and 4 orders of magnitude higher than SEFDE，PL-DE，and JADE，respectively.On the whole，in the singlepeak test function，GND-DE showed good optimization performance.

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(2) Multimodal function (f4～f10): Among the 7 multimodal functions，GND-DE achieves the best average results on the 4 multimodal functions，which has significant advantages over the other 5 comparison algorithms.In the standard deviation analysis，the GND-DE，JADE and CODE algorithms all achieved the optimal standard deviation of the two functions，and the performance of other algorithms is weak.Besides，GND-DE not only achieved the best standard deviation onf5andf7，but also the best mean value，which shows that the GND-DE algorithm has significant global optimization ability and solution accuracy on multimodal functions with good stability.

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(3) Mixed function (f11～f20): Among the 10 mixed functions，GND-DE obtains the optimal mean value on the 6 functions off11，f14，f15，f16，f17andf19，and is also the closest to the optimal mean value result on the functionf20.CODE achieved the optimal value in the other three mixing functions(f13，f18，f20)and JADE achieved the optimal mean value on f12，while SEFDE，PL-DE and SMGBDE did not achieve the optimal value.In terms of the stability analysis reflected by the standard deviation，the GND-DE algorithm achieves optimal stability in 6 mixed functions(f11，f12，f13，f15，f16，f19)，and CODE achieves optimal stability inf17，f18，andf20，and the other algorithms perform poorly.Compared with the other five types of algorithms，EDF not only has higher convergence accuracy but also maintains continuous high stability when dealing with mixed function optimization problems.

(4)Composite function(f21～f30): On the 10 composite functions，GND-DE achieved 5 optimal averages (f21，f21，f24，f26，f29).CODE achieved 4 best averages(f22，f27，f28，f30)，ranking second.It shows that the adaptive step factorFadopted by the GND-DE algorithm and the elite difference strategy feedback crossover will maintain diversity to a certain extent while promoting the accuracy of convergence，thereby improving the optimization performance in complex functions.In terms of the stability reflected by the standard deviation，GND-DE is inferior to other algorithms，and only obtains the best stability onf29.while JADE obtains the best stability on 5 functions(f21，f22，f23，f26，f27)，ranked first.SEFDE obtains the optimal stability of two functions on (f25，f28)and CODE on(f24，f30)respectively;PL-DE and SMGBDE do not obtain the optimal stability.From the experimental data，it can be concluded that JADE ranks first and obtains the best stability on 5 functions，while GND-DE is only the most stable on 1 function，indicating that the stability of JADE algorithm is much higher than that of GND-DE algorithm.JADE only obtains an optimal solution on the functionf22yet，that is，JADE is easy to get stuck in a certain position on the objective function，and it cannot be further searched to cause it to be “stable” on a non-optimal solution; while the smaller the mean of the GND and the larger the standard deviation indicate that the algorithm can often find a better solution than the mean.However，there are a small number of cases where the optimal solution is not found and the average value is pulled up.This shows that the use of neighborhood elite difference feedback mechanism can search for better solutions in a given range，which is better than other comparison algorithms.Local search capability.

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3.2.2 Convergence analysis

Where:Fis the step factor for adaptive change;F0is the upper bound of the step factor;F1is the lower bound of the step factor;F ∈[F1，F(xiàn)0].gis the current iteration number;Gmaxis the maximum number of iterations of the algorithm.It can be seen from the formula thatFis close to the upper limit of the step factor in the early stage of the iteration，realizing global exploration; as the number of iterations increases，F(xiàn)gradually decreases adaptively，and finally reaches the lower limit of the step factor，and local development is carried out to achieve fine search.

Due to space limitations，in order to verify the convergence performance of the GND-DE algorithm，we selected 12 representative test functions from 30 test functions to observe the convergence of GND-DE.Among them，there are 2 unimodal functions (f1andf2)，3 multimodal functions (f5，f6，f7)，3 mixed functions (f15，f16，f17) and 4 composite functions(f21，f22，f23，f24).The convergence comparison between GND-DE and the comparison algorithm is shown in Figure.3.In order to make the convergence graph clearer and more pronounced，the vertical axis in the graph uses logarithmic coordinates，and the abscissa represents the number of iterations.

It can be seen from Figure.3 that GND-DE has a strong convergence ability regardless of single-peak，multi-peak，mixed and composite functions.

Figure 3. Convergence curves of various algorithms on some functions.

In order to evaluate the performance of the GNDDE algorithm，the experiment selected 30 benchmark functions of IEEE CEC 2017，including single-peak，multi-peak，mixed and composite functions for multidirectional testing.Among them，the test functionsf1～ f3are unimodal functions.f4～ f10are multimodal functions.f11～f20are mixed functions.f21～f30composite functions.The experimental environment uses Inter i7-6700CPU，a PC with a main frequency of 3.40GHz and a memory of 8GB，and the operating system is 64-bit Windows 10 with the programming language of MATLAB2019B.The related parameters of the algorithm GND-DE in this paper are set as follows: The lower bound of the step size factorF1=0.1，the upper bound of the step size factorF0=0.8，the neighborhood radiusm=7，the hyperparameterv=0.5，the hyperparametert=0.7，and the crossover probabilityCR=0.5.The comparison algorithm selects the mainstream DE improvement algorithms in recent years，including the differential evolution algorithm based on the optimal learning strategy [13](PL-DE)，the adaptive skeleton differential evolution algorithm of the double mutation strategy[14](SMGBDE)，with optional external Archive adaptive differential evolution algorithm [19](JADE)，strategy adaptive differential evolution algorithm based on state estimation feedback [20](SEFDE) and differential evolution algorithm with compound test vector generation strategy and control parameters [21](CODE)，among which PL-DE，SMCBDE，and SEFDE are improved DE algorithms proposed in recent years，which have more advantages than other improved algorithms，while JADE and CODE algorithms are improvements of classic DE algorithms and have significant performance，used as a comparison algorithm in the experiment.

In the convergence graph of the multimodal functionsf5，f6，andf7，it can be seen that GND-DE is in the optimization process of the deep search and the breadth search alternately during the whole process，especially in the functionsf5andf7.The performance is relatively stable on the functionf6.

In the mixed functionsf15，f16，andf17，each algorithm can obtain the optimal value，but the GNDDE converges the fastest on the functionf15and can continue to be stable.Onf16，although the algorithms SMGBDE，CODE，SEFDE，PL-DE，and JADE all converged before the GND-DE algorithm，none of these algorithms found the optimal solution，especially SMGBDE converged in 200 generations and fell into a local extreme.Although the convergence speed of the GND-DE algorithm in the early stage is slightly slower than that of the comparison algorithm，it can maintain continuous mining until the algorithm converges to the global optimal solution and searches for the highest precision value.On the functionf17，the algorithm convergence process is similar to that onf16，so it will not be repeated here.

It can be seen from the composite functionsf21，f22，f23，andf24that the GND-DE algorithm has the best convergence performance in the composite function.On the functionf22，GND-DE and most algorithms can find the optimal solution，but the convergence speed is the fastest.Inf21，f23，andf24，GNDDE converges faster in the early stage of the algorithm，and the convergence rate tends to be flat in the middle of the algorithm.This is because the algorithm uses the difference of neighborhood elites to realize the disturbance of individual diversity and guide the population to implement local development，search breadth and increase diversity.When multiple local optima are searched，it enters the later stage of the algorithm，and executes the global elite difference strategy finally realizing deep search and fast convergence and obtaining the optimal solution.Through the above analysis，it can be seen that the GND-DE algorithm has excellent solution accuracy and convergence performance in single-peak，multi-peak，mixed and composite functions，especially in mixed and composite functions，which gain the upper hand.

3.3 Significance Test

(1)t-test: Here，the results of the GND-DE algorithm and other improved DE algorithms on the 30 test functions of CEC2017 are t-tested，and the test results are shown in Table 1.Explanation of symbols in the table: “+”，“-”，and “=” respectively represent the performance of GND-DE is“better，worse，and flat”compared to the comparison algorithm.t/w/l corresponds to the number of functions of“+”，“-”，and“=”respectively.The difference of tw indicates that the GNDDE algorithm is better than the number of comparison algorithms minus the number of inferior comparison algorithms.The larger the difference，the worst performance of the corresponding algorithm.

According to Table 1，compared with SEFDE，PLDE，JADE，CODE and SMGBDE algorithm，GNDDE has 25，24，21，15 and 26 functions which have more performance advantages.When comparing the significance，the significance level is 0.05，that is，when36 is known.The algorithm is significantly better than other algorithms，so this algorithm is significantly better than SEFDE and SMGBDE algorithms.This algorithm also has great advantages for PLDE and JADE.Compared with CODE algorithm，it also has a significant advantage over 15 standard test functions.From Table 1，CODE has achieved a sub-optimal ranking.Although GND-DE has better optimization ability，the significance of the comparison result is not obvious in the unimodal function.Therefore，we believe that the algorithm is global and there is still a long road for improvement in the setting of neighborhood elite differences.

Table 2. Friedman-test analysis of the mean result on the test function.

(2) Friedman-ranking: This paper performs on the 30 test functions of CEC2017 for 6 algorithms，as shown in Table 2.Friedman ranking is the average of the ranking of each algorithm on 30 test functions.The smaller the ranking，the better the comprehensive ability of the algorithm.Statistic represents the chi-square statistic.If the chi-square value is greater than 5.99，it is considered that there are obvious differences between the algorithms.It can be seen from Table 2 that on the functionsf1～ f30，the rank value of GND-DE is the smallest，and the chi-square statistic is greater than 5.99，which proves that there are significant differences between the algorithms.In general，GND-DE on 30 test functions has better performance.In addition，it ranked first in single-peak，multi-peak and mixed functions with CODE，JADE and CODE respectively，and ranked first in composite functions.Among them，the chi-square statistic of the unimodal function is less than 5.99，indicating that there is no significant difference between the algorithms in the unimodal function.The chi-square statistics on multimodal，mixed and composite functions are 15.490，27.657，and 29.097，respectively，indicating that there are significant differences in these three functions，which further proves that GND-DE has significant advantages over other algorithms.

He was smoking a long pipe, and from time to time he sipped3 a little coffee which a slave handed to him, and after each sip2 he stroked his long beard with an air of enjoyment4

3.4 WSN Node Coverage Optimization

The analysis of the experimental data in Tables 1 and 2 shows that the GND-DE algorithm is superior to other comparison algorithms in terms of optimization accuracy and convergence speed.Therefore，this algorithm is applied to the WSN network coverage optimization，aiming to find the WSN network nodes.optimal distribution，thereby improving resource utilization.Through the experiment simulation in matlab，select the second and third ranked CODE and JADE in Table 1 as the comparison algorithm of WSN network coverage optimization，and compare it with the GND-DE algorithm in this paper from the two dimensions of coverage and convergence speed.

The detection area is set to a two-dimensional plane of 1m×1mwith 40 sensor nodes set，0.1msensing radius，0.1mcommunication radius，and the number of iterations is 800.The corresponding optimization algorithm dimension is 40 and the number of populations is 100.In addition，the search range is [0，1]，and other parameters of the algorithm are the same as those set in Section 3.1.The algorithm target optimizes the WSN coverage rate to the maximum value.In order to facilitate the solution，the problem is converted into a minimum value optimization problem.The corresponding fitness value is (1-WSN coverage rate)，and the target change is to solve the minimum fitness optimization problem.The 40 dimensions of a single individual in the algorithm represent the posi-tions of 40 sensor nodes on a two-dimensional plane，and each dimension has x，y genes representing sensor coordinates，so as to obtain the fitness the lowest degree value corresponding to the individual algorithm，and the WSN node coverage optimization problem is modeled as a problem of finding the lowest point.

The optimized coverage rate of WSN node positions is shown in Table 3.The GND-DE algorithm ranks first，the final fitness function value is 0.05，and the coverage rate is 95%.Compared with the node coverage rate before optimization，the coverage rate is 78%，an increase of 17 percentage points.In comparison，the fitness function value of the comparison algorithm CODE is 0.11，and the coverage rate is 89%;the fitness function value of the JADE algorithm is 0.09，and the corresponding coverage rate is 91%.Figure.4 shows the convergence curves of the three algorithms in WSN network coverage optimization.From the figure，the overall convergence speed and convergence accuracy of the GND-DE algorithm are better than those of CODE and JADE.Among them，in the 0-200 generation，because GND-DE uses global elite differences for global exploration in the early stage，the convergence speed is slow，maintaining population diversity，and the role of local elite differences in the later stage.Development is carried out within the optimal individual neighborhood to accelerate convergence speed.So the algorithm is 600-800 generations，the GND-DE algorithm continues to converge，and the image appears to have a larger slope.Figure.5 shows the distribution of sensor nodes before network optimization.It can be clearly seen that the distribution of sensor nodes is uneven before convergence，and sensor nodes are crowded and overlapped，resulting in insufficient use of sensor node resources.Figures.6-8 are the distribution diagrams of sensor nodes after optimization of each algorithm.Obviously，the CODE algorithm in Figure.7 has the worst optimization effect，and the distribution of sensor nodes shows many blank areas.The GND-DE in Figure.6 is better optimized，the sensor nodes are more evenly distributed in the [0.1] interval，and the coverage rate is higher.The JADE optimization effect in Figure.8 is slightly weaker，and there are still some blank areas on the right side of the image，which also confirms that the coverage of JADE and CODE is lower than the node coverage after GND-DE optimization.

Table 3. Optimization of WSN network coverage by each algorithm.

Figure 4. Network coverage optimization convergence curve.

Figure 5. Distribution of sensor nodes before network optimization.

Figure 6. GND-DE optimized sensor node distribution map.

Figure 7. CODE optimized sensor node distribution diagram.

Figure 8. JADE optimized sensor node distribution map.

IV.CONCLUSION

In this paper，considering the problems of insufficient node coverage and waste of resources in WSN network，the improved differential evolution algorithm is applied to WSN node coverage optimization，and an elite differential feedback strategy is proposed to adjust the cross-link DE algorithm.This algorithm performs depth and breadth searches through the combined effect of global elite differences and neighborhood elite differences.Among them，the neighborhood elite difference introduces the individual diversity perturbation neighborhood construction strategy，maintains the population diversity through the neighborhood elite difference，and promotes the algorithm to perform ultra-fine search in multiple parts.Finally，through hyperparameter control，the traditional binomial crossover or the elite difference strategy control crossover is randomly selected in the crossover stage，which improves the ability of the algorithm to search.The experimental results show the effectiveness of the algorithm.It is worth noting that this algorithm still has shortcomings in terms of unimodal function and stability，and cannot automatically adjust the proportion of global elite difference and local elite difference according to the classification of the problem to be optimized，resulting in a slightly longer entire iterative process.The major difficulty is also the direction that the author of this paper needs to conduct in-depth research in the follow-up work.

ACKNOWLEDGMENTS

This work was supported by the National Key Research and Development Program Projects of China(No.2018YFC1504705)，the National Natural Science Foundation of China(No.61731015)，the Major instrument special project of National Natural Science Foundation of China (No.42027806)，the project of Natural Science Foundation in Shaanxi Province(No.2018JM6029)，and the Key Research and Development Program of Shaanxi(No.2022GY-331;2020GY-094).