Spanning Tree Research Articles

Better Hardness Results for the Minimum Spanning Tree Congestion Problem

AbstractIn the spanning tree congestion problem, given a connected graph G, the objective is to compute a spanning tree T in G that minimizes its maximum edge congestion, where the congestion of an edge e of T is the number of edges in G for which the unique path in T between their endpoints traverses e. The problem is known to be $$\mathbb{N}\mathbb{P}$$ N P -hard, but its approximability is still poorly understood, and it is not even known whether the optimum solution can be efficiently approximated with ratio o(n). In the decision version of this problem, denoted $${\varvec{K}-\textsf {STC}}$$ K - STC , we need to determine if G has a spanning tree with congestion at most K. It is known that $${\varvec{K}-\textsf {STC}}$$ K - STC is $$\mathbb{N}\mathbb{P}$$ N P -complete for $$K\ge 8$$ K ≥ 8 , and this implies a lower bound of 1.125 on the approximation ratio of minimizing congestion. On the other hand, $${\varvec{3}-\textsf {STC}}$$ 3 - STC can be solved in polynomial time, with the complexity status of this problem for $$K\in { \left\{ 4,5,6,7 \right\} }$$ K ∈ 4 , 5 , 6 , 7 remaining an open problem. We substantially improve the earlier hardness results by proving that $${\varvec{K}-\textsf {STC}}$$ K - STC is $$\mathbb{N}\mathbb{P}$$ N P -complete for $$K\ge 5$$ K ≥ 5 . This leaves only the case $$K=4$$ K = 4 open, and improves the lower bound on the approximation ratio to 1.2. Motivated by evidence that minimizing congestion is hard even for graphs of small constant radius, we also consider $${\varvec{K}-\textsf {STC}}$$ K - STC restricted to graphs of radius 2, and we prove that this variant is $$\mathbb{N}\mathbb{P}$$ N P -complete for all $$K\ge 6$$ K ≥ 6 .

Resilience enhancement for interdependent power systems by AI-assisted disaster response

The failure of urban Electrical Distribution Systems (EDSs) can lead to direct or indirect consequences for their interdependent critical infrastructures, for example, the hospitals, the fire department, etc., negatively affecting the well-being of citizens in the city. To improve the resilience of the coupled power network, the Operational Interdependence Simulator (OIS) is proposed to help prioritize the queue of failure recovery and Minimum Spanning Tree and Shortest Path Tree topology reconfiguration to minimize critical load downtime. Additionally, AI agents are introduced to assist decision-making processes. By utilizing machine learning and training in thousands of offline scenarios, the system can be used for real-time calculation of the optimal recovery paths when given the number and location of electrical faults. The IEEE 70-node distribution system case validated the high prediction accuracy and post-disaster topology reconfiguration while considering the interdependencies.

Genetic characterization of Neisseria meningitidis isolates recovered from patients with invasive meningococcal disease in Lithuania.

Neisseria meningitidis is a gram-negative bacterium responsible for life-threatening invasive infections known as invasive meningococcal disease and is associated with high fatality rates and serious lifelong disabilities among survivors. This study aimed to characterize N. meningitidis isolates cultured from blood and cerebrospinal fluid collected between 2009 and 2021 in Lithuania, assess their genomic relationships with European strains, and evaluate the possibility of using a cost-effective method for strain characterization, thus improving the national molecular surveillance of invasive meningococcal disease. In total, 321 N. meningitidis isolates were collected and analyzed using multilocus restriction typing (MLRT). Amplification of the penA gene and restriction fragment length polymorphism analysis were performed to identify the modified penA genes. Based on the MLRT genotyping results, we selected 10 strains for additional analysis using whole-genome sequencing. The sequenced genomes were incorporated into a dataset of publicly available N. meningitidis genomes to evaluate genomic diversity and establish phylogenetic relationships within the Lithuanian and European circulating strains. We identified 83 different strains using MLRT genotyping. Genomic diversity of N. meningitidis genomes analysed revealed 21 different sequence types (STs) circulating in Lithuania. Among these, ST34 was the most prevalent. Notably, three isolates displayed unique combinations of seven housekeeping genes and were identified as novel STs: ST16969, ST16901, and ST16959. The analyzed strains were found to possess virulence factors not commonly found in N. meningitidis. Six distinct penA profiles were identified, each with different frequencies. In the present study, we also identified N. meningitidis strains with new penA, NEIS0123, NEIS1320, NEIS1525, NEIS1600, and NEIS1753 loci variants. In our study, using the cgMLST scheme, Minimum Spanning Tree (MST) analysis did not identify significant geographic relationships between Lithuanian N.meningitidis isolates and strains from Europe. Discussion: To our knowledge, this is the first study to employ whole genome sequencing (WGS) method for a comprehensive genetic characterization of invasive N. meningitidis isolates from Lithuania. This approach provides a more detailed and precise analysis of genomic relationships and diversity compared to prior studies relying on traditional molecular typing methods and antigen analysis.

Enhanced Neonatal Brain Tissue Analysis via Minimum Spanning Tree Segmentation and the Brier Score Coupled Classifier

Automatic assessment of brain regions in an MR image has emerged as a pivotal tool in advancing diagnosis and continual monitoring of neurological disorders through different phases of life. Nevertheless, current solutions often exhibit specificity to particular age groups, thereby constraining their utility in observing brain development from infancy to late adulthood. In our research, we introduce a novel approach for segmenting and classifying neonatal brain images. Our methodology capitalizes on minimum spanning tree (MST) segmentation employing the Manhattan distance, complemented by a shrunken centroid classifier empowered by the Brier score. This fusion enhances the accuracy of tissue classification, effectively addressing the complexities inherent in age-specific segmentation. Moreover, we propose a novel threshold estimation method utilizing the Brier score, further refining the classification process. The proposed approach yields a competitive Dice similarity index of 0.88 and a Jaccard index of 0.95. This approach marks a significant step toward neonatal brain tissue segmentation, showcasing the efficacy of our proposed methodology in comparison to the latest cutting-edge methods.

A Study of the Multi-Objective Neighboring Only Quadratic Minimum Spanning Tree Problem in the Context of Uncertainty

A Study of the Multi-Objective Neighboring Only Quadratic Minimum Spanning Tree Problem in the Context of Uncertainty

Digging into the Interior of Hot Cores with ALMA (DIHCA). IV. Fragmentation in High-mass Star-forming Clumps

Fragmentation contributes to the formation and evolution of stars. Observationally, high-mass stars are known to form multiple-star systems, preferentially in cluster environments. Theoretically, Jeans instability has been suggested to determine characteristic fragmentation scales, and thermal or turbulent motion in the parental gas clump mainly contributes to the instability. To search for such a characteristic fragmentation scale, we have analyzed Atacama Large Millimeter/submillimeter Array (ALMA) 1.33 mm continuum observations toward 30 high-mass star-forming clumps taken by the Digging into the Interior of Hot Cores with ALMA survey. We have identified 573 cores using the dendrogram algorithm and measured the separation of cores by using the Minimum Spanning Tree technique. The core separation corrected by projection effects has a distribution peaked around 5800 au. In order to remove biases produced by different distances and sensitivities, we further smooth the images to a common physical scale and perform completeness tests. Our careful analysis finds a characteristic fragmentation scale of ∼7000 au, comparable to the thermal Jeans length of the clumps. We conclude that thermal Jeans fragmentation plays a dominant role in determining the clump fragmentation in high-mass star-forming regions, without the need to invoke turbulent Jeans fragmentation.

The Use of Probability and Edge Analysis to Solve the Multi-Period Degree Constrained Minimum Spanning Tree Problem

The goal of the Multiperiod Degree Constrained Minimum Spanning Tree (MPDCMST) problem is to determine the smallest weight-spanning tree that satisfies the vertex installation criterion for each period and maintains the degree requirement in each vertex. This issue emerges as a network connection problem. The degree requirement indicates the reliability of each vertex, and the vertex connection/installation requirement denotes the priority vertices that must be inserted in the network within a specific time frame. The installation is split up into multiple phases/stages. This is because of various considerations such as severe weather, budgetary limitations, etc. In this research, two algorithms for solving the MPDCMST using probability hybridized with Prim’s modification, and edge analysis are proposed. The algorithms are implemented on the undirected complete graph of orders 10 to 100. The solutions are compared with some heuristics which are already in the literature. The results show that the proposed algorithms perform better.

Genotyping and phylogenetic analysis of human Papillomaviruses from women with genital warts

Background and objectives. The pathophysiology of Human Papillomavirus (HPV) infections is dependent on the genotype of virus, the host immune system and the local environmental conditions. HPV infections are linked to a variety of benign and malignant disorders. Anogenital warts is a Sexually Transmitted Infection (STI) primarily caused by Low-Risk HPV (LR-HPV) genotypes 6 and 11. The aims were to Investigation of the prevalence and genotypic distribution of HPV in women with genital warts in Mosul/Iraq, HPV-DNA sequencing and Phylogenetic analysis including minimum spanning trees of amplified HPV L1 gene comparison the present sequences with similar genotypes in GenBank – National Center Biotechnology Information (NCBI) using mega-7-software. Materials and methods. A total of 150 samples were collected from women with genital wart infections. Polymerase Chain Reaction (PCR) was performed using GP+5/GP+6 and MY09/MY11 consensus primers. Results. About 84.6% prevalence of HPV genotypes. The highest frequency (32% and 28.6%) were observed in the 19-28 and 29-38 age groups. Conclusion. Women in Mosul, Iraq, who have genital warts are more likely to have different genotypes of HPV. HPV 11 type was found to be the most common, followed HPV6 and HPV10 as LR HPV, while HPV 16 and HPV45 were the most common as HR HPV. The analysis of the phylogenetic tree revealed a genetic relationship between our isolates and global isolates ranging from (10-100%) bootstrap ratio this indicating phylogenetic relationships (divergent and convergent evolution) this was meant the diversity in origin of the various genotypes common in Mosul, Iraq.

FRACTAL DIMENSION OF STAR CLUSTERS

Quantitative analysis of the structure of star clusters is crucial for understanding their formation and evolution. In this article, we explore the application of fractal dimension analysis to study the evolution of star clusters, also fractal dimension, a concept from fractal geometry, provides a quantitative measure of the complexity and self-similarity of geometric objects. By considering star clusters as complex networks, we employ the box covering method to calculate their fractal dimension. Our methodology combines the well-established Minimum Spanning Tree (MST) and Box-Covering (BC) methods and using these methods, the fractal structure of the clusters was determined. It was revealed that star clusters disintegrate at a fractal dimension of 1.3 and obey a power law. It should be noted that the obtained result was compared with the results of the McLuster.

One-ended spanning trees and definable combinatorics

Let ( X , τ ) (X,\tau ) be a Polish space with Borel probability measure μ \mu , and G G a locally finite one-ended Borel graph on X X . We show that G G admits a Borel one-ended spanning tree generically. If G G is induced by a free Borel action of an amenable (resp., polynomial growth) group then we show the same result μ \mu -a.e. (resp., everywhere). Our results generalize recent work of Timár, as well as of Conley, Gaboriau, Marks, and Tucker-Drob, who proved this in the probability measure preserving setting. We apply our theorem to find Borel orientations in even-degree graphs and measurable and Baire measurable perfect matchings in regular bipartite graphs, refining theorems that were previously only known to hold for measure preserving graphs. In particular, we prove that bipartite one-ended d d -regular Borel graphs admit Baire measurable perfect matchings.

Logarithmic Corrections to the Alexander–Orbach Conjecture for the Four-Dimensional Uniform Spanning Tree

We compute the precise logarithmic corrections to Alexander–Orbach behaviour for various quantities describing the geometric and spectral properties of the four-dimensional uniform spanning tree. In particular, we prove that the volume of an intrinsic n-ball in the tree is n2(logn)-1/3+o(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^2 (\\log n)^{-1/3+o(1)}$$\\end{document}, that the typical intrinsic displacement of an n-step random walk is n1/3(logn)1/9-o(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{1/3} (\\log n)^{1/9-o(1)}$$\\end{document}, and that the n-step return probability of the walk decays as n-2/3(logn)1/9-o(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{-2/3}(\\log n)^{1/9-o(1)}$$\\end{document}.

Forbidden Subgraphs Restricting Vertices of Degree Two in a Spanning Tree

Forbidden Subgraphs Restricting Vertices of Degree Two in a Spanning Tree

Minimum Spanning Tree Image Segmentation Model Based on New Weights

Minimum Spanning Tree Image Segmentation Model Based on New Weights

Order-sensitive competitive revenue maximization for viral marketing in social networks

The Competitive Influence Maximization (CIM) problem is a critical issue in viral marketing, focusing on selecting a set of influential individuals, known as seed users, for competitors to maximize their revenue. These seed users have significant sway in social networks and serve as valuable marketing resources provided by the platform. They are often displayed in a certain order launched by the platform and the potential information hidden in the order can profoundly affect the final marketing outcomes. However, current CIM research predominantly emphasizes designing effective algorithms for seed selection while ignoring the impact of the seed order launched by the platform. Therefore, this paper focuses on identifying the optimal seed order to maximize platform revenue in a competitive market environment. Specifically, we introduce a new problem called Order-Sensitive Competitive Revenue Maximization (OSCRM) to investigate the CIM problem from a new practical perspective. We prove the problem to be NP-hard and present a simple greedy algorithm with a 1/3-approximate ratio. To address it more efficiently, we further propose an enhanced greedy algorithm called GMST. This algorithm leverages the maximum spanning tree (MST) and achieves a 1/2-approximate ratio. Extensive experiments on four real-world datasets demonstrate the effectiveness of our proposed GMST algorithm.

Minimum Spanning Trees in Infinite Graphs: Theory and Algorithms

Minimum Spanning Trees in Infinite Graphs: Theory and Algorithms

Piercing Diametral Disks Induced by Edges of Maximum Spanning Trees

Let $P$ be a set of points in the plane and let $T$ be a maximum-weight spanning tree of $P$.For an edge $(p,q)$, let $D_{pq}$ be the diametral disk induced by $(p,q)$, i.e., the disk having the segment $\overline{pq}$ as its diameter. Let $\cal{D}_T$ be the set of the diametral disks induced by the edges of $T$. In this paper, we show that one point is sufficient to pierce all the disks in $\cal{D}_T$. Actually, we show that the center of the smallest enclosing circle of $P$ is contained in all the disks of $\cal{D}_T$, and thus the piercing point can be computed in linear time.

Improving data interpretability with new differential sample variance gene set tests.

Gene set analysis methods have played a major role in generating biological interpretations from omics data such as gene expression datasets. However, most methods focus on detecting homogenous pattern changes in mean expression and methods detecting pattern changes in variance remain poorly explored. While a few studies attempted to use gene-level variance analysis, such approach remains under-utilized. When comparing two phenotypes, gene sets with distinct changes in subgroups under one phenotype are overlooked by available methods although they reflect meaningful biological differences between two phenotypes. Multivariate sample-level variance analysis methods are needed to detect such pattern changes. We use ranking schemes based on minimum spanning tree to generalize the Cramer-Von Mises and Anderson-Darling univariate statistics into multivariate gene set analysis methods to detect differential sample variance or mean. We characterize these methods in addition to two methods developed earlier using simulation results with different parameters. We apply the developed methods to microarray gene expression dataset of prednisolone-resistant and prednisolone-sensitive children diagnosed with B-lineage acute lymphoblastic leukemia and bulk RNA-sequencing gene expression dataset of benign hyperplastic polyps and potentially malignant sessile serrated adenoma/polyps. One or both of the two compared phenotypes in each of these datasets have distinct molecular subtypes that contribute to heterogeneous differences. Our results show that methods designed to detect differential sample variance are able to detect specific hallmark signaling pathways associated with the two compared phenotypes as documented in available literature. The results in this study demonstrate the usefulness of methods designed to detect differential sample variance in providing biological interpretations when biologically relevant but heterogeneous changes between two phenotypes are prevalent in specific signaling pathways. Software implementation of the developed methods is available with detailed documentation from Bioconductor package GSAR. The available methods are applicable to gene expression datasets in a normalized matrix form and could be used with other omics datasets in a normalized matrix form with available collection of feature sets.

Analysis of the Association Between Antimicrobial Resistance Genes and Virulence Factors in ST11 and Non-ST11 CR-KP Bloodstream Infections in the Intensive Care Unit.

This study aims to investigate the association between antimicrobial resistance genes and virulence factors in ST11 and non-ST11 types of CR-KP in bloodstream infections in the intensive care unit, providing a theoretical basis for infection control and clinical diagnosis and treatment. From January 2021 to June 2023, samples of Klebsiella pneumoniae from bloodstream infections were collected at our hospital, focusing on those resistant to carbapenems. The resistance genes, housekeeping genes, and virulence genes were identified through PCR and analyzed using the GrapeTree software to perform MLST-based minimum spanning tree typing. Among the 85 CR-KP cases, 61.18% were of the ST11 type, predominantly of the KL64 capsular type; non-ST11 types were mainly ST15, accounting for 25.88%, predominantly of the KL5 capsular type. The carriage rates of virulence genes such as rmpA2, entB, silS, kpn, iucA, peg-344, and terB were significantly higher in the ST11 group than in the non-ST11 group. The primary carbapenemase identified was class A enzyme bla KPC-2, with a higher carriage rate in the ST11 group. Drug susceptibility tests showed that the resistance rates for cefepime, ertapenem, nitrofurantoin, amikacin, and gentamicin were also higher in the ST11 group, consistent with the resistance genotype findings. The study reveals that ST11 type CR-KP in intensive care unit bloodstream infections exhibits stronger resistance and higher virulence compared to non-ST11 types, posing significant challenges to clinical treatment. Thus, strict control over the use of carbapenem antibiotics is essential to prevent the spread of resistant plasmids.

New Dynamic Programming algorithm for the Multiobjective Minimum Spanning Tree Problem

The Multiobjective Minimum Spanning Tree (MO-MST) problem generalizes the Minimum Spanning Tree problem by weighting the edges of the input graph using vectors instead of scalars. In this paper, we design a new Dynamic Programming MO-MST algorithm. Dynamic Programming for a MO-MST instance requests solving a One-to-One Multiobjective Shortest Path (MOSP) instance and both instances have equivalent solution sets. The MOSP instance is defined on a so called transition graph. We study the original size of this graph in detail and reduce its size using cost-dependent arc pruning criteria. To solve the MOSP instance on the reduced transition graph, we design the Implicit Graph Multiobjective Dijkstra Algorithm (IG-MDA), exploiting recent improvements on MOSP algorithms from the literature. All in all, the new IG-MDA outperforms the current state of the art on a big set of instances from the literature. Our code and results are publicly available.

Comparison of data-driven thresholding methods using directed functional brain networks.

Over the past two centuries, intensive empirical research has been conducted on the human brain. As an electroencephalogram (EEG) records millisecond-to-millisecond changes in the electrical potentials of the brain, it has enormous potential for identifying useful information about neuronal transactions. The EEG data can be modelled as graphs by considering the electrode sites as nodes andthe linear and nonlinear statistical dependencies among them as edges (with weights). The graph theoretical modelling of EEG data results in functional brain networks (FBNs), which are fully connected (complete) weighted undirected/directed networks. Since various brain regions are interconnected via sparse anatomical connections, the weak links can be filtered out from the fullyconnected networks using a process called thresholding. Multiple researchers in the past decades proposed many thresholding methods to gather more insights about the influential neuronal connections of FBNs. This paper reviews various thresholding methods used in the literature for FBN analysis. The analysis showed that data-driven methods are unbiased since no arbitrary user-specified threshold is required. The efficacy of four data-driven thresholding methods, namely minimum spanning tree (MST), minimum connected component (MCC), union of shortest path trees (USPT), and orthogonal minimum spanning tree (OMST), in characterizing cognitive behavior of the normal human brain is analysed using directed FBNs constructed from EEG data of different cognitive load states. The experimental results indicate that both MCC and OMST thresholding methods can detect cognitive load-induced changes in the directed functional brain networks.