Path Length Research Articles

Path planning for unmanned aerial vehicles (UAVs) in mountainous environments requires satisfying terrain clearance and obstacle avoidance constraints while optimizing path length, flight time, and energy consumption. To address these challenges, this paper proposes EC-MOPSO (Epsilon-dominance and Crowding-distance-based Multi-Objective Particle Swarm Optimization). Inspired by the principle of symmetry, the algorithm integrates an adaptive parameter adjustment mechanism with a ε− dominance–crowding archiving strategy to balance global exploration and local exploitation through spatially symmetric archive management. A safety-repairable B-spline trajectory model ensures smooth and feasible flight paths under complex terrain conditions. Simulation results show that EC-MOPSO reduces path length by 10–40%, improves normalized hypervolume by over 25%, and decreases performance variance by 20–25%, confirming faster convergence and higher robustness compared with representative multi-objective optimization approaches. Ablation studies further verify that both the adaptive parameter mechanism and the ε− dominance–crowding strategy significantly enhance convergence stability and overall optimization performance. Overall, EC-MOPSO provides an adaptive and reliable optimization framework for generating efficient, safe, and energy-aware UAV trajectories in real-world mountainous rescue missions.

Tungsten oxide (WO 3 ) thin films were fabricated using a combination of electron beam evaporation and glancing angle deposition (GLAD) to study their gasochromic properties. The influence of deposition angle ( α between 0° and 88°) and post-process annealing on the morphology, crystallinity, and optical response to hydrogen was systematically analyzed. Structural characterization revealed that GLAD promotes the formation of inclined fibrous columns with enhanced porosity due to self-shadowing effects. Quantitative analysis combining SEM and optical modeling confirmed that porosity and refractive index correlate strongly with deposition angle. Gasochromic measurements showed that nanocrystalline films had better optical response and faster response times (∼5min) to hydrogen than amorphous films (∼11min). However, the gasochromic response decreased with an increasing deposition angle, which correlated with a reduced optical path length. The tunable morphology and refractive index achieved by GLAD highlight its potential for designing multifunctional optical coatings that combine hydrogen sensitivity with tailored optical performance.

Hemiplegic shoulder pain (HSP), a common complication of stroke, has a reported incidence of 34%-85% and a complex pathophysiology. This study aimed to explore differences in brain network topological properties between patients with and without HSP after stroke using resting-state functional magnetic resonance imaging. Fifty patients with hemiplegia after stroke were recruited and divided into two groups based on the presence of HSP. Resting-state functional magnetic resonance imaging data were acquired, and GRETNA was applied to calculate both global and regional network topological properties. Group differences were assessed between patients with and without HSP. At the global network level, both groups demonstrated a clear small-world organization (small-worldness index > 1). Compared with the non-pain group, patients with HSP showed significantly lower global efficiency (0.1733 ± 0.0047 vs. 0.1765 ± 0.0044; P = 0.02) and higher characteristic path length (0.3224 ± 0.0152 vs. 0.3170 ± 0.0089; P = 0.001). At the regional network level, patients with HSP showed reduced nodal degree centrality and nodal local efficiency in the opercular part of the right inferior frontal gyrus and in the orbital gyrus (P < 0.05). They also exhibited decreased nodal betweenness centrality in the right pallidum, left dorsolateral superior frontal gyrus, triangular part of the left inferior gyrus, and left Rolandic operculum (P < 0.05). Conversely, nodal betweenness centrality was increased in the left thalamus, right parahippocampal gyrus, left inferior occipital gyrus, and left angular gyrus (P < 0.05). The topological properties of brain networks in patients with HSP have shifted toward a weaker small-world organization. Nodal alterations were primarily concentrated in the executive control network, default mode network, basal ganglia, and language network, regions associated with pain processing and emotional regulation. These findings provide new insights into the central mechanisms of HSP after stroke and suggest potential neural targets for future research and therapeutic intervention.

Abstract Social networks formed via play interactions offer a powerful framework for investigating the social dynamics of animals and the functions of play behavior. Using this framework in mantled howler monkeys ( Alouatta palliata ), we examined whether social play might fulfil some of the affiliative and bonding functions typically addressed to allogrooming. We analyzed play networks in seven groups of mantled howler monkeys (89 individuals), analyzing group and individual-level patterns based on 1774 observation hours collected over approximately 3 years. We examined the relationship between group size and network complexity using edge density (the proportion of connections), modularity (the degree to which the network is subdivided into clusters), and average path lengths (the number of steps required to reach others in the network). We analyzed centrality across age-sex classes using betweenness, closeness, and eigenvector metrics. We tested which centrality measure best predicted play evenness, defined as the degree to which individuals distributed their playtime across partners. We found that network interconnectivity generally declined with group size, though average path lengths were not significantly different from those of random networks. Immatures showed the highest centrality values, while adult females exhibited the highest play evenness. However, centrality measures did not significantly predict play evenness. These patterns suggest age-specific functions of social play: immatures may benefit from frequent play to support development, while adult females may use evenly distributed play to navigate social competition. Our findings support the hypothesis that social play serves similar functions to allogrooming, contributing to social bonding.

A silicon (Si) slab waveguide resonator design with a circulating Gaussian-like cavity mode is described and characterized, for both all-pass and add-drop configurations and several different input/output coupling strengths. The circulating beam propagates in a slab waveguide with no lateral confinement. Three straight mirrors and one curved mirror define a folded two-dimensional Gaussian cavity. Light is coupled to and from the resonator by beam splitters formed by a narrow gap between a cavity mirror and the input slab waveguides. The coupling is determined by the gap width and is wavelength independent and lossless. For a L = 100 μm path length cavity, resonance line widths of λ 3 dB = 5 pm with Q-values of Q = 310000 were measured. The resonator drop spectrum exhibited a comb of almost identical resonance lines across a 100 nm tuning range. This resonator design is capable of broadband operation and is less susceptible to sidewall roughness and defect scattering-induced loss and mode splitting, when compared with Si rings formed from narrow single-mode waveguides.

Background: We used Social Network Analysis (SNA) to phenotypic and clinical characterization of tissue Doppler (TD)signals representing different phases of the cardiac cycle in heart failure with preserved ejection fraction (HFpEF). Methods: Retrospectively, 275 HFpEF patients (64±18 years, 54% females, EF: 57±7%, followed for median 2.8 years for heart failure hospitalization and mortality) were studied. TD signals were measured [Positive and negative pre systolic (PREp and PREn) and ejection systolic (S'), negative and positive post systolic velocities (POEn and POEp) and early diastolic (e’) velocities]. PREp/n, POEn/p, PREp/S’ and POEn/e’ were calculated to represent relationships between the biphasic PRE and POE and different parts of systole and diastole. SNA was done using Python and google Colab. The above parameters were treated as the edges and patients as the nodes. Cosine similarity and Louvain clustering were used for SNA.. Results: SNA yielded 6 clusters (edges: 2654, density 0.1, average degree: 19.3, average clustering coefficient: 0.42, assortativity: .165, path length: 2.77, figure 1, table 1). Cluster 1 (40 patients) and 2 (16 patients) had preserved TD signals relationships. Cluster 3 (54 patients) exaggerated PRE and POE with mild S’ depression, cluster 4 (44 patients) had depressed PRE and POE despite preserved S’ and e’. Cluster 5 (53 patients) and 6 (68 patients) showed all signals depressed. Clusters were not different for age, sex, risk factors or co-morbidities. NYHA class was worst in clusters 4 and 5, while clusters 1 and 5 had the worst diastolic profiles, and cluster 5 had intermediate diastolic profiles. Survival analyses suggested clusters 0,1, and 2 as low risk, cluster 3 was intermediate, and clusters 4 and 5 as high risk. Moreover, clusters were subdivided into smaller clusters that differentiate TD relationships further; in addition to specific clinical and echocardiographic features (Cluster 3 differentiates based on E/A ratio, Cluster 5 differentiates based on E/e’ and symptoms, Cluster 6 differentiates based on LAVi). Conclusions: In HFpEF patients PRE, ejection, POE, and early diastolic TD signals share complex clinically meaningful relationships that are difficult to express using traditional statistical techniques. Clustering using SNA can help computationally and visually stratify such relationships. Exploring such relationships in larger studies can unlock phenotypic characterization and management strategies.

Efficient and stable path planning in dynamic and obstacle-dense environments, such as large-scale structure assembly measurement, is essential for improving the practicality and environmental adaptability of mobile robots in measurement and quality inspection tasks. However, traditional reinforcement learning methods often suffer from inefficient use of experience and limited capability to represent policy structures in complex dynamic scenarios. To overcome these limitations, this study proposes a method named DPDQN-TER that integrates Transformer-based sequence modeling with a multi-branch parameter policy network. The proposed method introduces a temporal-aware experience replay mechanism that employs multi-head self-attention to capture causal dependencies within state transition sequences. By dynamically weighting and sampling critical obstacle-avoidance experiences, this mechanism significantly improves learning efficiency and policy performance and stability in dynamic environments. Furthermore, a multi-branch parameter policy structure is designed to decouple continuous parameter generation tasks of different action categories into independent subnetworks, thereby reducing parameter interference and improving deployment-time efficiency. Extensive simulation experiments were conducted in both static and dynamic obstacle environments, as well as cross-environment validation. The results show that DPDQN-TER achieves higher success rates, shorter path lengths, and faster convergence compared with benchmark algorithms including Parameterized Deep Q-Network (PDQN), Multi-Pass Deep Q-Network (MPDQN), and PDQN-TER. Ablation studies further confirm that both the Transformer-enhanced replay mechanism and the multi-branch parameter policy network contribute significantly to these improvements. These findings demonstrate improved overall performance (e.g., success rate, path length, and convergence) and generalization capability of the proposed method, indicating its potential as a practical solution for autonomous navigation of mobile robots in complex industrial measurement scenarios.

To improve the efficiency of multi-region multi-unmanned aerial vehicle (UAV) inspection, this paper proposes a composite task planning strategy integrating the K-Means++ genetic algorithm (KMGA) and the multi-neighborhood iterative dynamic programming (MNIDP) method. Firstly, the multi-region multi-UAV inspection problem is modeled as a multiple traveling salesmen problem with neighborhoods (MTSPN). Then, this problem is decomposed into two interrelated subproblems to mitigate the complexity inherent in the solution process: that is, the multiple traveling salesmen problem (MTSP) and multi-neighborhoods path planning (MNPP) problem. Based on this decomposition, the MTSP is solved by the KMGA by converting it into m spatially non-overlapping traveling salesmen problems (TSPs) and then these TSPs are solved to obtain the approximate optimal visiting sequences for the nodes in each TSP in a short time. Subsequently, the MNPP can be efficiently solved by an MNIDP which plans the paths between the corresponding neighborhood of each node based on the node visiting sequences, thus obtaining the approximate optimal path length of the MTSPN. The simulation results demonstrate that the proposed composite strategy exhibits advantages in computational efficiency and optimal path length. Specifically, compared to the baseline algorithm, the average tour length obtained by the KMGA decreased by 23.24%. Meanwhile, the average path lengths computed by MNIDP in three instances were reduced from 8.00% to 11.41% and from 6.46% to 10.08% compared to two baseline algorithms, respectively. It provides an efficient task and path planning solution for multi-region multi-UAV operations in power transmission line inspections, thereby enhancing inspection efficiency.

A high-flux, high-resolution Offner imaging spectrometer with a compact configuration is proposed and demonstrated. By introducing a custom-designed lens group between the entrance slit and the primary concave mirror, the effective slit is optically transformed into a virtual slit with extended divergence and path length. This design fundamentally alleviates astigmatism and enables a larger numerical aperture (NA = 0.3), while preserving the intrinsic aberration-free benefits of the Offner configuration. Theoretical modeling and optical analysis were conducted to derive aplanatic conditions and optimize focal power distribution. A three-lens assembly, including a toroidal surface, was employed to decouple meridional and sagittal focal lengths and correct residual aberrations. The optimized system covers the 400–1000 nm spectral range with a measured spectral resolution of 1.35 nm and maintains an MTF&gt;0.7 at 28 lp/mm across the full field of view. Compared to a conventional Offner spectrometer of equal resolution, the proposed design improves optical throughput by a factor of 2.25 and reduces system volume by 14.5%. Laboratory calibration using a mercury–argon lamp confirmed its high-resolution capabilities, and field experiments demonstrated excellent spectral and spatial imaging performance. The improved Offner spectrometer offers a compact, aberration-free, and high-resolution solution for applications requiring high-precision spectral imaging.

Background: Transcatheter aortic valve replacement (TAVR) has broadened treatment options for elderly patients with severe aortic valve stenosis (AVS), including those with challenging vascular anatomy such as increased aortic tortuosity. We previously demonstrated that a higher aortic tortuosity index (ATI), defined as the ratio of aortic length (from the sinotubular junction to the terminal aorta) to body height measured by 3D-computed tomography, was associated with complete atrioventricular block (CAVB). However, the anatomical mechanisms linking ATI to conduction disturbances (CD) requiring pacemaker implantation (PMI) remain unclear. Purpose: To investigate whether anatomical variations—specifically, aortic angle and membranous septum (MS) length—mediate the association between ATI and CD requiring PMI after TAVR. Methods: We analyzed 198 consecutive patients with severe AVS who underwent TAVR. ATI, aortic angle, and MS length were quantified using pre-procedural 3D-CT imaging. CD requiring PMI within 30 days post-TAVR was observed in 20 patients (10.2%). Patients were stratified by median ATI (3.24), aortic angle (49°), and MS length (3.3 mm). Structural equation modeling (SEM) was applied to evaluate both direct and indirect effects of ATI on CD through aortic angle and MS length. Results: Patients with higher ATI (&gt;3.24) had a significantly greater incidence of CD requiring PMI than those with lower ATI (18.2% vs. 3.0%, p = 0.001). SEM demonstrated a significant direct effect of higher ATI on CD (path coefficient = 0.26, p &lt; 0.01), along with a significant indirect effect mediated by aortic angle and MS length (path coefficient = 0.15, p &lt; 0.05). Conclusion: Higher ATI is associated with increased risk of CD requiring PMI after TAVR, and this relationship is mediated by steeper aortic angle and shorter MS length. Aortic tortuosity may reflect age-related anatomical remodeling that predisposes patients to conduction injury during TAVR.

Percutaneous puncture has become one of the most widely used minimally invasive techniques in clinical practice due to its advantages of low trauma, quick recovery and easy operation. However, incomplete needle tip movement, tissue barriers and complex distribution of sensitive organs make it difficult to balance puncture accuracy and safety. To this end, this paper proposes a new puncture path planning algorithm for flexible needles, which integrates gravitational guidance, bi-directional adaptive expansion, optimal node selection based on the A* algorithm, and path optimization strategies, with Bi-Rapid-Research Random Trees (Bi-RRTs) at its core, to significantly improve obstacle avoidance capability and computational efficiency. The simulation results of 2D and 3D complex scenes in MATLAB show that compared with the traditional RRT algorithm and Bi-RRT algorithm, the GBOPBi-RRT algorithm achieves significant advantages in terms of path length, computation time and node size. In particular, in the 3D environment, the GBOPBi-RRT algorithm shortens the planning path by 43.21% compared with RRT, 27.47% compared with RRT* and 30.33% compared with Bi-RRT, which provides a reliable solution for efficient planning of percutaneous puncture with flexible needles.

Background: Chemotherapy-related cognitive impairment (CRCI), commonly known as "chemobrain," frequently occurs during breast cancer treatment and has been linked to altered brain function. This resting-state functional magnetic resonance imaging study examined chemotherapy-related changes in functional brain activity, network connectivity, and associations with cognitive outcomes. Methods: Twenty-eight patients with breast cancer were assessed prechemotherapy (BB) and postchemotherapy (BBF), alongside 27 healthy controls of comparable age at baseline (BH) and follow-up (BHF). Mean fractional amplitude of low-frequency fluctuations (mfALFF) and mean regional homogeneity (mReHo) quantified functional brain activity. Graph theoretical analysis (GTA) assessed network topology; network-based statistics (NBS) evaluated interregional connectivity. Cognitive performance was evaluated through standardized assessments. Results: Postchemotherapy patients exhibited reduced anxiety and lower FACT-Cog scores. Voxel-wise analyses showed increased mfALFF in frontal regions and mReHo in superior temporal and inferior frontal gyri, alongside decreases in postcentral, lingual, and parahippocampal areas. Healthy controls showed increased activity in medial frontal and cingulate regions, with reductions in the temporal lobe and putamen. GTA revealed higher global efficiency and reduced modularity, path length, and network complexity in the BBF group compared with BHF. NBS showed weaker structural connectivity in motor and occipital regions prechemotherapy and decreased parietal and insular connectivity postchemotherapy. Multiple regression showed brain-behavior correlations: declines in FACT-Cog, Digit Symbol Substitution, and mood scores were linked to altered activity in frontal, parietal, cingulate, and occipital areas, while positive correlations suggested compensatory activation. Conclusions: Chemotherapy was associated with longitudinal alterations in brain activity, network organization, and connectivity in breast cancer survivors. Brain-behavior associations suggest disrupted neural networks may underlie CRCI.

The quintic polynomial algorithm is not suitable for the sudden change of road curvature in the roundabout overtaking scenario, resulting in large curvature fluctuation and velocity oscillation. A trajectory planning method is proposed based on non-uniform sampling (NUS) and dynamic constraint optimization (DCO). Based on the NUS mechanism, the curvature fluctuation can be effectively reduced by using different center angles and road radii for sampling. Additionally, arc-length parameterization and time mapping techniques are adopted to extract velocity data, thereby achieving real-time adjustment of interpolation points. Centripetal acceleration constraints are employed to suppress velocity oscillations under dynamic conditions, significantly improving the smoothness of trajectories and comfort of vehicle driving. The simulation results show that the path length and curvature are optimized, and the centripetal acceleration is reduced, which significantly improves trajectory smoothness and passenger comfort.

The Climate Crisis is reshaping not only ecosystems but also human cognition. While its psychological impact is increasingly acknowledged, little is known about how environmental degradation influences basic cognitive functions. Since spatial and temporal cognition provide the perceptual scaffolding for orientation and various decision-making processes, distortions in these dimensions may hinder adaptive responses to ecological change. This study examined whether simulated climate-related degradation affects spatial-temporal cognition and whether interoceptive awareness predicts variability in these effects. Using immersive Virtual Reality combined with an omnidirectional treadmill, participants walked along paths in verdant and arid landscapes and then estimated the duration and distance travelled on each path. The results showed that arid environments led to longer time and distance estimates than verdant ones, although there were no objective differences in path length or actual walking time. Furthermore, temporal judgements, but not spatial ones, were predicted by interoceptive attention regulation: participants with a higher capacity to regulate attention towards bodily sensations consistently provided shorter temporal estimates across all contexts. These findings demonstrate that spatial-temporal representations are sensitive to ecological quality and that interoceptive processes contribute to individual differences in temporal perception. This highlights the value of integrating cognitive processes and interoception into sustainability science, suggesting that environmental preservation supports not only ecological well-being but also the cognitive foundations through which humans perceive and adapt to their surroundings.

In new progress on conjectures of Stein, and Addario-Berry, Havet, Linhares Sales, Reed and Thomassé, we prove that every oriented graph with all in- and out-degrees greater than 5k/8 contains an alternating path of length k. This improves on previous results of Klimošová and Stein, and Chen, Hou and Zhou.

This study presents a fundamental theoretical investigation of gradient elution chromatography employing core-shell particles and variable mobile phase composition. An extended form of the general rate model (GRM) is developed to examine the influence of column overloading on elution performance. The linear solvent strength (LSS) model is incorporated to describe variations in Henry's constant and the nonlinearity coefficient with solvent composition, while accounting for intraparticle diffusion, film mass transfer resistance, and axial dispersion. Core-shell particles enhance separation efficiency by reducing the accessible pore volume and diffusion path lengths, thereby allowing higher flow rates. To approximate the resulting nonlinear model equations, a semidiscrete high-resolution finite volume scheme is adapted and applied. The numerical framework enables a detailed analysis of the effects of key model parameters on the behavior and shape of the elution profiles, providing valuable insights into chromatographic dynamics. Validation of the proposed model and evaluation of the numerical scheme are conducted through benchmark test problems. Specific performance metrics are employed to identify the most influential parameters. The study utilizes binary mixtures as a model system to establish a fundamental understanding of elution behavior, refine numerical strategies, and provide insights that support the optimization of experimental conditions. The findings offer a foundational framework for enhancing separation performance with broader implications for more complex systems.

Restless legs syndrome (RLS) is a sensorimotor disorder with evening-predominant symptoms; convergent models implicate brain iron dysregulation and alter dopaminergic/glutamatergic signaling. Because EEG provides millisecond-scale access to cortical dynamics, we synthesized waking EEG/ERP findings in RLS (sleep EEG excluded). A structured search across major databases (1980–July 2025) identified clinical EEG studies meeting prespecified criteria. Across small, mostly mid- to late-adult cohorts, four reproducible signatures emerged: (i) cortical hyperarousal at rest (fronto-central beta elevation with a dissociated vigilance profile); (ii) attentional/working memory ERPs with attenuated and delayed P300 (and reduced frontal P2), pointing to fronto-parietal dysfunction; (iii) network inefficiency (reduced theta/gamma synchrony and lower clustering/longer path length) that scales with symptom burden; and (iv) motor system abnormalities with exaggerated post-movement beta rebound and peri-movement cortical–autonomic co-activation, together with evening-vulnerable early visual processing during cognitive control. Dopamine agonist therapy partially normalizes behavior and ERP amplitudes. These converging EEG features provide candidate biomarkers for disease burden and treatment response and are consistent with models linking brain iron deficiency to thalamo-cortical timing failures. This mechanistic review did not adhere to PRISMA or PICO frameworks and did not include a formal risk-of-bias or quantitative meta-analysis; samples were small, heterogeneous, and English-only.

The Rapidly Exploring Random Tree (RRT) is widely employed in the field of intelligent vehicles, but traditional RRT has issues like inefficient blind expansion, tortuous/discontinuous paths, and slow convergence. Thus, a multi-dimensional optimized RRT is proposed. First, a heuristic search method is adopted to reduce blind sampling, guiding sampling toward the target and cutting irrelevant searches. Second, to fix RRT’s inability to adjust step size dynamically (limiting complex road adaptability), step size is optimized based on environmental information. Third, since treating vehicles as mass points leads to unreasonable paths, sampling points are expanded for practicality. Finally, redundant points are removed via a greedy strategy, and paths are smoothed with quasi-uniform cubic B-splines to meet ride comfort needs. MATLAB R2022b simulations validate the algorithm: in simple scenarios, optimized RRT reduces sampling points to 232 (24.4% of traditional RRT), runtime to 3.25 s (79.4% cut), path length to 673.84 m (15.6% reduction); in complex scenarios, 636 points (37.0%), 11.07 s runtime (58.8% cut), 699.61 m path (21.6% reduction), outperforming traditional RRT and Q-RRT*.

Background Following stroke, brain networks can be described by strength of local connections (clustering coefficient [Cw]) and strength of global interconnections (path length [Lw]) between nodes, and their balance (Small-worldness [Sw]). Objective. To identify electroencephalography (EEG) networks predicting clinical evolution in stroke through a multicenter cross-sectional study. Methods We consecutively recruited 87 anterior circulation ischemic stroke patients. We obtained resting-state EEG (31 electrodes, 10-10 system) within 24 hours from stroke (T0) and at discharge from stroke unit (4–10 days after stroke; T1). EEG data were elaborated with EEGLAB and Lagged Linear Coherence among cortical sources of EEG signals was analyzed using eLORETA. We performed a multiple linear regression with National Institutes of Health Stroke Scale (NIHSS) at T0 and T1 as dependent variables and Cw, Lw, and Sw of delta, theta, and alpha networks as independent variables. Results We found a negative association between alpha1 Sw and NIHSS at T0 (β = −.232, P = .04) meaning that the lower is alpha efficiency the higher is clinical severity and a positive association between delta Sw and NIHSS at T1 (β = .423, P < .001) meaning that the higher is delta efficiency the higher is clinical severity. We found positive association between delta Sw at T0 and NIHSS at T1 (β = .259, P = .02), meaning that the higher is delta efficiency in the hyperacute phase the higher is clinical severity at T1. Conclusions A higher delta Sw within 24 hours after stroke is associated to higher NIHSS within 10 days. Delta brain network rearrangement in the hyperacute phase is a potential neurophysiological measure to be integrated in multi-modal prognostic models.

Let G = (V (G), E(G)) be a simple and connected graph. The distance between two vertices u and vin G, is the length of a shortest path from u to v, denoted by d(u, v). Suppose S = {s1, s2, ...sk} is anordered subset of vertices of G, then the metric representation of a vertex u ∈ V (G) with respect to S,denoted by r(u|S), is the k−vector (d(u, s1), d(u, s2), ..., d(u, sk)). If every two nonadjacent vertices ofG have distinct metric representations with respect to S, then the set S is called a nonlocal resolvingset for G. A nonlocal resolving set with minimum cardinality is called a nonlocal metric basis. Thenonlocal metric dimension of G is the cardinality of the nonlocal metric basis of G and is denoted bynldim(G). In this paper, we obtained nonlocal metric dimension of windmill graph.