Geometric Spanner of Segments

One hundred and sixty-eight genotypes of cotton from the same growing region were used as a germplasm group to study the validity of different genetic distances in constructing cotton core subset. Mixed linear model approach was employed to unbiasedly predict genotypic values of 20 traits for eliminating the environmental effect. Six commonly used genetic distances (Euclidean, standardized Euclidean, Mahalanobis, city block, cosine and correlation distances) combining four commonly used hierarchical cluster methods (single distance, complete distance, unweighted pair-group average and Ward's methods) were used in the least distance stepwise sampling (LDSS) method for constructing different core subsets. The analyses of variance (ANOVA) of different evaluating parameters showed that the validities of cosine and correlation distances were inferior to those of Euclidean, standardized Euclidean, Mahalanobis and city block distances. Standardized Euclidean distance was slightly more effective than Euclidean, Mahalanobis and city block distances. The principal analysis validated standardized Euclidean distance in the course of constructing practical core subsets. The covariance matrix of accessions might be ill-conditioned when Mahalanobis distance was used to calculate genetic distance at low sampling percentages, which led to bias in small-sized core subset construction. The standardized Euclidean distance is recommended in core subset construction with LDSS method.

This paper investigates the impact of geometric semantic crossover operators in a wide range of symbolic regression problems. First, it analyses the impact of using Manhattan and Euclidean distance geometric semantic crossovers in the learning process. Then, it proposes two strategies to numerically optimize the crossover mask based on mathematical properties of these operators, instead of simply generating them randomly. An experimental analysis comparing geometric semantic crossovers using Euclidean and Manhattan distances and the proposed strategies is performed in a test bed of twenty datasets. The results show that the use of different distance functions in the semantic geometric crossover has little impact on the test error, and that our optimized crossover masks yield slightly better results. For SGP practitioners, we suggest the use of the semantic crossover based on the Euclidean distance, as it achieved similar results to those obtained by more complex operators.

Clustering is a data mining method that is widely used to group data based on similarity. This clustering process can be used to streamline data so as to facilitate the data ranking process. The purpose of this study was to make comparisons of distance measurements on the K-Means and TOPSIS methods to select students who would take part in industrial visit activities. The method used in this study is the K-Means Algorithm to carry out the clustering process whose results will be processed using the TOPSIS method, both of which use Euclidean, Manhattan and Minkowsky Distance. Based on the clustering process, there were 21 respondents who were eligible to be included, then with TOPSIS a ranking process was carried out. Of the three distance measurements used based on the Pearson Euclidean distance correlation test, the highest results were 0.992, Manhattan 0.982 and Minkowsky 0.980, with ratings one, two and three respectively. For the Spearman correlation, Eculidean is 0.972, Manhattan is 0.982 and Minkowski is 0.955. Thus, Euclidean distance gives the best correlation results, while for alternatives, Manhattan distance or Minkoesky distance can be used.

Semantic similarity, the task of determining whether two sentences convey the same meaning, is central to applications such as paraphrase detection, semantic search, and question answering. Despite the widespread adoption of transformer-based models for this task, their performance is influenced by both the choice of similarity measure and BERT (bert-base-nli-mean-tokens), RoBERTa (all-roberta-large-v1), and MPNet (all-mpnet-base-v2) on the Microsoft Research Paraphrase Corpus (MRPC). Sentence embeddings were compared using cosine similarity, dot product, Manhattan distance, and Euclidean distance, with thresholds optimized for accuracy, balanced accuracy, and F1-score. Results indicate a consistent advantage for MPNet, which achieved the highest accuracy (75.6%), balanced accuracy (71.0%), and F1-score (0.836) when paired with cosine similarity at an optimized threshold of 0.671. BERT and RoBERTa performed competitively but exhibited greater sensitivity to the choice of Similarity metric, with BERT notably underperforming when using cosine similarity compared to Manhattan or Euclidean distance. Optimal thresholds varied widely (0.334–0.867), underscoring the difficulty of establishing a single, generalizable cut-off for paraphrase classification. These findings highlight the value of fine-tuning of both Similarity metrics and thresholds alongside model selection, offering practical guidance for designing high-accuracy semantic similarity systems in real-world NLP applications.

The security and integrity of computer systems and networks highly depend on malware detection. In the realm of malware detection, the K-Nearest Neighbors (KNN) algorithm is a well-liked and successful machine learning algorithm. However, the choice of an acceptable distance metric parameter has a significant impact on the KNN algorithm's performance. This study tries to improve malware detection by adjusting the KNN algorithm's distance metric parameter. The distance metric greatly influences the similarity or dissimilarity between instances in the feature space. The KNN algorithm for malware detection can be more accurate and effective by carefully choosing or modifying the distance metric. This paper analyzes multiple distance metrics, including Minkowski distance, Manhattan distance, and Euclidean distance. These metrics account for the traits of malware samples while capturing various aspects of similarity. The effectiveness of the KNN algorithm is evaluated using the MalMem-2022 malware dataset, and the results are broken down into these three-distance metrics. The experimental findings show that, among the three distance metric parameters, the Euclidean and Minkowski distance metric parameters considerably produced the best outcomes with binary classification. While with multiclass classification, the KNN algorithm has achieved the highest outcomes using Manhattan distance.

Cultural relics contain unique and rich historical, artistic, economic and other diversified values. However, the imitation and counterfeiting of cultural relics have brought great challenges to the protection and research of cultural relics. In order to improve the problems of strong subjectivity and low precision of the traditional "ophthalmic" identification method, this paper combines spectroscopy with cultural relics to identify the authenticity of pottery and porcelain cultural relics. Under constant temperature and humidity conditions, four kinds of LED with different spectral bands were used as experimental light sources, two Jingdezhen moon flower winter melon porcelain vases with the same appearance and two songxianghu kiln white glazed small clay pots with the same appearance were used as cultural relics samples, and multiple feature points were selected for precision spectral detection. Treat the detection results as multidimensional vectors, calculate the Euclidean distance, Manhattan distance and Chebyshev distance between the vectors, and look for the appropriate threshold to identify the authenticity of the artifact. The results show that: First, the spectral difference between the feature points of cultural relics with the same appearance is significant, and the source of cultural relics can be accurately identified by using the spectral curve; Secondly, this paper calculates the thresholds based on Euclidean distance, Manhattan distance and Chebyshev distance for porcelain and pottery. Among them, the accuracy of identifying porcelain using a threshold based on Euclidean distance is the highest, which is 87.5%; The accuracy of identifying pottery using a threshold based on Manhattan distance was the highest, at 91.67%. By using the spectral curves and thresholds of cultural relics measured by precision spectrometers, this paper generates the spectral identity card of cultural relics. Compared with traditional cultural relics detection, the spectral dimension can better meet the needs of archaeologists for in-depth research on cultural relics.

Topologically associating domains (TADs) are the most fundamental elements and significant structures of the eukaryotic genome. Currently, algorithms have been developed to find the TADs. But few algorithms are reported to compare the similarity of TADs between genomes. In this study, mice Hi-C sequencing data of four contrasts were enrolled. Seventeen algorithms, including BPscore, Jaccard index (JI) distance, VI distance, image hash, image subtraction, image variance, and so on, were used to quantify the genomic similarity of TADs. Image subtraction, Euclidean distance, and Manhattan distance were significantly better for TAD difference detection than the others. Deferent Hash (dHash) with the best zoom size ranked the second, followed by improved Hamming distance algorithm and JI distance. Advantages and disadvantages of various algorithms for quantifying the similarity of TADs were compared. Our work could provide the fundament for TADs comparison.

This paper is concerned with a special case of the Generalized Minimum Spanning Tree Problem. The Generalized Minimum Spanning Tree Problem is de¯ned on an undirected graph, where the vertex set is partitioned into clusters, and non-negative costs are associated with the edges. The problem is to ¯nd a tree of minimum cost containing exactly one vertex in each cluster. We consider a geometric case of the problem where the graph is complete, all vertices are situated in the plane, and Euclidean distance de¯nes the edge cost. We prove that the problem admits PTAS if restricted to grid clustering.

Arrangements of planar curves are fundamental structures in computational geometry. We describe the recent developments in the arrangement package of Cgal, the Computational Geometry Algorithms Library, making it easier to use, to extend and to adapt to a variety of applications. This improved flexibility of the code does not come at the expense of efficiency as we mainly use generic-programming techniques, which make dexterous use of the compilation process. To the contrary, we expedited key operations as we demonstrate by experiments.

Lower envelopes are fundamental structures in computational geometry, which have many applications, such as computing general Voronoi diagrams and performing hidden surface removal in computer graphics. We present a generic, robust and efficient implementation of the divide-and-conquer algorithm for computing the envelopes of surfaces in IR3. To the best of our knowledge, this is the first exact implementation that computes envelopes in three-dimensional space. Our implementation is based on CGAL (the Computational Geometry Algorithms Library) and is designated as a CGAL package. The separation of topology and geometry in our solution allows for the reuse of the algorithm with different families of surfaces, provided that a small set of geometric objects and operations on them is supplied. We used our algorithm to compute the lower and upper envelope for several types of surfaces. Exact arithmetic is typically slower than floating-point arithmetic, especially when higher order surfaces are involved. Since our implementation follows the exact geometric computation paradigm, we minimize the number of geometric operations, and by that significantly improve the performance of the algorithm in practice. Our experiments show interesting phenomena in the behavior of the divide-and-conquer algorithm and the combinatorics of lower envelopes of random surfaces.

The well-separated pair decomposition (WSPD) is a fundamental structure in computational geometry. Given a set P of n points in d-dimensional space and a positive separation parameter s, an s-WSPD is a concise representation of all the O(n2) pairs of P requiring only O(sdn) storage. The WSPD has numerous applications in spatial data processing, such as computing spanner graphs, minimum spanning trees, shortest-path oracles, and statistics on interpoint distances. We consider the problem of maintaining a WSPD when points are inserted to or deleted from P. Worst-case arguments suggest that the addition or deletion of a single point could result in the generation (or removal) up to Ω(sd) pairs, which can be unacceptably high in many applications. Fortunately, the actual number of well separated pairs can be significantly smaller in practice, particularly when the points are well clustered. This suggests the importance of being able to respond to insertions and deletions in a manner that is output sensitive, that is, whose running time depends on the actual number of pairs that have been added or removed. We present the first output-sensitive algorithms for maintaining a WSPD of a point set under insertion and deletion. We show that our algorithms are nearly optimal, in the sense that these operations can be performed in time that is roughly equal to the number of changes to the WSPD.

Data clustering methods have been used extensively for image segmentation in the past decade. In our previous work, we had established that combining the traditional clustering algorithms with a meta-heuristic like Firefly Algorithm improves the stability of the output as well as the speed of convergence. In this paper, we have replaced the Euclidean distance formula with kernels. We have combined Intuitionistic Fuzzy C-Means (IFCM) with Firefly algorithm and replaced Euclidean distance with Gaussian, Hyper tangent and Radial Basis Function Kernels. The intuition behind using Kernels instead of Euclidean or Manhattan distance is explained in this paper. In order to prove that the kernel based counter part of IFCM performs better than its corresponding Euclidean distance based algorithm, we have used standard performance indices such as DB index and Dunn index.

K-means Algorithm is a popular method in cluster analysis, and it is most based on the Euclidean distance. In this paper, a modified version of the K-means algorithm based on the shape similarity distance (SSD-K-means) is presented. The shape similarity distance is one kind of non-metric distance measure for similarity estimation based on the characteristic of differences. To demonstrate the effectiveness of the method we proposed, this new algorithm has been tested on three shape data datasets. Experiment results prove that the performance of the SSD-K-means is better than those of the classical K-means algorithm based on the traditional Euclidean and Manhattan distances.

This work has three main parts: the bisectors of curves in the plane, the bisectors of surfaces in \(\mathbb{R}^{3}\), and the Voronoï diagram of a finite family of parallel half-lines in \(\mathbb{R}^{3}\). These subjects are closely related, and have applications in CAD/CAGD and Computational Geometry.We present a new approach to determine, using Cramer’s rule and suitable elimination steps, an exact algebraic parameterization (rational or non rational) of the bisector curve of two given planar rational curves. The approach is, then, generalized by using the generalized Cramer’s rule to determine an exact algebraic parameterization of the bisector surface of two low degree rational surfaces. We apply the method to obtain parametrizations of the bisector of two rational plane curves, when one of them is a circle or a line. On the other hand, we show how to easily obtain parametrizations of the bisector of the following pairs of surfaces: plane-quadric, plane-torus, circular cylinder-non developable quadric, circular cylinder-torus, cylinder-cylinder, cylinder-cone and cone-cone. These parametrizations are rational in most cases. In the remaining cases the parametrization involves one square root which is well-suited for a good approximation of the bisector.In addition, a different approach for the bisector curve problem will be presented. This new method uses dynamic color in GeoGebra for the computation of a geometric and numerical characterization of the bisector of two planar geometric objects (two curves, or a curve and a point). Even if it does not provide an algebraic representation, the method could lead to the computation of an approximate representation of the bisector curve.The Voronoï diagram (VD) is a fundamental data structure in computational geometry with various applications in theoretical and practical areas. We consider the VD of a set of parallel half-lines constrained to a compact domain \(\mathcal{D}_{0} \subset \mathbb{R}^{3}\), with respect to the Euclidean distance. This new kind of VD can be applied to the solution of some problems in the drilling industry, such as hydraulic or mining. We present an efficient approximate algorithm for computing such VD, using a box subdivision process, which produces a mesh representing the topology of the VD.KeywordsComputational GeometryInterval ArithmeticRational CurfRational SurfaceCompact DomainThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Voronoi diagram is one of the most fundamental data structures in computational geometry, which is concerned with the design and analysis of algorithms for geometrical problems. In this paper, a parallel algorithm for constructing the Voronoi diagram on CREW (Concurrent Read and Exclusive Write) model is proposed. This is an improved algorithm based on Preilowski and Mumbeck’s work. In their algorithm, they apply the Neighbor-Point-Theorem and present a parallel approach to check neighbor points. In this article, we propose an improved approach, Wave-Front algorithm, which is a quite different way to check neighbor points. The algorithm is then implemented in both sequential and multithreaded models.Since the Wave-Front algorithm has inherently concurrent tasks that can be executed simultaneously, multithreaded version was executed to observe the performance. Computational results indicate the effectiveness of the threaded model.KeywordsParallel AlgorithmVoronoi DiagramNeighbor PointConcurrent TaskVoronoi PolygonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.