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Keywords = Steiner point

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23 pages, 384 KiB  
Article
Robust Method for Confidence Interval Estimation in Outlier-Prone Datasets: Application to Molecular and Biophysical Data
by Victor V. Golovko
Biomolecules 2025, 15(5), 704; https://doi.org/10.3390/biom15050704 - 12 May 2025
Viewed by 811
Abstract
Estimating confidence intervals in small or noisy datasets is a recurring challenge in biomolecular research, particularly when data contain outliers or exhibit high variability. This study introduces a robust statistical method that combines a hybrid bootstrap procedure with Steiner’s most frequent value (MFV) [...] Read more.
Estimating confidence intervals in small or noisy datasets is a recurring challenge in biomolecular research, particularly when data contain outliers or exhibit high variability. This study introduces a robust statistical method that combines a hybrid bootstrap procedure with Steiner’s most frequent value (MFV) approach to estimate confidence intervals without removing outliers or altering the original dataset. The MFV technique identifies the most representative value while minimizing information loss, making it well suited for datasets with limited sample sizes or non-Gaussian distributions. To demonstrate the method’s robustness, we intentionally selected a dataset from outside the biomolecular domain: a fast-neutron activation cross-section of the 109Ag(n, 2n)108mAg reaction from nuclear physics. This dataset presents large uncertainties, inconsistencies, and known evaluation difficulties. Confidence intervals for the cross-section were determined using a method called the MFV–hybrid parametric bootstrapping (MFV-HPB) framework. In this approach, the original data points were repeatedly resampled, and new values were simulated based on their uncertainties before the MFV was calculated. Despite the dataset’s complexity, the method yielded a stable MFV estimate of 709 mb with a 68.27% confidence interval of [691, 744] mb, illustrating the method’s ability to provide interpretable results in challenging scenarios. Although the example is from nuclear science, the same statistical issues commonly arise in biomolecular fields, such as enzymatic kinetics, molecular assays, and diagnostic biomarker studies. The MFV-HPB framework provides a reliable and generalizable approach for extracting central estimates and confidence intervals in situations where data are difficult to collect, replicate, or interpret. Its resilience to outliers, independence from distributional assumptions, and compatibility with small-sample scenarios make it particularly valuable in molecular medicine, bioengineering, and biophysics. Full article
(This article belongs to the Topic Bioinformatics in Drug Design and Discovery—2nd Edition)
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34 pages, 614 KiB  
Article
Evolutionary Approach to the Euclidean Steiner Tree Problem in n-Space
by Michał Bereta
Appl. Sci. 2025, 15(3), 1413; https://doi.org/10.3390/app15031413 - 30 Jan 2025
Viewed by 881
Abstract
This article presents the application of a genetic algorithm for solving the Euclidean Steiner problem in spaces of dimensionality greater than 2. The Euclidean Steiner problem involves finding the minimum spanning network that connects a given set of vertices, including the additional Steiner [...] Read more.
This article presents the application of a genetic algorithm for solving the Euclidean Steiner problem in spaces of dimensionality greater than 2. The Euclidean Steiner problem involves finding the minimum spanning network that connects a given set of vertices, including the additional Steiner vertices, in a multi-dimensional space. The focus of this research is to compare several different settings of the method, including the crossover operators and sorting of the input data. The paper points out that significant improvement in results can be achieved through proper initialization of the initial population, which depends on the appropriate sorting of vertices. Two approaches were proposed, one based on the nearest neighbor method, and the other on the construction of a minimum spanning tree. Full article
(This article belongs to the Section Computing and Artificial Intelligence)
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13 pages, 309 KiB  
Article
The Cyclically Resolvable Steiner Triple Systems of Order 57
by Svetlana Topalova and Stela Zhelezova
Mathematics 2025, 13(2), 212; https://doi.org/10.3390/math13020212 - 10 Jan 2025
Viewed by 925
Abstract
A resolution of a Steiner triple system of order v (STS(v)) is point-cyclic if it has an automorphism permuting the points in one cycle. An STS(v) is cyclically resolvable if it has at least one point-cyclic resolution. Cyclically resolvable [...] Read more.
A resolution of a Steiner triple system of order v (STS(v)) is point-cyclic if it has an automorphism permuting the points in one cycle. An STS(v) is cyclically resolvable if it has at least one point-cyclic resolution. Cyclically resolvable STS(v)s have important applications in Coding Theory. They have been classified up to v=45 and before the present work v=57 was the first open case. There are 2,353,310 cyclic STS(57)s. We establish that 155,966 of them are cyclically resolvable yielding 3,638,984 point-cyclic resolutions which we classify with respect to their automorphism groups and to the availability of some configurations. Full article
(This article belongs to the Section E1: Mathematics and Computer Science)
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24 pages, 3764 KiB  
Article
Connectivity Recovery Based on Boundary Nodes and Spatial Triangle Fermat Points for Three-Dimensional Wireless Sensor Networks
by Hongsheng Chen and Ke Shi
Sensors 2024, 24(24), 7876; https://doi.org/10.3390/s24247876 - 10 Dec 2024
Viewed by 835
Abstract
In recent years, wireless sensor networks have been widely used, especially in three-dimensional environments such as underwater and mountain environments. However, in harsh environments, wireless sensor networks may be damaged and split into many isolated islands. Therefore, restoring network connectivity to transmit data [...] Read more.
In recent years, wireless sensor networks have been widely used, especially in three-dimensional environments such as underwater and mountain environments. However, in harsh environments, wireless sensor networks may be damaged and split into many isolated islands. Therefore, restoring network connectivity to transmit data effectively in a timely manner is particularly important. However, the problem of finding the minimum relay nodes is NP-hard, so heuristics methods are preferred. This paper presents a novel connectivity recovery strategy based on boundary nodes and spatial triangle Fermat points for three-dimensional wireless sensor networks. The isolated islands are represented as the boundary nodes, and the connectivity recovery problem is modeled as a graph connectivity problem. Three heuristics algorithms—the variant Kruskal algorithm, the variant Prim algorithm, and the spatial triangle Fermat point algorithm—are proposed to solve this problem. The variant Kruskal algorithm and the variant Prim algorithm connect the isolated islands by constructing the minimum spanning tree to link all the boundary nodes and placing relay nodes along the edges of this tree. We derive an accurate formula to determine the coordinates of spatial triangle Fermat points. Based on this formula, the spatial triangle Fermat point algorithm constructs a Steiner tree to restore network connectivity. Extensive simulation experiments demonstrate that our proposed algorithms perform better than the existing algorithm. Full article
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10 pages, 2599 KiB  
Article
Poncelet Porisms and Loci of Centers in the Isotropic Plane
by Ema Jurkin
Mathematics 2024, 12(4), 618; https://doi.org/10.3390/math12040618 - 19 Feb 2024
Cited by 1 | Viewed by 1296
Abstract
Any triangle in an isotropic plane has a circumcircle u and incircle i. It turns out that there are infinitely many triangles with the same circumcircle u and incircle i. This one-parameter family of triangles is called a poristic system of [...] Read more.
Any triangle in an isotropic plane has a circumcircle u and incircle i. It turns out that there are infinitely many triangles with the same circumcircle u and incircle i. This one-parameter family of triangles is called a poristic system of triangles. We study the trace of the centroid, the Feuerbach point, the symmedian point, the Gergonne point, the Steiner point and the Brocard points for such a system of triangles. We also study the traces of some further points associated with the triangles of the poristic family, and we prove that the vertices of the contact triangle, tangential triangle and anticomplementary triangle move on circles while the initial triangle traverses the poristic family. Full article
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16 pages, 4471 KiB  
Article
Weak Points of Double-Plate Stabilization Used in the Treatment of Distal Humerus Fracture through Finite Element Analysis
by Artur Kruszewski, Szczepan Piszczatowski, Piotr Piekarczyk, Piotr Cieślik and Krzysztof Kwiatkowski
J. Clin. Med. 2024, 13(4), 1034; https://doi.org/10.3390/jcm13041034 - 11 Feb 2024
Cited by 1 | Viewed by 1759
Abstract
Background: Multi-comminuted, intra-articular fractures of the distal humerus still pose a challenge to modern orthopedics due to unsatisfactory treatment results and a high percentage (over 50%) of postoperative complications. When surgical treatment is chosen, such fractures are fixed using two plates with locking [...] Read more.
Background: Multi-comminuted, intra-articular fractures of the distal humerus still pose a challenge to modern orthopedics due to unsatisfactory treatment results and a high percentage (over 50%) of postoperative complications. When surgical treatment is chosen, such fractures are fixed using two plates with locking screws, which can be used in three spatial configurations: either parallel or one of two perpendicular variants (posterolateral and posteromedial). The evaluation of the fracture healing conditions for these plate configurations is unambiguous. The contradictions between the conclusions of biomechanical studies and clinical observations were the motivation to undertake a more in-depth biomechanical analysis aiming to indicate the weak points of two-plate fracture stabilization. Methods: Research was conducted using the finite element method based on an experimentally validated model. Three variants of distal humerus fracture (Y, λ, and H) were fixed using three different plate configurations (parallel, posterolateral, and posteromedial), and they were analyzed under six loading conditions, covering the whole range of flexion in the elbow joint (0–145°). A joint reaction force equal to 150 N was assumed, which corresponds with holding a weight of 1 kg in the hand. The biomechanical conditions of bone union were assessed based on the interfragmentary movement (IFM) and using criteria formulated by Steiner et al. Results: The IFMs were established for particular regions of all of the analyzed types of fracture, with distinction to the normal and tangential components. In general, the tangential component of IFM was greater than normal. A strong influence of the elbow joint’s angular position on the IFM was observed, with excessive values occurring for flexion angles greater than 90°. In most cases, the smallest IFM values were obtained for the parallel plaiting, while the greatest values were obtained for the posteromedial plating. Based on IFM values, fracture healing conditions in particular cases (fracture type, plate configuration, loading condition, and fracture gap localization) were classified into one of four groups: optimal bone union (OPT), probable union (PU), probable non-union (PNU), and non-union (NU). Conclusions: No plating configuration is able to ensure distal humerus fracture union when the full elbow flexion is allowed while holding a weight of 1 kg in the hand. However, flexion in the range of 0–90° with such loadings is acceptable when using parallel plating, which is a positive finding in the context of the early rehabilitation process. In general, parallel plating ensures better conditions for fracture healing than perpendicular plate configurations, especially the posteromedial version. Full article
(This article belongs to the Special Issue Advances in Trauma and Orthopedic Surgery)
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14 pages, 323 KiB  
Article
Unveiling Insights: Harnessing the Power of the Most-Frequent-Value Method for Sensor Data Analysis
by Victor V. Golovko, Oleg Kamaev and Jiansheng Sun
Sensors 2023, 23(21), 8856; https://doi.org/10.3390/s23218856 - 31 Oct 2023
Cited by 7 | Viewed by 1438
Abstract
The paper explores the application of Steiner’s most-frequent-value (MFV) statistical method in sensor data analysis. The MFV is introduced as a powerful tool to identify the most-common value in a dataset, even when data points are scattered, unlike traditional mode calculations. Furthermore, the [...] Read more.
The paper explores the application of Steiner’s most-frequent-value (MFV) statistical method in sensor data analysis. The MFV is introduced as a powerful tool to identify the most-common value in a dataset, even when data points are scattered, unlike traditional mode calculations. Furthermore, the paper underscores the MFV method’s versatility in estimating environmental gamma background blue (the natural level of gamma radiation present in the environment, typically originating from natural sources such as rocks, soil, and cosmic rays), making it useful in scenarios where traditional statistical methods are challenging. It presents the MFV approach as a reliable technique for characterizing ambient radiation levels around large-scale experiments, such as the DEAP-3600 dark matter detector. Using the MFV alongside passive sensors such as thermoluminescent detectors and employing a bootstrapping approach, this study showcases its effectiveness in evaluating background radiation and its aptness for estimating confidence intervals. In summary, this paper underscores the importance of the MFV and bootstrapping as valuable statistical tools in various scientific fields that involve the analysis of sensor data. These tools help in estimating the most-common values and make data analysis easier, especially in complex situations, where we need to be reasonably confident about our estimated ranges. Our calculations based on MFV statistics and bootstrapping indicate that the ambient radiation level in Cube Hall at SNOLAB is 35.19 μGy for 1342 h of exposure, with an uncertainty range of +3.41 to 3.59μGy, corresponding to a 68.27% confidence level. In the vicinity of the DEAP-3600 water shielding, the ambient radiation level is approximately 34.80 μGy, with an uncertainty range of +3.58 to 3.48μGy, also at a 68.27% confidence level. These findings offer crucial guidance for experimental design at SNOLAB, especially in the context of dark matter research. Full article
(This article belongs to the Special Issue Advances in Particle Detectors and Radiation Detectors)
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9 pages, 281 KiB  
Brief Report
A Self-Similar Infinite Binary Tree Is a Solution to the Steiner Problem
by Danila Cherkashin and Yana Teplitskaya
Fractal Fract. 2023, 7(5), 414; https://doi.org/10.3390/fractalfract7050414 - 20 May 2023
Cited by 2 | Viewed by 1756
Abstract
We consider a general metric Steiner problem, which involves finding a set S with the minimal length, such that SA is connected, where A is a given compact subset of a given complete metric space X; a solution is called [...] Read more.
We consider a general metric Steiner problem, which involves finding a set S with the minimal length, such that SA is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension, which was an open question (the corresponding tree exhibits self-similar fractal properties). Full article
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8 pages, 299 KiB  
Article
Quantum Circuit Optimization for Solving Discrete Logarithm of Binary Elliptic Curves Obeying the Nearest-Neighbor Constrained
by Jianmei Liu, Hong Wang, Zhi Ma, Qianheng Duan, Yangyang Fei and Xiangdong Meng
Entropy 2022, 24(7), 955; https://doi.org/10.3390/e24070955 - 9 Jul 2022
Cited by 2 | Viewed by 2291
Abstract
In this paper, we consider the optimization of the quantum circuit for discrete logarithm of binary elliptic curves under a constrained connectivity, focusing on the resource expenditure and the optimal design for quantum operations such as the addition, binary shift, multiplication, squaring, inversion, [...] Read more.
In this paper, we consider the optimization of the quantum circuit for discrete logarithm of binary elliptic curves under a constrained connectivity, focusing on the resource expenditure and the optimal design for quantum operations such as the addition, binary shift, multiplication, squaring, inversion, and division included in the point addition on binary elliptic curves. Based on the space-efficient quantum Karatsuba multiplication, the number of CNOTs in the circuits of inversion and division has been reduced with the help of the Steiner tree problem reduction. The optimized size of the CNOTs is related to the minimum degree of the connected graph. Full article
(This article belongs to the Special Issue Quantum Computation and Quantum Information)
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13 pages, 323 KiB  
Article
On Some Properties of the First Brocard Triangle in the Isotropic Plane
by Vladimir Volenec, Zdenka Kolar-Begović and Ružica Kolar-Šuper
Mathematics 2022, 10(9), 1381; https://doi.org/10.3390/math10091381 - 20 Apr 2022
Viewed by 2458
Abstract
In this paper we introduce the first Brocard triangle of an allowable triangle in the isotropic plane and derive the coordinates of its vertices in the case of a standard triangle. We prove that the first Brocard triangle is homologous to the given [...] Read more.
In this paper we introduce the first Brocard triangle of an allowable triangle in the isotropic plane and derive the coordinates of its vertices in the case of a standard triangle. We prove that the first Brocard triangle is homologous to the given triangle and that these two triangles are parallelogic. We consider the relationships between the first Brocard triangle and the Steiner axis, the Steiner point, and the Kiepert parabola of the triangle. We also investigate some other interesting properties of this triangle and consider relationships between the Euclidean and the isotropic case. Full article
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34 pages, 1133 KiB  
Review
Symmetries and Geometries of Qubits, and Their Uses
by A. R. P. Rau
Symmetry 2021, 13(9), 1732; https://doi.org/10.3390/sym13091732 - 18 Sep 2021
Cited by 8 | Viewed by 4266
Abstract
The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such [...] Read more.
The symmetry SU(2) and its geometric Bloch Sphere rendering have been successfully applied to the study of a single qubit (spin-1/2); however, the extension of such symmetries and geometries to multiple qubits—even just two—has been investigated far less, despite the centrality of such systems for quantum information processes. In the last two decades, two different approaches, with independent starting points and motivations, have been combined for this purpose. One approach has been to develop the unitary time evolution of two or more qubits in order to study quantum correlations; by exploiting the relevant Lie algebras and, especially, sub-algebras of the Hamiltonians involved, researchers have arrived at connections to finite projective geometries and combinatorial designs. Independently, geometers, by studying projective ring lines and associated finite geometries, have come to parallel conclusions. This review brings together the Lie-algebraic/group-representation perspective of quantum physics and the geometric–algebraic one, as well as their connections to complex quaternions. Altogether, this may be seen as further development of Felix Klein’s Erlangen Program for symmetries and geometries. In particular, the fifteen generators of the continuous SU(4) Lie group for two qubits can be placed in one-to-one correspondence with finite projective geometries, combinatorial Steiner designs, and finite quaternionic groups. The very different perspectives that we consider may provide further insight into quantum information problems. Extensions are considered for multiple qubits, as well as higher-spin or higher-dimensional qudits. Full article
(This article belongs to the Special Issue Symmetry in Quantum Systems)
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9 pages, 295 KiB  
Article
Estimating the Tour Length for the Close Enough Traveling Salesman Problem
by Debdatta Sinha Roy, Bruce Golden, Xingyin Wang and Edward Wasil
Algorithms 2021, 14(4), 123; https://doi.org/10.3390/a14040123 - 12 Apr 2021
Cited by 4 | Viewed by 3836
Abstract
We construct empirically based regression models for estimating the tour length in the Close Enough Traveling Salesman Problem (CETSP). In the CETSP, a customer is considered visited when the salesman visits any point in the customer’s service region. We build our models using [...] Read more.
We construct empirically based regression models for estimating the tour length in the Close Enough Traveling Salesman Problem (CETSP). In the CETSP, a customer is considered visited when the salesman visits any point in the customer’s service region. We build our models using as many as 14 independent variables on a set of 780 benchmark instances of the CETSP and compare the estimated tour lengths to the results from a Steiner zone heuristic. We validate our results on a new set of 234 instances that are similar to the 780 benchmark instances. We also generate results for a new set of 72 larger instances. Overall, our models fit the data well and do a very good job of estimating the tour length. In addition, we show that our modeling approach can be used to accurately estimate the optimal tour lengths for the CETSP. Full article
(This article belongs to the Special Issue Algorithms for Travelling Salesperson Problems)
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15 pages, 446 KiB  
Article
Steiner Configurations Ideals: Containment and Colouring
by Edoardo Ballico, Giuseppe Favacchio, Elena Guardo, Lorenzo Milazzo and Abu Chackalamannil Thomas
Mathematics 2021, 9(3), 210; https://doi.org/10.3390/math9030210 - 21 Jan 2021
Cited by 3 | Viewed by 2427
Abstract
Given a homogeneous ideal Ik[x0,,xn], the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs m,rN [...] Read more.
Given a homogeneous ideal Ik[x0,,xn], the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs m,rN, I(m)Ir holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne–Huneke Conjecture, and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in Pkn. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly’s Conjectures. Moreover, given a hypergraph H, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to H. We apply these results in the case that H is a Steiner System. Full article
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26 pages, 4766 KiB  
Article
Urban Free-Space Optical Network Optimization
by Revital Marbel, Boaz Ben-Moshe and Tal Grinshpoun
Appl. Sci. 2020, 10(21), 7872; https://doi.org/10.3390/app10217872 - 6 Nov 2020
Cited by 4 | Viewed by 2832
Abstract
This paper presents a set of graph optimization problems related to free-space optical communication networks. Such laser-based wireless networks require a line of sight to enable communication, thus a visibility graph model is used herein. The main objective is to provide connectivity from [...] Read more.
This paper presents a set of graph optimization problems related to free-space optical communication networks. Such laser-based wireless networks require a line of sight to enable communication, thus a visibility graph model is used herein. The main objective is to provide connectivity from a communication source point to terminal points through the use of some subset of available intermediate points. To this end, we define a handful of problems that differ mainly in the costs applied to the nodes and/or edges of the graph. These problems should be optimized with respect to cost and performance. The problems at hand are shown to be NP-hard. A generic heuristic based on a genetic algorithm is proposed, followed by a set of simulation experiments that demonstrate the performance of the suggested heuristic method on real-life scenarios. The suggested genetic algorithm is compared with the Euclidean Steiner tree method. Our simulations show that in many settings, especially in dense graphs, the genetic algorithm finds lower-cost solutions than its competitor, while it falls behind in some settings. However, the run-time performance of the genetic algorithm is considerably better in graphs with 1000 nodes or more, being more than twice faster in some settings. We conclude that the suggested heuristic improves run-time performance on large-scale graphs and can handle a wider range of related optimization problems. The simulation results suggest that the 5G urban backbone may benefit significantly from using free-space optical networks. Full article
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15 pages, 3315 KiB  
Article
Minimum Connected Dominating Set Algorithms for Ad Hoc Sensor Networks
by Xuemei Sun, Yongxin Yang and Maode Ma
Sensors 2019, 19(8), 1919; https://doi.org/10.3390/s19081919 - 23 Apr 2019
Cited by 15 | Viewed by 3896
Abstract
To achieve effective communication in ad hoc sensor networks, researchers have been working on finding a minimum connected dominating set (MCDS) as a virtual backbone network in practice. Presently, many approximate algorithms have been proposed to construct MCDS, the best among which is [...] Read more.
To achieve effective communication in ad hoc sensor networks, researchers have been working on finding a minimum connected dominating set (MCDS) as a virtual backbone network in practice. Presently, many approximate algorithms have been proposed to construct MCDS, the best among which is adopting the two-stage idea, that is, to construct a maximum independent set (MIS) firstly and then realize the connectivity through the Steiner tree construction algorithm. For the first stage, this paper proposes an improved collaborative coverage algorithm for solving maximum independent set (IC-MIS), which expands the selection of the dominating point from two-hop neighbor to three-hop neighbor. The coverage efficiency has been improved under the condition of complete coverage. For the second stage, this paper respectively proposes an improved Kruskal–Steiner tree construction algorithm (IK–ST) and a maximum leaf nodes Steiner tree construction algorithm (ML-ST), both of which can make the result closer to the optimal solution. Finally, the simulation results show that the algorithm proposed in this paper is a great improvement over the previous algorithm in optimizing the scale of the connected dominating set (CDS). Full article
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