List Of Vertices Research Articles

METHOD OF CREATING A MINIMAL SPANNING TREE ON AN ARBITRARY SUBSET OF VERTICES OF A WEIGHTED UNDIRECTED GRAPH

Context. The relevance of the article is determined by the need for further development of models for optimal restoration of the connectivity of network objects that have undergone fragmentation due to emergency situations of various origins. The method proposed in this article solves the problematic situation of minimizing the amount of restoration work (total financial costs) when promptly restoring the connectivity of a selected subset of elements of a network object after its fragmentation.
 The purpose of the study is to develop a method for creating a minimal spanning tree on an arbitrary subset of vertices of a weighted undirected graph to minimize the amount of restoration work and/or total financial costs when promptly restoring the connectivity of elements that have a higher level of importance in the structure of a fragmented network object.
 Method. The developed method is based on the idea of searching for local minima in the structure of a model undirected graph using graph vertices that are not included in the list of base vertices to be united by a minimal spanning tree. When searching for local minima, the concept of an equilateral triangle and a radial structure in such a triangle is used. In this case, there are four types of substructures that provide local minima: first, those with one common base vertex; second, those with two common base vertices; third, those with three common base vertices; fourth, those without common base vertices, located in different parts of the model graph. Those vertices that are not included in the list of basic ones, but through which local minima are ensured, are added to the basic ones. Other vertices (non-basic) along with their incident edges are removed from the structure of the model graph. Then, using one of the well-known methods of forming spanning trees, a minimal spanning tree is formed on the structure obtained in this way, which combines the set of base vertices.
 Results. 1) A method for creating a minimal spanning tree on an arbitrary subset of vertices of a weighted undirected graph has been developed. 2) A set of criteria for determining local minima in the structure of the model graph is proposed. 3) The method has been verified on test problems.
 Conclusions. The theoretical studies and several experiments confirm the efficiency of the developed method. The solutions developed using the developed method are accurate, which makes it possible to recommend it for practical use in determining strategies for restoring the connectivity of fragmented network objects.

Trapping Sets Search Using the Method of Mixed Integer Linear Programming with a Priori List of Variable Nodes

Purpose of research is to develop a new high-speed method for searching trappin sets in graph codes, ensuring the completeness of the search.Methods. There are two approaches to finding trappin sets. The first, based on the Monte Carlo method with a biased probability estimation using Importance Sampling, involves the use of a decoder. The advantage of this approach is its high performance. The disadvantages are the dependence on decoder parameters and channel characteristics and the finite probability of missing trappin sets. The second approach is based on the use of linear programming methods. The advantage of this approach is the completeness of the resulting list of trappin sets, due to its independence from the decoder parameters and channel characteristics. The disadvantage of this approach is its high computational complexity. In the article, within the framework of the second approach, a new method for searching trappin sets with less computational complexity is proposed. The method involves solving a mixed integer linear programming problem using an a priori list of code vertices participating in the shortest cycles in the code graph. Results. Using the proposed method, a search for trappin sets was performed for several low-density codes. For this purpose, the mathematical linear programming package IBM CPLEX version 12.8 was used, which was run on 32 threads of a 16-core AMD Ryzen 3950X processor with 32GB of RAM (DDR4). In the Margulis code (2640, 1320), using the proposed method, the trappin set TS(6,6) was found in a time of 0.53 s. The speedup provided by the method proposed in the paper compared to the Velazquez-Subramani method is 8252.415 times. Thanks to the high speed and completeness of the search, trappin sets were found for the first time TS(62,16) and TS(52,14) in the Margulis code (4896, 2474 ).Conclusion. The paper proposes a new method for searching trapping sets by solving a mixed integer linear programming problem with an a priori list of code. The method is fast and provides completeness of the search.

Efficient 3D Surface Patch Compression and Reconstruction using Parametric Descriptions and Transform Techniques

This paper proposes and demonstrates novel methods for compressing and reconstructing 3D surface patches typically obtained from scanners that rely on stereo vision, structured light, or time-of-flight techniques. The methods involve applying a polygon reduction to the mesh to get a set of vertices lying in structured planes of a sparse, regular grid. The data in each plane is then described parametrically, and a comparative analysis is conducted using the Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT), and Discrete Wavelet Transform (DWT). Quality factors are employed to further process the transform coefficients, resulting in substantial reductions in the amount of data saved to disk. The paper also defines file formats with the necessary parameters for a complete reconstruction of the sparse mesh. Finally, elliptic Partial Differential Equations (PDE) are used to represent the reconstructed data, and the Laplace equation is iteratively solved between adjacent planes to recover the vertex density of the original mesh. Experimental results demonstrate the effectiveness of the proposed methods, achieving compression rates of over 98% compared to the OBJ file format and over 91% compared to a list of vertices in ASCII format.

Sums of distances on graphs and embeddings into Euclidean space

AbstractLet be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices , take to be any vertex maximizing the sum of distances to the vertices already chosen and iterate, keep adding the “most remote” vertex. The frequency with which the vertices of the graph appear in this sequence converges to a set of probability measures with nice properties. The support of these measures is, generically, given by a rather small number of vertices . We prove that this suggests that the graph G is, in a suitable sense, “m‐dimensional” by exhibiting an explicit 1‐Lipschitz embedding with good properties.

A fast method for obtaining convex combination coefficients

This technical communiqué presents a novel approach for fast computation of convex combination coefficients based on ray shooting. Such a problem arises in the real-time implementation of many set-based control techniques, which require a computationally efficient solution to cope with systems of high dimension or fast dynamics. The proposed method poses no constraints on the polytope structure and requires only a list of vertices and boundary hyperplanes. In test cases involving polytopes of various dimensions and numbers of vertices, the proposed method provided up to seven-fold reductions in the 95% percentile of computation time, compared with a state-of-the-art technique.

Conformal Marked Bisection for Local Refinement of n-Dimensional Unstructured Simplicial Meshes

We present an n -dimensional marked bisection method for unstructured conformal meshes. We devise the method for local refinement in adaptive n -dimensional applications. To this end, we propose a mesh marking pre-process and three marked bisection stages. The pre-process marks the initial mesh conformingly. Then, in the first n − 1 bisections, the method accumulates in reverse order a list of new vertices. In the second stage, the n -th bisection, the method uses the reversed list to cast the bisected simplices as reflected simplices, a simplex type suitable for newest vertex bisection. In the final stage, beyond the n -th bisection, the method switches to newest vertex bisection. To allow this switch, after the second stage, we check that under uniform bisection the mesh simplices are conformal and reflected. These conditions are sufficient to use newest vertex bisection, a bisection scheme guaranteeing key advantages for local refinement. Finally, the results show that the proposed bisection is well-suited for local refinement of unstructured conformal meshes. • A new multi-stage bisection for n-dimensional unstructured conformal meshes. • Suitable for local refinement. • Leads to conformal and reflected meshes after n uniform refinements. • In the third stage, the method favors conformity, finiteness, stability, and locality.

LG-Tree: An Efficient Labeled Index for Shortest Distance Search on Massive Road Networks

With the development of mobile Internet technology, the road network in the real world is becoming larger and more complex, and the real-time response to the shortest distance query has become a high requirement for many industrial applications. Many existing approaches, such as G-tree or G*-tree, have been proposed to answer such search problems, however, these approaches are not efficient enough in dealing with very large graphs. To this end, we propose a novel index called LG-tree, which partitions the large graph into sub-graphs, and then indexes these subgraphs using a balanced tree. For each leaf node of LG-tree, a Distance Inverted File (DIF) is constructed, and these DIFs preserve all the connectivity information between the border vertices on the original graph. Inspired by the state-of-the-art label methods, we propose a novel hierarchy computing approach for each border vertex of LG-tree. Based on DIF and the hierarchy of border vertices, the border vertex list for each border is established, which is convenient for us to calculate the shortest distance and path between a query vertex <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v_{q}$ </tex-math></inline-formula> and a target vertex <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> . Specifically, the number of levels of a vertex hierarchy increases dramatically as the graph size increases, and calculating vertex labels on a very large graph is an NP-hard problem. In order to further improve the calculation efficiency of the shortest distance between vertices, a heuristic method is proposed to limit the level of vertex hierarchy. In addition, a stage-based dynamic programming search method is proposed to divide all search situations between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v_{q}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> into three cases to guarantee the efficiency of the shortest distance search. Extensive experiments are conducted to show that LG-tree and the stage-based method have better performance than the state-of-the-art approaches.

MeshWalker

Most attempts to represent 3D shapes for deep learning have focused on volumetric grids, multi-view images and point clouds. In this paper we look at the most popular representation of 3D shapes in computer graphics---a triangular mesh---and ask how it can be utilized within deep learning. The few attempts to answer this question propose to adapt convolutions &amp; pooling to suit Convolutional Neural Networks (CNNs). This paper proposes a very different approach, termed MeshWalker to learn the shape directly from a given mesh. The key idea is to represent the mesh by random walks along the surface, which "explore" the mesh's geometry and topology. Each walk is organized as a list of vertices, which in some manner imposes regularity on the mesh. The walk is fed into a Recurrent Neural Network (RNN) that "remembers" the history of the walk. We show that our approach achieves state-of-the-art results for two fundamental shape analysis tasks: shape classification and semantic segmentation. Furthermore, even a very small number of examples suffices for learning. This is highly important, since large datasets of meshes are difficult to acquire.

A Cell Reconstruction Tool to Deploy Binary Pyroelectric Sensor Arrays

Due to the advantages of being low cost and low power consumption, pyroelectric infrared sensors are widely employed in many applications, such as target location and tracking. In such applications, reference structure which is used to modulate views of binary sensors can help improve spatial resolutions by segmenting monitoring spaces into many cells which are identified by states (called signatures). A spatial resolution determines localization or tracking accuracy, which depends on the sizes, the shapes, and the signatures of the cells, and is drastically impacted by deployment of sensors and reference structure. However, in order to obtain a better deployment formation and a better spatial resolution, researchers have to spend lots of time to deploy and measure the deployment results of cells (such as their locations, sizes, shapes, and signatures), i.e., a huge amount of measurements which cost time and money in practice. Hence, in this paper, we propose a tool to visually and efficiently represent and analyze the deployment formation and spatial resolution. Generally, the location of a cell is represented by a list of vertices scattered in a cartesian coordinate system and can help researchers get its size, shape, and signature. A cell reconstruction algorithm is proposed to reconstruct cells generated by the deployment formation in which lists of vertices are generated by regarding cells as shortest path rings. We further provide some theoretical analysis and computational complexity analysis of the proposed algorithm. We conduct experiments using our tool to study sensor efficiency and spatial resolution easily and effectively.

A new watermarking method for vector map data: a study on information capacity of the carrier image

The paper presents a study of the watermarking method for vector map data based on the cyclic shift of the vertex list of polygonal objects. We propose a method modification to provide an accurate authentication procedure, as well as to enhance the method robustness against the map contents alteration. The main idea of the improved method is to use a noise-like image as a secondary container for a watermark represented in the form of a bit vector. An algorithm for the construction of a carrier image on the basis of the watermark sequence, as well as an algorithm for extraction of such a sequence, are given. An experimental study on the information capacity of the carrier image and its robustness against distortions simulating the embedding into real cartographic data, such as quantization and noise addition, has been carried out.

New method for GIS vector data protection based on the use of secondary watermark

The paper provides a study on the watermarking method for vector map data based on a cyclic shift of a polygon vertex list. We propose a method modification allowing to automate authentication procedure, as well as to increase robustness against map contents modification. The main idea of the improved method is to use a noise-like image as a secondary carrier for a watermark, represented in the form of a bit sequence. An algorithm for construction of a noise-like image carrying a watermark sequence, as well as an algorithm for extraction of such a sequence, are given. An experimental study explores the information capacity of the carrier image and its robustness against quantization and interpolation noise adding, i.e. distortions simulating the embedding into map objects. The efficiency of the method is also demonstrated on real cartographic data. Conclusions comprise recommendations for rational choice of embedding parameters depending on the number of polygons on the vector map.

Breaking Permutation-Based Mesh Steganography and Security Improvement

Permutation-based steganography in polygonal meshes can provide considerably large embedding capacities for hiding secret messages. However, corresponding steganalysis techniques against such methods have never been studied. This paper identifies the essential differences between naturally generated meshes and meshes produced by permutation-based steganography methods. It is found that the two types of meshes differ significantly in the distribution of topological distances between consecutive mesh elements. Therefore, by measuring the orderliness of the vertex list and the face list of meshes, we develop solutions for several mesh steganalysis problems. These solutions are effective, leading to high detection accuracy; and they are also universal, requiring no knowledge such as which steganography method is used and what data embedding rate is adopted for the detection mechanism to work. Moreover, this paper also presents a security-improved permutation-based mesh steganography approach, by taking advantage of the connectivity information of polygonal meshes and establishing a good trade-off between embedding capacity and undetectability. Without bringing global changes, our approach embeds secret messages into local neighborhoods on meshes. As a result, meshes generated by the proposed steganography approach tend to have natural structures that are unlikely to draw suspicions to steganalyzers.

Cubic interaction vertices for massive/massless continuous-spin fields and arbitrary spin fields

We use light-cone gauge formalism to study interacting massive and massless continuous-spin fields and finite component arbitrary spin fields propagating in the flat space. Cubic interaction vertices for such fields are considered. We obtain parity invariant cubic vertices for coupling of one continuous-spin field to two arbitrary spin fields and cubic vertices for coupling of two continuous-spin fields to one arbitrary spin field. Parity invariant cubic vertices for self-interacting massive/massless continuous-spin fields are also obtained. We find the complete list of parity invariant cubic vertices for continuous-spin fields and arbitrary spin fields.

Linear criterion for testing the extremity of an exact game based on its finest min-representation

A game-theoretical concept of an exact (cooperative) game corresponds to the notion of a discrete coherent lower probability, used in the context of imprecise probabilities. The collection of (suitably standardized) exact games forms a pointed polyhedral cone and the paper is devoted to the recognition of extreme rays of that cone, whose generators are called extreme exact games. We give a necessary and sufficient condition for an exact game to be extreme. Our criterion leads to solving a simple linear equation system determined by a certain min-representation of the game. It has been implemented on a computer and a web-based platform for testing the extremity of an exact game is available, which works with a modest number of variables.The paper also deals with different min-representations of a fixed exact game μ, which can be compared with the help of the concept of a tightness structure (of a min-representation) introduced in the paper. The collection of tightness structures (of min-representations of μ) is shown to be a finite lattice with respect to a refinement relation. We give a method to obtain a min-representation with the finest tightness structure, which construction comes from the coarsest standard min-representation of μ given by the (complete) list of vertices of the core (polytope) of μ. The newly introduced criterion for exact extremity is based on the finest tightness structure.

Cubic interaction vertices for continuous-spin fields and arbitrary spin massive fields

Light-cone gauge formulation of relativistic dynamics of a continuous-spin field propagating in the flat space is developed. Cubic interaction vertices of continuous-spin massless fields and totally symmetric arbitrary spin massive fields are studied. We consider parity invariant cubic vertices that involve one continuous-spin massless field and two arbitrary spin massive fields and parity invariant cubic vertices that involve two continuous-spin massless fields and one arbitrary spin massive field. We construct the complete list of such vertices explicitly. Also we demonstrate that there are no cubic vertices describing consistent interaction of continuous-spin massless fields with arbitrary spin massless fields.

Reversible data hiding for 3D mesh models with three-dimensional prediction-error histogram modification

This study proposes a reversible data hiding (RDH) algorithm for 3D mesh models based on the optimal three-dimensional prediction-error histogram (PEH) modification with recursive construction coding (RCC). Firstly, we design a double-layered prediction scheme to divide all the vertices of 3D mesh model into the “embedded” set and the “referenced” set, according to the odd-even property of indices into the vertex list. Thanks to the geometrical similarity among neighboring vertices, we obtain the prediction errors (PEs) with a sharp histogram. Then we combine every three adjacent PEs into one prediction-error triplet (PET) and construct the three-dimensional PEH with smaller entropy than one-dimensional PEH by utilizing the correlation among PEs. Next we project the three-dimensional PEH into one-dimensional space for scalar PET sequence which is suitable for using RCC. And also, we define the distortion metrics for 3D mesh models, by which we can estimate the optimal probability transition matrix (OTPM) indicating the optimal PEH modification manner. After that, we modify the PET sequence and embed data by RCC according to OTPM. The experimental results show that our method is superior to two state-of-the-art spatial-domain RDH algorithms for 3D mesh models a lot.

A new watermarking method for vector map data

In this paper, we propose a new digital watermarking method for protecting of vector data in geographic information systems. The method is based on a combination of two novel approaches. Firstly, the watermark information is embedded into the polygon map objects by cyclically shifting the list of vertices of each polygon. Secondly, instead of a bit vector, a raster image superimposed on the map is considered as a watermark. The major advantage of this method is that it does not distort the map by altering the coordinate values. A simple example demonstrates the method efficiency, as well as the robustness of the embedded watermark against geometric transformations usually performed when working with map data: translation, scaling, rotation and cropping.

Improvement of the methods for determining optimal characteristics of transportation networks

An improved method of approaching the calculation of maximum flow is developed, which implies applying the method of trees and capacities of tabular processors. The solution can be extended for a problem with several sources and runoffs. This will solve the problem on the optimization of transportation networks with and without limitations in throughput capacity. We designed an improved method for calculating the shortest path, which is resolved by using the Minty Dijkstra's algorithm. By solving the shortest path problem, we receive the shortest route and a list of vertices that it passes. By having indicators of freight traffic from each vertex to all others, we build a tree of the shortest paths. Going from one vertex to another vertex, we obtain density of traffic in the network without limitation in the throughput capacity. When the network has a throughput capacity limitation, imposing flows on the network is a bit complicated. In this case, it is necessary to subtract each elementary flow from the existing throughput capacity of the arc, on which it is imposed. For finding the shortest path, it is possible to correct the flows manually. The process of transforming network models for the process of cargo transportation to the matrix models is presented, through the use of the modified Dijkstra's algorithm. Elements of transportation networks in this case are set in the form of directed graphs. Graphic representation of the results of solving a network traffic problem is given.

Efficient k-edge connected component detection through an early merging and splitting strategy

Computing k-edge connected components can be used to capture closely related vertices in a graph. It is an important problem with many applications. Due to the high time complexities of traditional algorithms for computing k-edge connected components, it is difficult for them to be applied to efficiently process large scale graphs. In this paper, an early merging and splitting based maximal k-edge connected subgraph detection algorithm, named MSK, is proposed. It computes the k-edge connected components of a graph with an order list of vertices which is decided according to the connectivity of vertices in the graph. During the processing of the vertex list, if two vertices are k-edge connected, they are merged into a super-vertex. On the contrary , if the vertex pair does not satisfy the condition of k-edge connectivity, the related vertices and edges are removed from the graph, and the graph is decomposed into several subgraphs. The above steps are performed iteratively in the obtained subgraphs until each subgraph become a k-edge connected component. In order to further improve the time efficiency, an approximate algorithm PMSK is also proposed. The experimental results on a large number of real and synthetic graphs show that the proposed algorithms are accurate and more efficient than the state-of-the-art algorithms.

Classifying Resolving Lists by Distances between Members

Let G be a connected graph and let w1, · · ·wr be a list of vertices. We refer to the choice of a triple (r;G;w1, · · ·wr), as a metric selection. Let ρ be the shortest path metric ofG. We say that w1, · · ·wr resolves G (metricly) if the function c : V (G) −→ Z given by x 7→ (ρ(w1, x), · · · , ρ(wr, x)) is injective. We refer to c as the code map, and to its image as the codes of the triple (r;G;w1, · · · , wr). This paper proves basic results on the following questions: 1. What sets can be the image of a code map? 2. Given the image of a graph’s code map, what can we determine about the graph?