# Algorithm Based on Morphological Operators for Shortness Path Planning
## 1 Introduction  
Path planning in a graph represents one of the most fundamental tasks in a graph space. In the literature, several approaches deal with this task by limiting and conditioning it to particular contexts, rendering it computable in closed scenarios [1; 2; 3; 4; 5]. In this context, the main approaches developed to deal with the problem of best-searching path planning are the following. _Global search approaches:_ This approach considers the full graph as the solution space, where the origin and destination are known, and it uses a global directional cost function such that it minimizes/maximizes in terms of the best direction. The traveling path across the graph is performed by local navigation using the most reliable direction provided by the global cost function. The global approach has the disadvantage that it might grow, becoming difficult to analyze with specific computer resources [6; 7; 8]; however, it has the advantage that it is always looking for the best path solution, making it less likely to fall into loops or diverge from the destination. These approaches produce a general representation with a low level of detail, prioritizing the global searching reference, making it suitable for scenarios with fixed computational resources [9; 10; 11]. Local search approachesThese approaches assume that local optimization might drive the global solution. They use a local neighborhood, and it begins by locating the best solution. Each step forward updates the local searching neighborhood by adding new evidence to search for the best path solution. However, these approaches are computationally restricted; they might fall into loops, diverge from the global solution, or locate a sub-optimal solution. The computational complexity depends on the size of the neighborhood and the criterion function cost [12; 13; 14]. These approaches are helpful when there is no previous knowledge about the scenario, making them feasible in uncertain environments. The low complexity makes it possible to obtain subgraph partitions and instantiate in several initial states [15; 16; 17]. Hybrid approachesThese combine the above two types of approaches with the local analysis neighborhood, which results in a method that is computationally tractable. Adding some restrictions helps to reflect the global direction to locate the solution. Additionally, prior knowledge can be used several times to define additional criteria as a heuristic for specific scenarios and contexts. Most studies, including [3; 18; 19], have proposed mixed strategies that combine well-accepted local and global approaches. The cost function can be modified with the introduction of reset criteria whenever partial solution are reached. Nature-inspired approachesFinally, this type of approach involves using optimization techniques that are inspired by the principles of nature, including the theory of evolution and the behaviors of living organisms, to efficiently solve problems across various fields of application. This is achieved by imitating the ways in which nature tackles these challenges. Some examples of such techniques are the particle swarm algorithm [20; 21], evolutionary algorithms [22], ant colonies [8], and neural networks [23]. On the other hand, mathematical morphology provides a conceptual foundation for the morphological analysis (inner structures) of abstract objects, such as numbers, shapes, or graphs. The morphological analysis consists of defining two basic operators named dilation and erosion. The mixture of basic operators defines other new operators and filters, which are used to detect structures of interest, e.g., in the case of image applications [24] or volumetric reconstructions [25; 26; 27; 28]. Nowadays, morphology represents a generalized framework to manage finite structures disposed into a given space [29; 30]. In this context, it has been used in a wide range of image analysis applications [31] where the spaces are expressed as grid structures that might be associated with lattices using morphological operators [32].