Modular design and algorithmic complexity optimization for large-scale software systems

Early in the pre-design stage of the system in order to obtain user requirements, the acquisition methods mainly include market research, which is the most basic acquisition method, and can be divided into inquiry survey and observation survey. Network media acquisition, including network questionnaire surveys, the collection of web-related data. Resource sharing of cooperative enterprises, collecting customer information through channel cooperation, achieving win-win cooperation with cooperative enterprises through resource sharing, and other ways. In the field of intelligent software engineering, with the deepening of informatization and intelligence, there are more and more ways to acquire user requirements, and the acquisition of demand information has the advantages of real-time, accuracy, convenience and so on.

With the rapid expansion of big data, it is increasingly important for the modular design of large-scale software systems, the modular design process of the system needs to have big data thinking, extraction, storage and analysis of external relevant big data. It can also organize and analyze the user-system database in the process of system design, effectively analyze user information, system information and behavioral data, and then get the user’s demand information.

Through the clustering analysis of user requirements, complex requirements can be simplified, analyzed and processed into multiple categories, each category contains its own unique detailed requirements under each category, with a low degree of similarity of the requirements between different categories and a high degree of similarity of the requirements within the same category, which helps to carry out the system design work accurately, reduces the development cost, and improves the competitiveness of the system. In this paper, K-Means clustering algorithm is introduced on the basis of the weighted network model, and the user demand weights are obtained by normalizing the comprehensive eigenvalues of the demand nodes, the number of clusters is determined by using the elbow method, and the initial clustering centers are determined according to the size of the user demand weights, and then the clustering of the user demands is realized [21].

The system design quality house is the core of quality function unfolding [22], and its role is to convert fuzzy user requirements into technical requirements that can be understood by engineers and technicians, and then waterfall decompose them into system part characteristics, development characteristics and quality characteristics, which is an effective tool for user requirements conversion. The complete system design quality house is shown in Fig. 1, the user requirements as inputs to the system design quality house is a hierarchical list of discrete and fuzzy user requirements collected from market research summarized, classified and processed in terms of importance, which is the key to the implementation of QFD.

Figure 1.

Structure of system design house of quality

In recent years, with the increasing maturity of artificial intelligence technology, large-scale software system design is becoming more and more intelligent, which provides users with a lot of convenience. 2021, despite the overall consumer market performance downturn, but the survey data shows that the household appliances based on large-scale software system still shows growth, which set the self-cleaning technology of sweeping robots in the cleaning appliances market is in the core position. Compared with the past, the sweeping machine in the technical innovation makes it in each function has been greatly improved and upgraded, in the navigation and obstacle avoidance, combined with a variety of technologies to solve the past sweeping robots, “chaotic collision”, “leakage of sweeping” and other common problems, the use of modularized design and other artificial intelligence large-scale software system design. Modular design and other artificial intelligence large-scale software system design technology to improve the intelligent cleaning ability of the sweeping robot and the planning ability of the sweeping route to reduce the burden of labor, in addition to mopping, water tank capacity and other aspects have been greatly improved. In this paper, combining the previous industry reports of sweepers and consulting relevant experts and industry personnel, numerous demands are organized, and the detailed user demands are shown in Table 2.

For the determined user requirements, the triangular fuzzy number is firstly used to realize the transformation from semantic evaluation to interval value, and then the precise value of the relationship between each user requirement is obtained according to the defuzzification of the triangular fuzzy number. In order to ensure the accuracy and objectivity of the user demand relationship, in addition to selecting professional researchers, users are invited to participate in the evaluation of the demand relationship. In this paper, 10 experts and 10 users are selected to form an evaluation group to score the semantic evaluation of the interrelationships between demands from different perspectives and get the comprehensive evaluation value of the triangular fuzzy number as shown in Table 3.

For the triangular fuzzy number comprehensive evaluation value given by the expert group, according to the formula of its defuzzification processing, to get the corresponding exact value, after the defuzzification of the user demand relationship comprehensive evaluation value as shown in Table 4, in which each specific value represents the degree of interaction between the user’s needs, the larger the value, it means that the two needs the greater the degree of mutual influence, when the value of the comprehensive evaluation of the value of 0, then it means that the two demands When the composite evaluation value is 0, it means that the two needs do not affect each other and are independent of each other.

According to the formula for the eigenvalue of each attribute of the weighted network, the eigenvalue of the parameters of each demand node is calculated separately, and the weight of user demand is obtained by normalizing the comprehensive eigenvalue, and the results of the calculation of each parameter are shown in Table 5. Through the establishment of the demand network model, the mutual influence relationship between user demands is considered, and more accurate demand weights are obtained, thus laying the foundation for the clustering of demands and the transformation of demands.

As can be seen from Table 5, the top-ranked demand nodes in terms of weight are SR₄, SR₁, SR₃, SR₂ and SR₆, whose corresponding user demands are anti-fall collision, strong sweeping ability, fast cleaning speed, monitoring fault information, and movable operation and control, i.e., they are the methods that users are most concerned about, and prioritizing these kinds of demands in the design of large-scale software systems can provide users with a better experience and solution and helps to improve the system’s market competitiveness and user satisfaction.In the process of user demand clustering, this no shopping user demand as the initial clustering center candidate points, using the elbow method to determine the number of clusters, input the weight of the user demand used, apply MATLAB to calculate the output about the number of classifications k and the sum of squares of the error within the cluster, through which you can determine the optimal number of clusters of demand, and then use clustering algorithms to classify and group all the user demand, which helps to better understanding of user requirements, understanding the real needs and key pain points of users, and providing strong support and guidance for the development of a large-scale software system sweeper that is more composite of user expectations and needs.

Figure 2 shows the result graph of demand data processing through the elbow method, and there is an obvious inflection point at the classification number k = 3, which corresponds to the optimal number of clusters. According to the weight ordering of each demand and the optimal number of chiron classes in Table 5, the initial clustering center of the K-means algorithm is determined to be the demand node 4, node 1, and node 3, and the user demand is clustered through the K-means algorithm, and the clustering results are shown in Figure 3. The user requirements are clustered into three categories, respectively {SR₁,SR₂,SR₃,SR₄}, {SR₅,SR₆,SR₇,SR₈,SR₉,SR₁₀,SR₁₁,SR₁₂}, {SR₁₃,SR₁₄,SR₁₅,SR₁₆,SR₁₇,SR₁₈,SR₁₉,SR₂₀}. It can be seen that the first category of requirements are functional requirements, which are closer to the actual use of large-scale software system products and satisfy the basic large-scale functionality. The second category of requirements is emotional requirements, which are more satisfying to the users. The third category of requirements are additional requirements, which are important requirements to be considered behind the realization of system functions.

Figure 2.

Clustering deviation of different K values

Figure 3.

User demand clustering results

In establishing the QFD model, the independence of each function of the product needs to be taken into account, and the independence of each function is guaranteed by using the independence axiom in axiomatic design. By analyzing the home sweeper, its main functions include sweeping and cleaning function, monitoring and planning function, intelligent control function, optimization function and human-machine interaction function. The improved QFD is used to establish the quality house of user requirements-product functions, and the degree of correlation between the parameters in the requirements-functions relationship matrix is represented by the numbers “1”, “3”, “5”, “7”, “9”, the larger the value indicates that the stronger the relationship, with ‘0’ indicates that there is no relationship between the two parameters, the function of the sweeper with FR₁, FR₂, FR₃, FR₄, FR₅, respectively. The quality configuration relationship between the user needs of household floor sweepers and product features is shown in Table 6.

Table 6 gives the relationship matrix of user demand - product function, the absolute importance of each function is calculated by the demand weight and demand - function relationship matrix, the specific calculation process is for the column where the demand weight is located and the column where each product function is located, each row of data is weighted and summed to get the absolute importance of the product function, and then normalization is done to get the relative importance, the results are shown in Table 7. The sweeping and cleaning function is the most important in household sweepers, followed by the intelligent control function monitoring and planning function, which needs to be focused on, and finally the optimization function and human-computer interaction function, respectively. The design and implementation of optimization function and human-computer interaction function can improve the user experience and user satisfaction, and the calculation results of the importance of each function in Table 7 are in line with the engineering reality, which provides a strong data support for the modular design of large-scale software system that meets the user’s needs.

Module partitioning is the first step in modular design, which determines how to decompose the system into a number of relatively independent modules. Common ways of module division include division by function, division by hierarchy and division by domain. The goal of module partitioning is to reasonably allocate system functions to each module, so that each module can independently complete a specific function and the dependency relationship between modules is minimized.

Module interface is a bridge for communication between modules, which defines the interaction between modules. The design of module interfaces should follow the principles of simplicity, stability and medical use to ensure efficient and reliable communication between modules. Common interface design methods include abstract class or interface based design, event or message based design, and service or API based design.

Module implementation refers to how the functionality of a module is concretely realized. Common module implementation methods include object-oriented programming, functional programming and componentized programming. Object-oriented programming implements module functionality by defining classes and objects, and utilizes features such as encapsulation, integration, and polymorphism to ensure module independence. Functional programming implements the functionality of a module by means of pure and higher-order functions, emphasizing statelessness and invariance. Componentized programming implements the functionality of modules by defining components or plug-ins, supporting modular assembly and dynamic replacement.

Module combination is the combination of individual modules into a complete system. Methods of module combination include dependency injection, mediator patterns, and process or state machine combination methods. Dependency injection manages the dependencies between modules through the framework, realizing the dynamic combination and replacement of modules. Intermediary pattern manages the communication between modules through intermediary objects to realize decoupling and collaboration of modules. Process or state machine manages the execution order and state transitions of modules by defining processes or state machines to realize the orderly combination and control of modules.

Particle swarm optimization algorithm is a global stochastic search algorithm based on group intelligence by simulating the migration and flocking behaviors of bird flocks during foraging [23]. Solving the software modular design problem using particle swarm optimization algorithm has to include key processes such as coding and initialization, position update, etc.

For the modular design problem of a software system, each particle represents a feasible modularized design solution, represented by a modular design matrix. The initialization of the population in the elementary particle swarm algorithm is done in a random initialization manner, which does not make use of the a priori knowledge of the problem to be solved, and the initial solution tends to be poor. In the software system method network diagram, the methods in which direct calling relationships occur are connected to each other by edges. The more frequent the calling relationship, the closer the two methods are connected. The two methods that have indirect calling relationships are measured by the minimum number of edges passing between two vertices in the graph to measure the degree of connection between the two vertices. Therefore, in this paper, taking into account the characteristics of the software modular design problem, the probabilistic choice of initialization is combined with the principle of the shortest path of the graph, and the initialization is carried out in this way to improve the quality of the initial solution.

The call relationship graph of a software system is expressed as G = (V,R), assuming that the software system is designed as m modules, then in Figure G to find the first m degree of the largest and not directly adjacent to each other as the center of the software module vertices, find the m center vertices are expressed as V′={v′1,v′2,⋯,v′k,⋯,v′m}(1≤k≤m)\[{V}'=\{{{{v}'}_{1}},{{{v}'}_{2}},\cdots ,{{{v}'}_{k}},\cdots ,{{{v}'}_{m}}\}(1\le k\le m)\], and V′⊆V, respectively, calculate the shortest path from each vertex to the center of each module can be expressed as dist(vi,v′k)(1≤k≤m)\[dist({{v}_{i}},{{{v}'}_{k}})(1\le k\le m)\], compare dist(vi,v′1),dist(vi,v′2),⋯dist(vi,v′m)\[dist({{v}_{i}},{{{v}'}_{1}}),dist({{v}_{i}},{{{v}'}_{2}}),\cdots dist({{v}_{i}},{{{v}'}_{m}})\], with the shortest path as a probability of choosing the calculation of the vertex to which the v_i belongs to the module center.

Assuming that the position of the w st particle at the time of iteration to step t is denoted as ClusterMt=[ aM11taM12t⋯aM1ntaM21taM22t⋯aM2nt⋮⋮⋮⋮aMm1taMm2t⋯aMmnt ]$Cluster_{M}^{t}=\left[ \begin{matrix} a_{M11}^{t} & a_{M12}^{t} & \cdots & a_{M1n}^{t} \\ a_{M21}^{t} & a_{M22}^{t} & \cdots & a_{M2n}^{t} \\ \vdots & \vdots & \vdots & \vdots \\ a_{Mm1}^{t} & a_{Mm2}^{t} & \cdots & a_{Mmn}^{t} \\ \end{matrix} \right]$, it is necessary to determine the position code ClusterMt+1$Cluster_{M}^{t+1}$ at the time of iteration to step t+1, i.e., it is necessary to determine the module to which each method in particle w belongs at the time of iteration to step t+1. Suppose that to determine the module to which method f_j(1≤j≤n) belongs, the value of the i(1≤i≤m)th row and j(1≤j≤n)th column of ClusterMt$Cluster_{M}^{t}$ is 1, and the other values of the jth column are 0, i.e., aMijt=1$a_{Mij}^{t}=1$, aM1jt=0$a_{M1j}^{t}=0$, aM2jt=0$a_{M2j}^{t}=0$, aM3jt=0,⋯$a_{M3j}^{t}=0,\cdots $, aMmjt=0$a_{Mmj}^{t}=0$. From which the matrix is constructed: (15) Aωti=[ αω11tαω12t⋯0⋯αω1ntαω21tαω22t⋯0⋯αω2nt⋮⋮⋮⋮αωi1tαωi2t⋯1⋯αωint⋮⋮⋮⋮αωm1tαωm2t⋯0⋯αωmnt ] $A_{\omega }^{{{t}_{i}}}=\left[ \begin{matrix} \alpha _{\omega 11}^{t} & \alpha _{\omega 12}^{t} & \cdots & 0 & \cdots & \alpha _{\omega 1n}^{t} \\ \alpha _{\omega 21}^{t} & \alpha _{\omega 22}^{t} & \cdots & 0 & \cdots & \alpha _{\omega 2n}^{t} \\ \vdots & \vdots & {} & \vdots & {} & \vdots \\ \alpha _{\omega i1}^{t} & \alpha _{\omega i2}^{t} & \cdots & 1 & \cdots & \alpha _{\omega in}^{t} \\ \vdots & \vdots & {} & \vdots & {} & \vdots \\ \alpha _{\omega m1}^{t} & \alpha _{\omega m2}^{t} & \cdots & 0 & \cdots & \alpha _{\omega mn}^{t} \\ \end{matrix} \right]$

In total, m such matrix ClusterMti(1≤i≤m)$Cluster_{M}^{{{t}_{i}}}\left( 1\le i\le m \right)$ can be constructed, and the fitness value for each positional code ClusterMti(1≤i≤m)$Cluster_{M}^{{{t}_{i}}}\left( 1\le i\le m \right)$ is calculated according to Eq. (15) fitnessMi*\[fitnes{{s}_{M_{i}^{*}}}\] the probability value for selecting each positional code ClusterMti$Cluster_{M}^{{{t}_{i}}}$ is calculated separately prob_Mi based on the calculated fitness value: (16) probxi= fitnessxi∑k=1mfitnessxk \[pro{{b}_{{{x}_{i}}}}=\frac{\text{ }fitnes{{s}_{{{x}_{i}}}}}{\sum\limits_{k=1}^{m}{fitnes{{s}_{{{x}_{k}}}}}}\]

Then method f_j selects module i with probability prob_i, thus determining the module to which method f_j belongs, and determining the module to which each method of the wth particle belongs, i.e., the value of Clus−terxt+1$Clus-ter_{x}^{t+1}$.

Traditional particle swarm optimization algorithms are not globally convergent due to the iterative formula traversal being too directional. In this paper, we give a global convergence theorem for hybrid optimization algorithms.

Theorem 1.An iterative evolutionary algorithm must converge globally to the optimal solution with probability 1 if the iterative algorithm has the following 2 conditions.

It is easy to see that the probabilistic selection particle swarm algorithm satisfies the 2 conditions stated in Theorem 1 above, so it globally converges to the optimal solution with probability 1. This is illustrated as follows: firstly, in the elite retention strategy of the particle swarm algorithm, the one with the largest value of the fitness is selected as gbest every time, which satisfies Condition 1. Secondly, the algorithm selects each possible location with a probabilistic value, and any two locations in the feasible solution space are reachable, at any moment, the probability of arriving at the other location from one location is greater than zero, with the ability to get rid of the local optimum, satisfying condition 2.

This section from the theoretical data and the actual case of 2 aspects of the experimental analysis, and in the follow-up on the design results and other algorithms for comparison, to verify the reasonableness and effectiveness of this paper’s method.

Theoretically, and with a set of test cases, that is, reflecting the large-scale software system amount of source code, can obviously analyze the modular design situation. Then use the algorithm proposed in this paper to carry out modularization design, the two to verify the reasonableness of the algorithm in this paper. Among them, merge and Carch are theoretical data. Merge is mainly a merger of 2 independent programs. Modular design, obviously 2 independent programs are divided into their own modules, 2 sub-modules and then based on the modularity of the design of a finer-grained sub-module.Carch is the implementation of this paper’s algorithm. The experimental results of the design carried out for the software system in Table 8 are shown in Table 9.

In fact, real open source software systems are also experimentally analyzed in this paper. In addition, a number of other open source software systems are also verified by extensive experiments. Unlike theoretical data, large open source systems cannot visualize their information, and the experimental data are analyzed with the help of source code analysis tools (e.g., Understand, Sourcenav), including the size of the system’s source code, the number of classes, and the inter-class calls within the source code, to get the size of the experimental data and its complexity.

In this section, based on the resultant modules designed from the above experiments, the design results of the hMetis algorithm are compared with the design results of the hMetis algorithm to analyze the goodness of the design results in this paper. The so-called good and bad design results are balanced between a tighter association within the module and a lower coupling between the module and the module. In order to intuitively see the difference between the two design results, the visualization of the design results is realized, i.e., the design result modules and inter-module associations are displayed, and the average module degree between the design modules derived from the 2 algorithms is calculated. Taking the case 4 listed in this paper as an example, the design results based on hMetis and the design results based on this paper’s method are displayed as shown in Fig. 5, respectively. (a) and (b) are the modular design results of the hMetis algorithm and this paper’s algorithm, respectively, where the dots represent the resultant modules, and the size of which indicates how many vertices are contained in the module, respectively, and the various dots are labeled by different colors, which indicate that the connection tightness within the module degree, the darker the color, the stronger the degree of cohesion, in a way, this is a better design, it is recommended, the modules are connected by different thickness of the line, representing the degree of coupling between the modules, the thicker the line, the greater the degree of inter-module coupling, which is not recommended. In Fig. 5(a), the modules seem to be more uniform in size, but the connecting lines between the modules are thicker compared to Fig. 5(b), i.e., there is a large degree of inter-module coupling in Fig. 5(a).

Figure 5.

Module division results

The values of segmented expected modularity obtained based on the 2 algorithms are shown in Fig. 6, and it can be seen that the design results obtained based on this paper’s algorithm have an expected modularity greater than that of the design based on the hMetis algorithm, i.e., this paper’s algorithm is better than the hMetis as a whole, and the reasonableness and effectiveness of this paper’s algorithm is verified.

Figure 6.

Expectation module degree contrast

The PSPSO algorithm has been widely applied to many large-scale software system modular design problems, but there are few studies on the time complexity of the algorithm. In this section, we will initially explore the relationship between the time complexity of the PSPSO algorithm and the size of the problem and the population size model.

To facilitate the analysis, the concept of pheromone is introduced in the PSPSO algorithm to characterize the biological properties of the particle population. In this section, the time complexity of the simplified PSPSO algorithm will be analyzed based on the pheromone rate, which is a function of pheromone, and the concept of pheromone rate is briefly introduced below. For (i^*,j^*)∈s^*, the pheromone rate is shown in the following equation: (18) c(i,t)=τα(i*,j*,t)∑k∈Diτα(i*,k,t) \[c(i,t)=\frac{{{\tau }^{\alpha }}\left( {{i}^{*}},{{j}^{*}},t \right)}{\sum\limits_{k\in {{D}_{i}}}{{{\tau }^{\alpha }}}\left( {{i}^{*}},k,t \right)}\] Where τ(i^*,j^*,t) is the pheromone on the optimal path (i^*,j^*) at the moment of t, and D_i is the set of all the paths that can be selected by the particle swarm located at point i.

Because the PSPSO algorithm is more complex, and its time complexity is more difficult to realize through theoretical research, the probability selection function and pheromone update strategy of the PSPSO algorithm are simplified as follows.

Probability selection function: (19) P(i,j,t)={ τα(i,j,t)∑k∈Diτα(i,k,t)if(i,j)∈Di0otherwise $P\left( i,j,t \right)=\left\{ \begin{array}{*{35}{l}} \frac{{{\tau }^{\alpha }}\left( i,j,t \right)}{\sum\limits_{k\in {{D}_{i}}}{{{\tau }^{\alpha }}}\left( i,k,t \right)} & if\left( i,j \right)\in {{D}_{i}} \\ 0 & otherwise \\ \end{array} \right.$

In the probability selection function of the above equation, the effect of heuristic information is not considered, so the analysis of complexity in this section mainly considers the effect of pheromone rate c(i,t) on complexity. From Eq. (19), it can be seen that the probability selection function is a bounded function, which is in line with the definition of the probability selection function in the PSPSO algorithm.

The pheromone update strategy is as follows: (20) τ(i,j,tm+1)=τ(i,j,tm)+μ(i,j,tm) \[\tau \left( i,j,{{t}_{m+1}} \right)=\tau \left( i,j,{{t}_{m}} \right)+\mu \left( i,j,{{t}_{m}} \right)\]

The effect of the pheromone residual rate function is not considered in the pheromone update strategy.

In the following, the time complexity of the PSPSO algorithm will be initially analyzed based on the simplified model of the algorithm given above.

PSPSO-1: Set the number of particle swarms k = 1, ∀(x,y)∈s, s∈S_iter(t)(t = 1,2,⋯), the pheromone increment function μ(i,j,t_m) is variable, and the probability selection function and pheromone updating strategy are shown in Eqs. (19) and (20).

Let ξ^IGACO−1(t) be the stochastic process of PSPSO-1 algorithm, and the expectation of pheromone rate at the moment of t+1 is E{c(i,t+1)|ξ^tGACo−1a(t)}. The relationship between this expectation and pheromone rate c(i,t) is derived and simplified by the following formula.

The derivation process is as follows: (21) E{ c(i,t+1)|ξIGACO−1(t) }=∑〈 i−1,y 〉∈D(i−1)−{ 〈 i−1,i 〉 }[ τ(i−1,y,t)∑〈 i−1,x 〉∈D(i−1)τ(i−1,x,t)⋅τ(i−1,i,t)l ]+τ(i−1,i,t)∑〈 i−1,x 〉∈D(i−1)τ(i−1,x,t)⋅τ(i−1,i,t)+μ(i−1,i,t)g=∑〈 i−1,y 〉∈D(i−1)−{ 〈 i−1,i 〉 }[ τ(i−1,i,t)∑〈 i−1,x 〉∈D(i−1)τ(i−1,x,t)⋅τ(i−1,y,t)l ]+τ(i−1,i,t)∑〈 i−1,x 〉∈D(i−1)τ(i−1,x,t)⋅τ(i−1,i,t)+μ(i−1,i,t)g \[\begin{align} & E\left\{ c\left( i,t+1 \right)\left| {{\xi }^{IGACO-1}} \right.\left( t \right) \right\} \\ & =\sum\limits_{\left\langle i-1,y \right\rangle \in D\left( i-1 \right)-\left\{ \left\langle i-1,i \right\rangle \right\}}{\left[ \frac{\tau \left( i-1,y,t \right)}{\sum\limits_{\left\langle i-1,x \right\rangle \in D\left( i-1 \right)}{\tau }\left( i-1,x,t \right)}\cdot \frac{\tau \left( i-1,i,t \right)}{l} \right]} \\ & +\frac{\tau \left( i-1,i,t \right)}{\sum\limits_{\left\langle i-1,x \right\rangle \in D\left( i-1 \right)}{\tau }\left( i-1,x,t \right)}\cdot \frac{\tau \left( i-1,i,t \right)+\mu \left( i-1,i,t \right)}{g} \\ & =\sum\limits_{\left\langle i-1,y \right\rangle \in D\left( i-1 \right)-\left\{ \left\langle i-1,i \right\rangle \right\}}{\left[ \frac{\tau \left( i-1,i,t \right)}{\sum\limits_{\left\langle i-1,x \right\rangle \in D\left( i-1 \right)}{\tau }\left( i-1,x,t \right)}\cdot \frac{\tau \left( i-1,y,t \right)}{l} \right]} \\ & +\frac{\tau \left( i-1,i,t \right)}{\sum\limits_{\left\langle i-1,x \right\rangle \in D\left( i-1 \right)}{\tau }\left( i-1,x,t \right)}\cdot \frac{\tau \left( i-1,i,t \right)+\mu \left( i-1,i,t \right)}{g} \\ \end{align}\]

In Eq. (21), to simplify the equation, let: (22) l=μ(i−1,y,t)+∑〈 i−1,x 〉∈D(i−1)τ(i−1,x,t) \[l=\mu \left( i-1,y,t \right)+\sum\limits_{\left\langle i-1,x \right\rangle \in D\left( i-1 \right)}{\tau }\left( i-1,x,t \right)\] (23) g=μ(i−1,i,t)+∑〈 i−1,x 〉∈D(i−1)τ(i−1,x,t) \[g=\mu \left( i-1,i,t \right)+\sum\limits_{\left\langle i-1,x \right\rangle \in D\left( i-1 \right)}{\tau }\left( i-1,x,t \right)\] (24) h=∑〈 i−1,y 〉∈D(i−1)−{ 〈 i−1,i 〉 }τ(i−1,y,t)⋅[μ(i−1,i,t)−μ(i−1,y,t)]l \[h=\sum\limits_{\left\langle i-1,y \right\rangle \in D\left( i-1 \right)-\left\{ \left\langle i-1,i \right\rangle \right\}}{\frac{\tau \left( i-1,y,t \right)\cdot [\mu \left( i-1,i,t \right)-\mu \left( i-1,y,t \right)]}{l}}\]

Based on the concept of pheromone rate, equation (21) can be further simplified as follows: (25) E{ c(i,t+1)|ξIGACO−1(t) }=[ τ(i−1,i,t)+μ(i−1,i,t)g+∑〈 i−1,y 〉∈D(i−1)−{ 〈 i−1,i 〉 }τ(i−1,y,t)l ]⋅c(i,t)=[ τ(i−1,i,t)+μ(i−1,i,t)g+∑〈 i−1,y 〉∈D(i−1)−{ 〈 i−1,i 〉 }τ(i−1,y,t)g+hg ]⋅c(i,t)=[ μ(i−1,i,t)+∑〈i−1,x〉∈D(i−1)τ(i−1,i,t)g+hg ]⋅c(i,t)=(gg+hg)⋅c(i,t)=(1+hg)⋅c(i,t) \[\begin{align} & E\left\{ c\left( i,t+1 \right)\left| {{\xi }^{IGACO-1}} \right.(t) \right\} \\ & =\left[ \frac{\tau \left( i-1,i,t \right)+\mu \left( i-1,i,t \right)}{g}+\sum\limits_{\left\langle i-1,y \right\rangle \in D\left( i-1 \right)-\left\{ \left\langle i-1,i \right\rangle \right\}}{\frac{\tau \left( i-1,y,t \right)}{l}} \right]\cdot c\left( i,t \right) \\ & =\left[ \frac{\tau \left( i-1,i,t \right)+\mu \left( i-1,i,t \right)}{g}+\sum\limits_{\left\langle i-1,y \right\rangle \in D\left( i-1 \right)-\left\{ \left\langle i-1,i \right\rangle \right\}}{\frac{\tau \left( i-1,y,t \right)}{g}+\frac{h}{g}} \right]\cdot c\left( i,t \right) \\ & =\left[ \frac{\mu \left( i-1,i,t \right)+\sum\limits_{\langle i-1,x\rangle \in D(i-1)}{\tau }\left( i-1,i,t \right)}{g}+\frac{h}{g} \right]\cdot c\left( i,t \right) \\ & =\left( \frac{g}{g}+\frac{h}{g} \right)\cdot c\left( i,t \right) \\ & =\left( 1+\frac{h}{g} \right)\cdot c\left( i,t \right) \\ \end{align}\]

From equations (22), (23), and (24), l, g, and h are functions of variables i and t, and in order to conveniently represent and highlight the relationship between the expectation of the pheromone rate at the next moment and the current pheromone rate, let the function ψ(i,t)=hg$\psi \left( i,t \right)=\frac{h}{g}$, then: (26) E{ c(i,t+1)|ξIGACO−1(t) }=[ 1+ψ(i,t) ]⋅c(i,t) \[E\left\{ c\left( i,t+1 \right)\left| {{\xi }^{IGACO-1}} \right.\left( t \right) \right\}=\left[ 1+\psi \left( i,t \right) \right]\cdot c\left( i,t \right)\]

From equation (26), for i = 1,2,⋯,n, t = 1,2,⋯,E{c(i,t+1)|ξ^IGACO−1a(t)} is proportional to c(i,t). Also, according to the concept of pheromone rate, c(i,t) is proportional to the pheromone increment function μ(i–1,i,t–1), then it can be concluded that E{c(i,t+1)|ξ^IGACO−1a(t)} is also proportional to the pheromone increment function μ(i–1,i,t–1).

If ψ(i,t)≥ψ>0, then: (27) E{ c(i,t)|ξIGACO−1(t−1) }≥(1+ψ)⋅c(i,t−1) \[E\left\{ c\left( i,t \right)\left| {{\xi }^{IGACO-1}} \right.\left( t-1 \right) \right\}\ge \left( 1+\psi \right)\cdot c\left( i,t-1 \right)\] (28) E{ c(i,t)−c(i,t−1)|ξIGACO−1(t−1) }≥ψ⋅c(i,t−1) \[E\left\{ c\left( i,t \right)-c\left( i,t-1 \right)\left| {{\xi }^{IGACO-1}} \right.\left( t-1 \right) \right\}\ge \psi \cdot c\left( i,t-1 \right)\]

Based on the analysis of the convergence rate of the PSPSO algorithm, it can be seen that { ξIGACO−1(t) }t=0+∞$\left\{ {{\xi }^{IGACO-1}}\left( t \right) \right\}_{t=0}^{+\infty }$ is an absorbing state Markov process, so: (29) c(i,t)≈E{ c(i,t)|ξIGACO-1(0),ξIGACO-1(1),⋯,ξIGACO-1(t−1) }=E{ c(i,t)|ξIGACO-1(t−1) } \[\begin{align} & c\left( i,t \right)\approx E\left\{ c\left( i,t \right)\left| {{\xi }^{IGACO-1}} \right.\left( 0 \right),{{\xi }^{IGACO-1}}\left( 1 \right),\cdots ,{{\xi }^{IGACO-1}}\left( t-1 \right) \right\} \\ & =E\left\{ c\left( i,t \right)\left| {{\xi }^{IGACO-1}}\left( t-1 \right) \right. \right\} \end{align}\]

From equation (27): (30) E{ c(i,t)|ξIGACO−1(0),ξIGACO−1(1),⋯,ξIGACO-1(t−1) }=E{ c(i,t)|ξIGACO−1(t−1) }≥(1+ψ)⋅c(i,t−1)≈(1+ψ)⋅E{ c(i,t)|ξIGACO−1(t−2) }≥(1+ψ)2⋅c(i,t−2)⋮≥(1+ψ)t⋅c(i,0) \[\begin{align} & E\left\{ c\left( i,t \right)\left| {{\xi }^{IGACO-1}} \right.\left( 0 \right),{{\xi }^{IGACO-1}}\left( 1 \right),\cdots ,{{\xi }^{IGACO-1}}\left( t-1 \right) \right\} \\ & =E\left\{ c\left( i,t \right)\left| {{\xi }^{IGACO-1}} \right.\left( t-1 \right) \right\} \\ & \ge \left( 1+\psi \right)\cdot c\left( i,t-1 \right) \\ & \approx \left( 1+\psi \right)\cdot E\left\{ c\left( i,t \right)\left| {{\xi }^{IGACO-1}} \right.\left( t-2 \right) \right\} \\ & \ge {{\left( 1+\psi \right)}^{2}}\cdot c\left( i,t-2 \right) \\ & \vdots \\ & \ge {{\left( 1+\psi \right)}^{t}}\cdot c(i,0) \\ \end{align}\]

From Eq. (27), the pheromone rate at moment t = 0 is c(i,0) = 1/n, so Eq. (30) can be further simplified as: (31) E{ c(i,t)|ξIGACO−1(0),ξIGACO−1(1),⋯,ξIGACO−1(t−1) }≥(1+ψ)tn \[E\left\{ c\left( i,t \right)\left| {{\xi }^{IGACO-1}} \right.\left( 0 \right),{{\xi }^{IGACO-1}}\left( 1 \right),\cdots ,{{\xi }^{IGACO-1}}\left( t-1 \right) \right\}\ge \frac{{{\left( 1+\psi \right)}^{t}}}{n}\] (32) t≤log1+ψ(n⋅E{ c(i,t)|ξIGACO−1(0),ξIGACO−1(1),⋯,ξIGACO−1(t−1) }) \[t\le {{\log }_{1+\psi }}\left( n\cdot E\left\{ c\left( i,t \right)\left| {{\xi }^{IGACO-1}} \right.\left( 0 \right),{{\xi }^{IGACO-1}}\left( 1 \right),\cdots ,{{\xi }^{IGACO-1}}\left( t-1 \right) \right\} \right)\]

Set c_low>c(i,j,0) = 1/n, and when t≥t₀, c(i,t)≥c_low, and when 0<t≤t₀, c(i,t)≤c_low. According to equation (32): (33) E[ t0|ξIGACO−1(0) ]≤log1+ψ(n⋅E{ c(i,t)|ξIGACO−1(0),ξIGACO−1(1),⋯,ξIGACO−1(t−1) }) \[\begin{align} & E\left[ {{t}_{0}}\left| {{\xi }^{IGACO-1}} \right.\left( 0 \right) \right] \\ & \le {{\log }_{1+\psi }}\left( n\cdot E\left\{ c\left( i,t \right)\left| {{\xi }^{IGACO-1}} \right.\left( 0 \right),{{\xi }^{IGACO-1}}\left( 1 \right),\cdots ,{{\xi }^{IGACO-1}}\left( t-1 \right) \right\} \right) \\ \end{align}\]

It is also obtained from equation (29): (34) E[ t0|ξIGACO−1(0) ]=log1+ψ(n⋅E{ c(i,t)|ξIGACO−1(t−1) })=log1+φ(n⋅c(i,t))≤log1+φ(n⋅clow) \[\begin{align} & E\left[ {{t}_{0}}\left| {{\xi }^{IGACO-1}} \right.\left( 0 \right) \right]={{\log }_{1+\psi }}\left( n\cdot E\left\{ c\left( i,t \right)\left| {{\xi }^{IGACO-1}} \right.\left( t-1 \right) \right\} \right) \\ & ={{\log }_{1+\varphi }}\left( n\cdot c\left( i,t \right) \right) \\ & \le {{\log }_{1+\varphi }}\left( n\cdot {{c}_{low}} \right) \end{align}\]

From the probabilistic selection function of the PSPSO-1 algorithm Eq. (29), which defines ω(c_low,t₀) as an upper bound for Eλ according to Eq. (18), we have: (35) ω(clow,t0)=[ 1−ϕ(0) ][ 1−(1−clown)m ]−1 \[\omega \left( {{c}_{low}},{{t}_{0}} \right)=\left[ 1-\phi \left( 0 \right) \right]{{\left[ 1-{{\left( 1-c_{low}^{n} \right)}^{m}} \right]}^{-1}}\]

For t′ = 1,2,⋯, i = 1,2,⋯,n, let t′ = t–₀t, c(i,t′)≥c_low(t₀)>0. For when t>t₀, c(i,t)>c_low>0, by Eq. (35): (36) Eλ≤t0+ω(clow,t0) \[E\lambda \le {{t}_{0}}+\omega \left( {{c}_{low}},{{t}_{0}} \right)\]

Because Eλ=∑t=0+∞[ (1−ϕ(0))∏i=1i(1−pi) ]$E\lambda =\sum\limits_{t=0}^{+\infty }{\left[ (1-\phi (0))\prod\limits_{i=1}^{i}{\left( 1-{{p}_{i}} \right)} \right]}$, it follows from the definition of the process model of the PSPSO algorithm that ξ^IGACO−1(0) = (S(0),T(0)), S(0) = {S_bs(0)∪S_iter(0)} = ∅ because during the initialization of the algorithm, S(t) = {S_bs(t)∪S_iter(t)} = ∅, so ϕ(0) = 0. Then Eλ=∑t=0+∞[ ∏i=1t(1−pi) ]$E\lambda =\sum\limits_{t=0}^{+\infty }{\left[ \prod\limits_{i=1}^{t}{\left( 1-{{p}_{i}} \right)} \right]}$.

From ϕ(0) = 0, it follows that P{ξ^IGACO−1(0)∈Y^*} = 0, so Eλ=∑t=0+∞[ ∏i=1t(1−pi) ]$E\lambda =\sum\limits_{t=0}^{+\infty }{\left[ \prod\limits_{i=1}^{t}{\left( 1-{{p}_{i}} \right)} \right]}$ is independent of ξ^IGACO−1(0). This gives E[Eλ|ξ^IGACO−1(0)] = Eλ.

According to equation (35), ω(c_low,t₀) is independent of ξ^IGACO−1(0), so E[ω(c_low,t₀)|ξ(0)] = ω(c_low,t₀). Summarizing the above, we have: (37) Eλ≤E[ t0|ξIGACO-1(0) ]+ω(clow,t0) \[E\lambda \le E\left[ {{t}_{0}}\left| {{\xi }^{IGACO-1}} \right.\left( 0 \right) \right]+\omega \left( {{c}_{low}},{{t}_{0}} \right)\]

Since the number of particle swarms in the PSPSO-1 algorithm is 1, m = 1 in Eq. (35) and ϕ(0) = 0, the following equation is obtained: (38) ω(clown)=clow−n \[\omega \left( c_{low}^{n} \right)=c_{low}^{-n}\] can be obtained from Eqs. (34), (37), and (38): (39) EλIGACO−1≤log(1+ψ)(n⋅clow)+clow−n \[E{{\lambda }^{IGACO-1}}\le {{\log }_{\left( 1+\psi \right)}}\left( n\cdot {{c}_{low}} \right)+c_{low}^{-n}\]

According to the above equation, it can be concluded that if the problem size n becomes larger, more running time is required.

Based on the above case study of the PSPSO algorithm it can be concluded that the time complexity of the PSPSO algorithm increases as the problem size increases and decreases as the number of particle swarms increases.

The following is an analysis of the space complexity of the particle swarm optimization algorithm based on probabilistic selection. Taking n large-scale software system modular design problem as an example, m particle swarms are used in the PSPSO algorithm, then through the analysis of each step of the particle swarm optimization algorithm based on probabilistic selection, its space complexity can be obtained as: (40) S(n)=O(n2)+O(n⋅m) \[S\left( n \right)=O\left( {{n}^{2}} \right)+O\left( n\cdot m \right)\]

Dynamic programming is a method of solving complex problems by decomposing them into subproblems. It stores the solution for each subproblem to avoid repeated computations. Dynamic programming is suitable for problems with overlapping subproblems and optimal substructure properties, such as Fibonacci series, shortest path problem and knapsack problem. The time complexity of dynamic programming is usually O(n²) or O(n³), but lower complexity can be achieved with appropriate optimization. For example, the geodesic formula for the Fibonacci series can be optimized to a time complexity of O(n) by memetic recursion.

A greedy algorithm is a method for solving globally optimal problems by choosing the current optimal solution at each step. Greedy algorithms are suitable for problems that satisfy the greedy selection property and the optimal substructure property, such as minimum number of generators, single-source shortest paths, and active selection. The time complexity of a greedy algorithm is usually O(nlog(n)) or O(n²), depending on the specific implementation. For example, the time complexity of the Kruskal algorithm for solving the minimum number of generators is O(E log v), where E is the number of edges and v is the number of vertices. The advantage of the greedy algorithm is its simplicity and efficiency, but it may not be able to obtain a globally optimal solution in a certain write situation.

Partitioning is a method of solving complex problems by recursively decomposing them into subproblems. The partition method is suitable for problems with independent subproblems and merged solution properties, such as fast sorting, merge sorting and large integer multiplication. The time complexity of the partition method is usually O(nlog(n)) or O(n²), depending on the specific decomposition and merging strategies. For example, fast sorting has an average time complexity of O(nlog(n)), but in the worst case may degenerate to O(n²). The advantage of the partitioning method is its efficiency and flexibility for many different types of problems.