Connecting Cities: A Case Study on the Application of Morphological Shortest Paths

Abstract

Navigatingdensely connected networks can be complex due to the different connection structures present within a network. No explicit algorithms are designed specifically for this navigation, so heuristic approaches and existing network systems are often employed. However, this task can become computationally asymmetrical, as the complexity of creating a representation of the city is lower than the complexity involved in identifying a set of feasible paths in a combinatorial order. This paper extends the applicability of morphological approaches to compute the shortest path in smart cities, driven by the complexity and size of the vital communication infrastructure. As is well known, this communication infrastructure changes dynamically, particularly with the evolving connection paths due to continuous population growth. Consequently, efficient communication trajectories can quickly become obsolete. The challenge of computing the best trajectories to respond more quickly to the growing population comes with high computational complexity. This paper presents an application that uses a discrete algorithm designed to compute the shortest path through a morphological approach. Specifically, it seeks to identify the best trajectory within a densely populated city based on a complex density graph. By incorporating morphological approaches into path-search algorithms, we can define a new family of methods that operate in discrete spaces with a morphological representation, resulting in approaches that have lower computational requirements. Other well-known applications in this context include the delivery of resources, such as managing electrical power consumption or minimizing time delays in resource delivery. This task is essential but classified as an NP problem, making it an appropriate scenario for applying the proposed algorithm to navigate a dense graph. The paper highlights the well-known problem of finding the shortest path as one of the potential applications of the introduced algorithm. The algorithm aims to identify the optimal path trajectory within a graph representing a dense city’s real scenario. This discussion compares and contrasts the proposal with other established approaches, highlighting the advantages and characteristics of the proposed method.

1. Introduction

Navigating through a graph is a crucial task in graph theory. Finding the most appropriate path trajectories is one of the critical aspects of this task. Many approaches in the literature deal with this task by limiting and conditioning it to particular contexts, making it computable only in closed scenarios [1,2,3,4,5]. The task of computing the most suitable paths in a graph can be grouped into the following general approaches.

The approaches and techniques are classified into three large groups:

• Global search approach.
• Local search approach.
• Hybrid or bio-inspired search approach.

The first approach, global search, focuses on considering the entire graph (or search space) to determine the best possible solution to the problem. This approach usually requires initial conditions and satisfaction criteria to find the best solution in the entire search space. Such methods typically employ a global directional cost function, which guides the process through the whole search space toward the best possible solution. However, this approach has the disadvantage of analyzing the entire search space, significantly increasing the computational resources needed to obtain a solution [6,7]. This approach becomes unsuitable; even if it computes the best path, the resources required to represent the search space require an enormous amount of computer memory. This is, the combinatorial of possible paths in a fully connected graph has a factorial growth and not viable to compute in graphs with few nodes.

To mitigate this difficulty, criteria are applied that allow the search space to be reduced or divided into manageable sections, adapting to the available computational resources [8,9,10,11]. This helps prevent the algorithm from falling into cycles or moving away from the optimal solution. Strategies based on metaheuristic search algorithms are frequently employed, such as Evolutionary Algorithms (EAs) [12], Branch and Bound (B&B) [13], Particle Swarm Optimization (PSO) [14], and Simulated Annealing (SA) [15].

The second approach, known as local search, assumes that it is possible to find an optimal solution that, at best, could coincide with the global solution of the problem. This method is based on iteratively modifying an initial solution and exploring neighboring solutions, which in most cases leads to a local optimum [16]. These techniques can be classified according to the search dynamics they employ. On the one hand, there are exact deterministic methods, such a B&B, which follow a fixed sequence of steps to ensure an exact optimal solution is obtained. On the other hand, there are non-deterministic stochastic methods, such SA and PSO, which incorporate random variables in the search process, allowing for exploring solutions in a more flexible and adaptive way.

This approach is suitable when prior information, whether complete or partial, about the search space, is available. This is its main advantage as it requires fewer computational resources compared to the global approach. However, its limitation lies in the fact that, when reaching local optima, there is a risk of incorrectly assuming that the global solution of the problem has been found [17,18,19]. In the literature, one can find heuristic local optimization methods that follow this approach, such as the Tabu Algorithm (TA) [20], and the Gradient Descent Search (GDS) [21].

Finally, hybrid approaches combine the global and local search methodologies described above, resulting in computationally tractable solutions by implementing constraints that always lead to a global solution to the problem. This approach requires, in most cases, prior information on the search space, which allows for the formulation of criteria, such as heuristics, applicable to specific problems within given contexts [22,23,24].

Algorithms bio-inspired by natural or biological processes are frequently found in the literature, aiming to improve search efficiency. In addition to EA, others stand out, such as Ant Colony Optimization (ACO) [25,26,27], Memetic Algorithms (MAs) [28], and the Sparrow Search Algorithm (SSA) [29].

Determining the best path within a graph is an asymmetrical task, and the complexity of creating a representation for this task is relatively low. However, finding the most suitable path is an NP-complete problem due to the combinatorial space it generates. Therefore, it is essential to develop alternative solutions that can help identify appropriate paths while avoiding computationally intractable challenges. The specific context and operations involved impose constraints that can make the problem more manageable.

Mathematical morphology (MM) features two fundamental operations: dilation and erosion. These can be combined to create new operators and filters [30,31,32,33,34]. Today, morphological techniques are a generalized framework for managing grid structures or arbitrary graph spaces associated with lattices through morphological operators [35,36]. These operators act as predecessor and successor functions within an induced relation in a given graph space.

This work presents a generalized framework for graph traversal based on morphological operators. In this framework, users must specify the initial and final conditions of the workspace as inputs. The approach defines morphological operators starting from the minimum connected subgraph, the infimum, and the overall space, referred to as the supremum. This methodology has proven to be efficient in finding optimal routes in graph-based spaces [37].

Cities and human populations tend to proliferate in various contexts. This growth often leads to a shortage of essential goods city residents need. Several expanding cities have implemented automated monitoring systems that enhance resource delivery, communication, and surveillance to address this issue. However, these systems require various resources, including electronic devices, communication infrastructure, data storage, and data analysis capabilities [38,39,40].

Historically, many studies have employed technological approaches to detect and monitor urban variables [41]. By analyzing these variables, cities can make the most effective decisions in response to different events [42,43]. Communication is crucial for sustaining our society and economy, as it helps balance required resources with timely responses [44,45,46].

Additionally, finding the most reliable routes for transportation within the city is essential. This task may involve transporting people, traffic, electrical supplies, or other goods across a specific network.

A dense city can be represented as a graph of interconnections, and finding the best path within this graph is a challenging task that requires NP algorithms [47,48,49]. The foundation for modeling a city as a dense graph is less complex than inspecting the solution space in search of feasible required solutions, making it an asymmetrical task. Consequently, this proposal expands upon this idea by representing a dense city as a high-dimensional graph with the objective of identifying feasible paths for delivering the city’s resources. The computational complexity is managed through a mathematical approach [37] designed to reduce this complexity.

The remainder of the paper is structured into different sections. Section 2 briefly introduces the theoretical concepts of MM and graph theory. Section 3 describes the experimental process, presents the findings obtained from the experimental method, and analyzes them. Finally, Section 4 summarizes the findings, addresses the limitations of the methodology used, and describes possible lines for future research.

2. Fundamentals of Graph Theory and Mathematical Morphology

This section discusses the theoretical foundations of graph theory, which allow information to be represented in an abstract structure with a high level of meaning. This theory is particularly useful for modeling real problems, such as the intricate way streets and avenues are interconnected, facilitating their analysis and optimization. Furthermore, the proposed solution is based on a reference framework based on tools offered by morphological mathematics, specifically on dilation, considered as a base operator. This operator allows the search space defined by the graph to be covered iteratively, generating an order relation that determines the path of the nodes in the graph.

2.1. Fundamental Definitions

As a fundamental part of modeling the base problem of searching for an optimal route in a discrete space, it is necessary to define the theoretical model based on a graph. In this article, we will consider a graph with three main characteristics: it will be undirected, connected, and weighted. The literature defines a graph as a structure consisting of $N\times E$, mathematically defined as an ordered pair $G=(N,E)$ where $N=\{n_1,\dots ,n_k\}$ and represents a non-empty set of nodes or vertices, and $E=\{e_1,\dots ,e_l\}$ is the set of edges, which represent the connections between pairs of nodes; in this sense, the definition in extensive way is represented by an enumerated set $G=\left\{\right(n,e\left)\right|n\in N,e\in E\}$ [31,50]. A graph in this context could represent a search space of an encoded scenario. In these terms, a scenario illustrates the description of nodes and connected routes to move between two states (see Figure 1).

Let $\mathcal{L}$ be a lattice constructed from G, which expresses the new order relation with an initial reference subgraph space $x_0\subseteq G$, and the set of subgraphs generated sequentially, which must cover the graph space G respecting the established order ${\le }_G$, is represented as a process composed of a set of finite number of steps. At the end of the process, the subgraph $x_0$ covers the search space in its entirety, where the intermediate steps are defined by the homomorphic structure defined in $\lambda$, such that the order ${\le }_G$ is preserved through said correspondence between two sets in a binary operation. Furthermore, any i-th intermediate step constructs a subgraph $x_i\subseteq G$ describing how consecutive finite states cover the entire search space G. Now, initially consider these two subgraphs as follows: $x_0,x^{*}\in \mathcal{L}$ such that $x_0\subseteq G$ and $x^{*}\subseteq G$.

In this way, the subgraph $x^{*}$ can be reached starting from the subgraph $x_0$ by successively applying the dilation operator to it, even if $x^{*}$ does not satisfy the order relation defined for G. The approximation is completed by composing the dilating operator naming as covering operator:

$${\mathrm{cover}}_{\lambda }(x_0,x^{*})\leftarrow {\delta }_{\lambda }^k·\dots ·{\delta }_{\lambda }^1{x}_0$$

(1)

where k represents the number of times the operator dilation ${\delta }_{\lambda }$ is successively composed and denotes the compositional operator. The k-th dilation represents the first step forward until the condition $({\delta }_{\lambda }^k·\dots ·{\delta }_{\lambda }^1{x}_0)\cap x^{*}=x^{*}$ becomes true.

For all nodes involved to reach $x^{*}$, the number of times each node becomes dilated as the result of ${\mathrm{cover}}_{\lambda }(x_0,x^{*})$ is denoted for a set of nodes as $|{\mathrm{cover}}_{\lambda }(x_0,x^{*})|$. The function becomes $|·|:2^G\to \{G\times f\}$ such that for all nodes $n_j\in x_i$, the cumulative frequency $f_j$ for a particular node is expressed as a $(n_j,f_j)$ tuple. Each $f_j$ counter is computed for all nodes in a given graph.

$$f_j=\sum _{l=1}^{k}b(n_j,{\delta }_{\lambda }^l·\dots ·{\delta }_{\lambda }^1{x}_0)$$

(2)

where b is a function defined as follows: $b(n_j,{\delta }_{\lambda }^j·\dots ·{\delta }_{\lambda }^1{x}_0)=\left\{\begin{array}{cc}+1\hfill & \mathrm{if} n_j\in {\delta }_{\lambda }^j·\dots ·{\delta }_{\lambda }^1{x}_0\hfill \\ 0\hfill & \mathrm{Other} \mathrm{case}\hfill \end{array}\right.$ where $n_j$ is a node. According to the notation, the expression $|{\mathrm{cover}}_{\lambda }{(x_0,x^{*})}_{n_j}|$ refers to the particular frequency $f_j$ for the node $n_j$.

Figure 2 illustrates the search process in the graph space using frequency-associated covering, performed by successive dilations. The process starts from a reference or origin node (represented in black) and extends to the nodes that have not yet been covered. This process ends when the target node is reached, represented by a double circle. Figure 2 illustrates the progress of the process, where the dilated nodes (represented in gray) cover the search space, applying only to the nodes that have not yet been processed. The frequency is defined by successive dilations, increasing its counter as the covering process progresses. It can be observed that the frequency nodes with the highest natural value become the new reference and the limits become the frequencies with the lowest associated natural value. Finally, the process of morphological covering by dilation stops before reaching the objective node (see Figure 2). The initial reference node shows the cost frequency value to reach the objective.

2.2. Search Strategy

The strategy used in this work is based on a morphological search, implemented using fundamental operators of Morphological Mathematics: dilation and erosion, which generate a structure in the form of a lattice. The process starts from a reference state represented by a subgraph of the graph that models the complete search space, thus defining the starting point for calculating the final trajectory. From the first dilation applied to the reference node, a set of new subgraphs is generated that make up a lattice, representing all the possible routes that are candidates to be a trajectory solution. When the target node is reached, the optimal path trajectory is calculated by following the natural order defined by the coverage process through successive morphological dilations, taking into account the frequency associated with these operations. Finally, the optimal trajectory is estimated by tracing a reverse path from the target node to the initial reference node.

In [37], a morphological shortness algorithm is presented based on the analysis of local neighborhoods. This algorithm is suitable for our proposal because it operates on discrete topological spaces, with neighborhoods defined by the topological structure of the structuring element. The ability to modify the shape of this element allows for the inclusion of navigation constraints within the graph space. Furthermore, the covering operator establishes an order relation with low computational complexity for the best navigating paths.

Algorithm described in Algorithm 1 employs a morphological covering operator, beginning from the node $x_0$ and continuing until the successive dilations reach the target D. Typically, the intersection of $x_0$ and D is an empty set, represented as $x_0\cap D=\{\varnothing \}$. Semantically, when the intersection of these two elements becomes a non-empty set, it indicates that the graph’s starting point and destination are the same. The covering operator employs a dilation operator as the primary method for examining the jump-neighborhood within the graph. The shape of the dilation and structural elements determines the algorithm’s strategy for navigating the graph. Larger structural elements indicate a more aggressive search strategy, which can converge quickly with fewer iterations, instead of increasing local complexity to identify the path. In contrast, smaller structural elements facilitate a more detailed search strategy while minimizing computational complexity.

The covering dilation results in a graph denoted by $x^{*}$, such that $x^{*}\cap (x_0\cup D)=x_0\cup D$. This means that the destination D has been reached through successive dilations starting from $x_0$.

Algorithm 1 Morphological search with the constraint of $x_0\subseteq x^{*}$ and $D\subseteq x^{*}$ [37].

1: procedure SearchProcess($x_0,D,x^{*},G,\lambda$) 2:     $M\leftarrow {\mathrm{cover}}_{\lambda }(x_0,D)$    ▹ Iterates covering the search space $x^{*}$ until the goal D is reached. 3:     $Mn=\left|\right|M\left|\right|$ or $Mn=\left|M\right|$               ▹ Cost of traveling across ${\le }_G$ order. 4:     Starting in any $s_i\in S_0$ 5:     $C\leftarrow M{n}_{x_0}$                   ▹ Initial cost with the weight of node $n_i$. 6:     $P\leftarrow \{\varnothing \}$                           ▹ Initial path trajectory. 7:     while a node $n_i\in D$ is not reached do         ▹ Repeat until a solution is found. 8:         $n_{i+1}\leftarrow C\left(n_i\right)$               ▹ Advance in the best path according to C. 9:         $P=P\cup \left\{(n_i,n_{i+1})\right\}$                   ▹ Adds current traveling node. 10:       $n_i\leftarrow n_{i+1}$                     ▹ Further state becomes current state. 11:       $C\leftarrow C+M{n}_{n_i}$                          ▹ Update cost $x_i$. 12:     end while 13: return P, C                     ▹ Return path trajectory P and cost C. 14: end procedure

The Algorithm 1 takes as an input a G graph space, $x^{*}\subseteq G$ represents the covering by dilation to reach destination $D\subseteq G$ starting from $x_0\subseteq s^{*}$ that represents the initial position (or reference), and $D\subseteq x^{*}$ repents the destination. The last input is a structural graph-connection neighborhood. It represents a homomorphic structure for the lattice navigation $\lambda \subseteq x^{*}$. The algorithm outputs a path that follows the search criterion and the total cost associated with displacing along this path. The complete Algorithm is described in Algorithm 1. This algorithm computes the best trajectory under the assumption that there are one or more solutions.

The function C criterion (line 8 of Algorithm 1) takes as input a given node n and a set of adjacent nodes expressed as $N=\{e_i\mathrm{such}\mathrm{that}(n,e_i)\in G\}$. The output is a node that denotes the best transition from the current node n to any node belonging to its neighborhood. The function C analyzes the neighborhood, selecting the best node (minima or maxima) to construct the path.

The Algorithm 1 implementation assumes that the destination is a subgraph denoted by $D\subseteq x^{*}$. In a blind search, the assumption that there is at least one path from $x_0$ to D becomes complicated. In the particular case that $D\cap x^{*}=\{\varnothing \}$, Algorithm 1 becomes undecidable (falls into an infinite loop), which is considered a disadvantage.

2.3. Implementation Issues

The expected connections between its nodes primarily influence the implementation of a graph G. To analyze the graph, it is treated as a finite list of nodes, and a sequential process is applied to each individual element. There are three main stages for addressing computational complexity:

• The use of morphological operators such as dilation and its frequency associated with its weight calculations.
• Computing and building the optimal path trajectory.
• Evaluating the process to select the best forward step from a given objective.

For the first stage, the complexity depends on the number of nodes $|x^{*}|$, which represents the set $x^{*}$ cardinality. It is also influenced by the expected order of all nodes involved in $x^{*}$, expressed by the constant $o^{*}$, which denotes the structural element size $\left|\lambda \right|$. Consequently, the complexity can be expressed as $O\left(\right|x^{*}|\times o^{*}\times |\lambda \left|\right)$.

The computational complexity for frequency and weighted operators is the same, denoted as $|·|$ or $\left|\right|·\left|\right|$. In real scenarios, covering and frequency/weighted calculations can be performed simultaneously, having the required computation time. Each iteration, the list of nodes for dilation, can be processed separately since the dilation process is independent for each node. This allows for the possibility of segmenting the process into m chunks (or sublists) to implement a parallel approach, thereby reducing the final complexity by a factor of ${\displaystyle \frac{1}{m}}$. When the graph order is homogeneous across all elements, the computational complexity for a given dimensionality becomes $O\left(n^{2m}\right)$, where m denotes the spatial dimensionality. These spaces typically represent structures such as images, volumes, or grid-like arrangements.

In the second stage, the loop for constructing the best trajectory involves backward analysis of the topological structure provided by the frequency/weighted calculations. The minimum number of iterations required to reach the goal D is represented by $|M_{x_0}|$; however, this may vary depending on how the C criterion is implemented. For practical purposes, the lower bound of the complexity is expressed as $O\left(\right|M_{s_0}\left|\right)$ multiplied by the complexity of the C criterion.

Finally, (3) refers to the criterion C for the sting forward to the destination D. This function requires a complexity of $O\left(\right|N\left|\right)$ computations for determining the best choice. Then, the total complexity from stages (1) and (2) is $O\left(\right|M_{x_0}\left|\right)\times O\left(\right|N\left|\right)$. To conclude, with these issues, the total complexity of the algorithm becomes expressed as $\underset{\mathrm{morphological} \mathrm{operators}}{\underbrace{O\left(\tau \right|x^{*}|\times o^{*}\times |\lambda |\times k)}}+$$\underset{\mathrm{best} \mathrm{path} \mathrm{building}}{\underbrace{O\left(\right|M_{x_0}\left|\right)\times O\left(\right|N\left|\right)}}$.

3. Experimental Process and Results Analysis

This section introduces an experimental model designed to assess the algorithm’s performance and applicability in practical situations within large cities with complex road systems. The results will be discussed, along with the implications of using this model as a new method for evaluating feasible paths represented as discrete scenarios.

3.1. Experimental Process in Real Scenarios

The proposed method was tested in Querétaro, Mexico City, representing a densely populated and continually growing urban environment. The road infrastructure in this city is complex due to unplanned growth over time, leading to multiple arterial routes that concentrate the main traffic flow and communication. As a significant urban center, Querétaro is one of the most important cities in Mexico, serving as a key communication hub between the northern and southern regions of the country. Consequently, the logistic problems caused by saturated arterial resources pose a significant challenge in efficiently utilizing resources to improve the quality of life. Figure 3 illustrates the geographical position of Querétaro and lists the towns with the strongest economies in the region. The experimental process is structured in three main stages:

• Coding processes in a complex graph. This stage involves coding an urban area comprising interconnected streets and avenues as a connected graph. To do this, the intersections of streets and avenues are identified and represented as nodes of the graph, which are added and form its structure. In this stage, maps of Querétaro are used, specifically of the downtown area.
• Graph weighing process. The process of assigning weights to transitions between adjacent nodes is performed based on the distance in a metric space, which allows for the effective representation of the connections between them. Node detection is achieved by identifying the maximum values of the distance transformation, which correspond to the areas where the intersections between avenues are located. Interconnected streets define the arcs that connect the pairs of nodes detected.
• A comparison with Dijkstra’s approaches. This process compares computational complexity between search methods using another method as a reference.

3.2. Discussion and Results

Search Algorithm 1 is based on a morphological operator approach to determine the best path within a graph. It was tested using the experimental process described in the previous sections.

In the first stage of the methodology, a coding process converts a geographic map of a city, densely interconnected by streets and avenues, into a coded graph representation of nodes and arcs. The intersections of these avenues are considered nodes, while the connections between them, defined by the avenues, form the arcs that link each ordered pair of nodes.

The coding process utilizes a distance transform, followed by detecting and isolating identified maximum zones. This abstraction process requires minimal computational resources and provides a consistent representation of the geographic layer that reflects the typical interconnected structure of avenues in any city. The crossings between avenues indicate local maximum distances, which result from applying a distance transformation [52]. The locations of all detected maxima are obtained by applying the extended morphological distance.

The detection of maxima can be affected due to the intrinsic structure of the flood geometry, particularly the thickness of the floods and the resolution. This effect becomes more evident when two nearby maxima merge. As mentioned in previous sections, the detection of areas with maxima is performed using the morphological extended maxima transformation [53,54]. This morphological tool offers a reliable process to detect maxima in discrete search spaces.

The information about the city’s avenues can be represented as a graph. This graph can be used to generate a search path, as illustrated in Figure 4 based on the sample used in [37], starting from two positions (see Figure 4a), the starting point $x_0$ and the destination point D. The subsequent graphs in Figure 4b depict the process of covering the graph. The nodes are expanded from the starting node until they reach the destination set D. A dilation operator influences the number of times a node is used to calculate the path solution. The frequency of this occurrence establishes a natural order relation that determines the shortest trajectory based on the connections made to the reached nodes.

Once the destination node is reached, the computed solution is highlighted in red (see Figure 4). In this case, reaching the destination requires 17 iterations (successive dilations). The proposed method performs a breadth-first search, which provides a significant advantage by achieving a higher convergence speed. However, combining breadth-first and depth-first search strategies makes it possible to obtain an efficient solution without the need to exhaustively explore the entire search space in the best case.

3.3. A Comparison with Other Well-Accepted Approaches

A wide range of well-documented methodologies is discussed in the literature.

Table 1. Comparison of techniques for computing optimal path trajectories.

Approach	Workspace	Complexity	Weighted	Remarks
Dijkstra with referenced graph [9]	Graph	$O(mn+nlogn)$	Yes	It is an algorithm focused on the use of a priori information on nodes and arcs, where the concepts of popular location and popular traversal are introduced, together with a dynamic cost associated with the potentially optimal route.
Improved PSO [14]	Grid	$O(N·G·D)$	Yes	This approach is applied in collaborative robotics to assign individual trajectories, optimizing the updated weight strategy and improving convergence over traditional PSO methods. The convergence of the method is improved by classifying particles into three categories: elite, high-quality, and low-quality, which optimizes the shortest path calculation process.
Improved A* Algorithm [55]	Grid	$O\left(n^2\right)$	No	The improved A* algorithm, named EBHSA*, proposes an innovative approach that optimizes three key aspects in route planning: (i) increasing the expansion distance for obstacle detection, (ii) implementing bidirectional search from the two nodes of interest simultaneously, and (iii) optimizing the classical heuristic function used in the base algorithm.
Dijkstra enhancement [56]	Graph	$O(m+n{c}_{max}^{+}{c}_{max}^{-})$	Yes	This approach adds penalties for vertex transfers in an expanded graph. It is sometimes slow compared to the first version because it duplicates the graph to remove edges from the adjacency list.
L* Algorithm [57]	Graph	$O\left(n\right)$	Yes	This approach is based on an improvement of Algorithm A*, where the weight is expressed as a floating point number applied in global searches. The proposal proves to be functional and provides good results in terms of computation time, especially when the distance between the nodes of interest is considerably short.
Dijkstra enhancement [58]	Graph	$O\left(\right(n+m)logn)$	Yes	It introduces the use of an adjacent node table, priority queue, and traffic impedance analysis, reducing the number of cycles compared to the traditional Dijkstra approach.
Dijkstra enhancement [7]	Graph and Grid	$O\left(n^2\right)$	No	This proposal improves Dijkstra’s algorithm by redefining the data structure for better storage and providing a means to search for nodes ordered by priority. It also limits the search area by restricting the area where a better solution to the optimal route problem can be found.
Dijkstra enhancement [11]	Graph	$O\left(\right(V\times logV)+E)$	Yes	This work implements Dijkstra’s algorithm using graph theory, optimizing the scheduling of industrial tasks by assigning extra weights to tasks with priority and solving by combinations.
Morphological search [37]	Graph and Grid	$O\left(c_1\right|x^{*}{|}^2\times \lambda )$	Yes	This paper describes a comprehensive framework using mathematical MM and introduces two novel operators based on dilation and erosion that establish a natural connection to determine the most reliable path.

summarizes the main characteristics of these methods, which allow the optimal trajectory to be calculated in various contexts and specific applications. These works can be compared mainly through three factors: (1) the type of search space, (2) the limitations associated with the application context, and (3) the approach used to find the best trajectory.

Table 1 presents a summary of the different approaches that address the shortest path problem in a search space, whether based on grids or graphs, to highlight and delimit the scope of the proposed method. In the first case of analysis, two main classes can be identified, as detailed in the ’workspace’ column of Table 1: (i) approaches based on structured spaces, such as grid representations and (ii) approaches using unstructured spaces, which set upper bounds to mitigate the risk of combinatorial explosion. Although the size of the search space restricts grid-based methods, unstructured approaches provide greater flexibility to handle more complex and irregular workspaces.

Column feature (2) is a technique used to reduce the search space to make it computable and tractable. However, it introduces an extra layer of complexity associated with the segmentation of the search space. On occasions, this extra complexity can significantly increase the overall complexity, especially in hybrid approaches [59,60,61] and mixture approaches [6,62,63]. These approaches impose additional constraints to make them more tractable in the search space.

The methods above only help narrow the search space and do not contribute to how the trajectory is computed. In contrast, column feature (3), which is used by the majority of approaches, employs a Dijkstra-like algorithm [3,64,65,66,67,68] to compute the best trajectory. It would be worthwhile to conduct a detailed comparison with the Dijkstra algorithm as our proposal provides a new theoretical framework for dealing with this problem.

3.4. Comparison with Dijkstra Algorithm

Table 1 indicates several methods similar to the method that uses Dijkstra’s algorithm as its basis but with some modifications based on context or heuristics. To assess the proposed method’s performance, we conducted a case study on a network where we compared how the method performs with the base Dijkstra algorithm without a priority queue. We aimed to analyze the performance and main differences between the two algorithms in the given area. Figure 5 illustrates the execution stages of both algorithms. The first column shows the execution of the algorithm that uses the morphological coverage operator. In contrast, the second column shows how the Dijkstra algorithm operates the search without utilizing the priority queue.

In the morphological search, we start with the first dilation of the reference node in iteration $k=1$. Each node is associated with a frequency value that denotes the number of times it is dilated. In the subsequent iterations $(k=2\mathrm{to}k=4)$, we apply dilation successively to include new nodes until the dilated nodes cover the destination. When we reach iteration $k=5$, the goal has been achieved. The minimum solutions are all the paths that follow a natural order of increasing frequency, starting from the destination and ending at the origin.

In contrast, the Dijkstra algorithm analyzes each adjacent node in the first iteration $(k=1)$ and calculates the cost for each connected node. Refer to Figure 5 in the second column for an example of this analysis from iteration $k=1$ to $k=4$, which only considers the decision to select the node with the minimum cost, number 3. As a combinatorial problem, the algorithm’s computational time is affected by the degree of connection that a node maintains. However, it is essential to note that modifications such as adding a priority queue can further increase the algorithm’s computational complexity. Despite this, the worst-case scenario remains at $O\left(\right|E|+|N\left|log\right|N\left|\right)$.

$${O\left(\right|V|}^2+\left|A\right|)={O\left(\right|V|}^2)$$

(3)

where V expresses the non-empty set of vertices (nodes), A is the set of arcs, and $|·|$ represents the cardinality operators. The complexity mentioned above refers to the basic algorithm without a priority queue. In the literature, many authors agree that the computational complexity of Dijkstra’s algorithm is $O\left(\right|A\left|log\right|V\left|\right)$ when a priority queue is implemented [69]. However, this complexity is often considered incomplete since the additional cost associated with generating and maintaining the priority queue needs to be included.

However, in order to compare the methodology with the proposal of the classical Dijkstra algorithm, its computational complexity was analyzed [69]. In the first analysis, the complexity of the algorithm is expressed in its initial form by Equation (3)

The morphological approach involves a level of complexity where $\tau$, $o^{*}$, $\lambda$, and k are constant values. In the worst case, the k parameter can take values equal to $|x^{*}|$, with $x^{*}$ representing the number of vertices. The parameters $\tau$ and $o^{*}$ take on small values, which we can simplify by rewriting as a constant value $c_1$. The parameter $\lambda$ indicates the neighborhood expressed by the structural element.

Assuming that the complexity for path generation is constant with $O\left(\right|N\left|\right)=c_2$ and in the worst case $O\left(\right|M_{s_0}\left|\right)$ has a frequency of $|x^{*}|$, we can rewrite the complexity Equation as follows:

$$O(c_1{\left|x^{*}\right|}^2\times \lambda )+c_2\times \left|x^{*}\right|=O(c_1{\left|x^{*}\right|}^2\times \lambda )$$

(4)

The final expression becomes simplified to $O\left(n^2\right)$; the complexity is comparable to the Dijkstra algorithm without a priority queue [37].

The results indicate that the approach is a reliable general framework for developing search algorithms in a graph. This framework provides a theoretical basis for defining a search process, allowing for a simplified representation of search methods in discrete spaces. The structural element and neighborhood connectivity parameters establish the criteria for covering the graph space, influencing the search speed; higher values lead to faster coverage. In grid spaces, symmetrical structural elements outperform irregular shapes, reflecting a trend in step-forward structures that guide the search towards the solution.

The covering dilation operator offers a travel criterion within the graph that helps prevent loops in the search space due to the induced lattice. Typically, a morphological approach is a first-in-width search strategy, meaning that the first solution reached, based on the width of tree paths generated from the lattice, is considered optimal. By incorporating weights into the graph, we can adjust the cost of traveling between nodes, enabling the analysis of non-uniform graph spaces.

To supplement the experimental process, we compared both algorithms. The experimental process uses a city map to calculate the shortest path, randomly selecting start and end points for the trajectory. However, it is essential to note that the two algorithms are not directly equivalent, leading to the following comparison:

In our proposed method, we measure the percentage of marked states to determine the total number of visited states needed to reach the destination. A low percentage indicates that the approach requires only a small portion of the total space to be visited to compute the optimal path.

For the Dijkstra algorithm, we illustrate the distribution of the number of states that comprise the optimal path. This distribution reveals the number of states that form the solution and indicates how often the priority queue needs to be processed with all states conforming to the system. In the testing scenario, 261,597 states are involved, as shown in Figure 6a. Each marked pixel within the avenues represents a unique state. Here are the results obtained:

Our proposal advocates for a uniform distribution of the usage percentage within the solution space, as illustrated in Figure 6a. Statistically, this distribution has a mean of only half of the total number of states. On average, our approach requires only about half as many states to compute the optimal path compared to the Dijkstra algorithm [69].

Figure 6b depicts the length distribution of the optimal path. Both approaches yield equivalent paths, but there is a notable difference between them [37]. The path length distribution for Dijkstra reveals the frequency with which the priority queue needs to be constructed to identify the optimal path, involving interactions with all system states to locate and sort the most probable solutions. The maximum values indicated by the arrows (at 2000 and 3000) represent the anticipated number of times the entire solution space is analyzed. This higher frequency is a consequence of constructing the priority queue, which exceeds our proposed method’s.

This test confirms that the proposed method can be more cost-effective than the classical Dijkstra algorithm in certain applications.

This proposed method clearly defines the town’s strategic analysis of potential solutions for logistics planning, specifically designed to optimize delivery and transport routes for goods and services within the community. We will proceed with a focused analysis to identify the most effective delivery paths, ensuring we address time-sensitive conditions and constraints effectively.

In summary, the proposal advances the field of graph search by introducing a morphological approach to describing search graph algorithms. This approach simplifies the modeling and description of these algorithms. It offers several advantages, including low computational complexity, the ability to express algorithms equivalent to the best in the state of the art, and a reference description suitable for discrete environments. Additionally, it can accommodate parallel processing by implementing morphological operators.

4. Conclusions

This work presents a method for determining the shortest path for resource delivery in densely populated cities. One key advantage of this approach is its low complexity, which enables the analysis of large urban areas as discrete spaces. By utilizing a morphological framework, we create a compact representation of the city that simplifies the search space.

This feature makes it practical for implementation in various densely populated cities, where the effectiveness of communication infrastructure can change frequently. This approach is particularly effective when combined with other methodologies, such as Dijkstra’s algorithm, as it does not necessitate examining the entire graph space. The results indicate that the amount of space inspected to find the solution is approximately $5\%$.

Further works: This proposal outlines a morphological searching approach. Future research will focus on criteria for parallelization to enhance algorithm performance on multi-core hardware. Additionally, strategies for isolating processes and executing sub-processes in parallel will be explored.

We also propose a criterion for weighted graphs, where magnitudes define connections. This addresses issues related to probability, neural networks, and search algorithms. While the experimental analysis utilized only one coding scheme, it could be expanded to automatically generate graphs based on urban variables such as traffic flow density and velocity.

Furthermore, the search framework enables the identification of path trajectories even when the destination graph is unreachable. This feature introduces a new connectivity criterion for covering dilation connections in graph-dense clustering analysis, which aids in developing techniques for clustering and labeling data.

Applications that benefit from this approach include task routing, which optimizes network traffic and establishes the most efficient path for connections, ultimately reducing latency. This process involves determining optimal connection ratios in urban areas to analyze short paths, thereby decreasing response times for goods delivery. In cellular networks, leveraging topology is an effective strategy for selecting the best cells to balance bandwidth communication. Additionally, there are medical applications involving the use of catheters in surgical procedures.