The problem of finding the best path trajectory in a graph is highly complex due to its combinatorial nature, making it difficult to solve. Standard search algorithms focus on selecting the best path trajectory by introducing constraints to estimate a suitable solution, but this approach may overlook potentially better alternatives. Despite the number of restrictions and variables in path planning, no solution minimizes the computational resources used to reach the goal. To address this issue, a framework is proposed to compute the best trajectory in a graph by introducing the mathematical morphology concept. The framework builds a lattice over the graph space using mathematical morphology operators. The searching algorithm creates a metric space by applying the morphological covering operator to the graph and weighing the cost of traveling across the lattice. Ultimately, the cumulative traveling criterion creates the optimal path trajectory by selecting the minima/maxima cost. A test is introduced to validate the framework’s functionality, and a sample application is presented to validate its usefulness. The application uses the structure of the avenues as a graph. It proposes a computable approach to find the most suitable paths from a given start and destination reference. The results confirm that this is a generalized graph search framework based on morphological operators that can be compared to the Dijkstra approach.

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.

DBSCAN is widely used to identify structured regions in unlabeled data, but its performance depends critically on the selection of the neighborhood parameter $\epsilon$. Traditional heuristics for estimating $\epsilon$ often become unreliable in high-dimensional or varying-density settings because they rely heavily on local geometric criteria and may fail under smooth transitions or topological ambiguity. This work presents a three-level perspective on DBSCAN hyperparameter selection. At the algorithmic level, $\epsilon$ controls neighborhood connectivity and structural transitions in clustering. At the modeling level, the ordered k-distance signal is approximated through a surrogate dynamical estimation framework inspired by a mass–spring–damper system. At the causal level, the resulting estimator is interpreted through interventions on its internal threshold-selection mechanism. The proposed method models the variation of $\epsilon$ using ordinary differential equations defined on the ordered k-distance signal, enabling analysis of structural transitions in density organization via a surrogate dynamical representation. System identification is performed using L-BFGS-B optimization on the smoothed k-distance curve, while the system dynamics are solved with the fourth-order Runge–Kutta method. The resulting estimator identifies transition regions that are structurally informative for $\epsilon$ selection in DBSCAN. To analyze the estimator at the intervention level, Pearl’s do-calculus is used to compute the Average Causal Effect (ACE). The method was evaluated on synthetic benchmarks and on the Covtype dataset, including scenarios with multi-density overlap and dimensionality up to ${\mathbb{R}}^{10}$. The resulting ACE values, $+0.9352$, $+0.5148$, and $+0.9246$, indicate that the proposed estimator improves intervention-based $\epsilon$ selection relative to the geometric baseline across the evaluated datasets. Its practical computational cost is dominated by nearest-neighbor search, behaving approximately as $O(NlogN)$ under favorable indexing conditions and degrading toward $O\left(N^2\right)$ in high-dimensional or weak-pruning regimes.

Wind dynamics in urban environments exhibit non-stationarity and marked spatial variability, complicating stochastic modeling when a single global distribution is assumed. This article discusses the estimation of wind density under quasi-stationary regimes at the local level using SICABI, a two-phase framework: (i) Stationary Region Identification (ISR) estimates, through spectral power analysis, a specific dominant period for each location and validates the induced subsampling using the Augmented Dickey–Fuller (ADF) test, and (ii) RAPID adjusts an adaptive parametric density by recursively updating the mixture parameters and creating new components when a normalized membership distance exceeds a threshold. The analysis uses wind speed records collected from eight stations in the Metropolitan Area of Queretaro, Mexico, during the period from 1 January 2023 to 31 December 2023, aggregated at a 10 min resolution, from which $X_{\delta ,s}$ is constructed for each site. RAPID is compared against Gaussian Kernel Density Estimation (KDE) with Silverman bandwidth and EM-fitted Gaussian mixtures with BIC-based selection ($K_{max}=12$). The resulting densities were compared with an empirical density estimated from a histogram over a fixed grid ($m=50$) using the MISE and RMSE metrics. The results reveal marked site-dependent differences in dominant periodicity and residual behavior, including asymmetry and heavy tails. ISR identified dominant periods ranging from 37 to 166 days, and RAPID adapted its complexity with $K_s\in [5,10]$ without fixing the number of mixture components in advance. Quantitatively, RAPID achieved the lowest RMSE at 6/8 sites and the lowest MISE at 5/8 sites, while also exhibiting shorter execution times than KDE and MoG under the same input $X_{\delta ,s}$. The results support RAPID as a competitive adaptive method for site-specific density estimation in non-stationary urban climate signals. In this context, local regimes can be viewed as approximate invariants under time translation in the weak stochastic sense, while deviations from this assumption are reflected in increased distributional complexity across sites.