Spanning Tree Research Articles

Hopcroft’s Problem, Log* Shaving, Two-dimensional Fractional Cascading, and Decision Trees

We revisit Hopcroft’s problem and related fundamental problems about geometric range searching. Given n points and n lines in the plane, we show how to count the number of point-line incidence pairs or the number of point-above-line pairs in O ( n 4/3 ) time, which matches the conjectured lower bound and improves the best previous time bound of \(n^{4/3}2^{O(\log ^*n)}\) obtained almost 30 years ago by Matoušek [ 58 ]. We describe two interesting and different ways to achieve the result: The first is randomized and uses a new two-dimensional version of fractional cascading for arrangements of lines; the second is deterministic and uses decision trees in a manner inspired by the sorting technique of Fredman [42]. The second approach extends to any constant dimension. Many consequences follow from these new ideas: For example, we obtain an O ( n 4/3 )-time algorithm for line segment intersection counting in the plane, O ( n 4/3 )-time randomized algorithms for distance selection in the plane and bichromatic closest pair and Euclidean minimum spanning tree in three or four dimensions, and a randomized data structure for halfplane range counting in the plane with O ( n 4/3 ) preprocessing time and space and \(O(n^{1/3})\) query time.

On Single-Objective Sub-Graph-Based Mutation for Solving the Bi-Objective Minimum Spanning Tree Problem.

We contribute to the efficient approximation of the Pareto-set for the classical NP-hard multiobjective minimum spanning tree problem (moMST) adopting evolutionary computation. More precisely, by building upon preliminary work, we analyze the neighborhood structure of Pareto-optimal spanning trees and design several highly biased sub-graph-based mutation operators founded on the gained insights. In a nutshell, these operators replace (un)connected sub-trees of candidate solutions with locally optimal sub-trees. The latter (biased) step is realized by applying Kruskal's single-objective MST algorithm to a weighted sum scalarization of a sub-graph. We prove runtime complexity results for the introduced operators and investigate the desirable Pareto-beneficial property. This property states that mutants cannot be dominated by their parent. Moreover, we perform an extensive experimental benchmark study to showcase the operator's practical suitability. Our results confirm that the sub-graph-based operators beat baseline algorithms from the literature even with severely restricted computational budget in terms of function evaluations on four different classes of complete graphs with different shapes of the Pareto-front.

Sharp threshold for embedding balanced spanning trees in random geometric graphs

AbstractA rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph . In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.

Optimizing Collaborative Crowdsensing: A Graph Theoretical Approach to Team Recruitment and Fair Incentive Distribution.

Collaborative crowdsensing is a team collaboration model that harnesses the intelligence of a large network of participants, primarily applied in areas such as intelligent computing, federated learning, and blockchain. Unlike traditional crowdsensing, user recruitment in collaborative crowdsensing not only considers the individual capabilities of users but also emphasizes their collaborative abilities. In this context, this paper takes a unique approach by modeling user interactions as a graph, transforming the recruitment challenge into a graph theory problem. The methodology employs an enhanced Prim algorithm to identify optimal team members by finding the maximum spanning tree within the user interaction graph. After the recruitment, the collaborative crowdsensing explored in this paper presents a challenge of unfair incentives due to users engaging in free-riding behavior. To address these challenges, the paper introduces the MR-SVIM mechanism. Initially, the process begins with a Gaussian mixture model predicting the quality of users' tasks, combined with historical reputation values to calculate their direct reputation. Subsequently, to assess users' significance within the team, aggregation functions and the improved PageRank algorithm are employed for local and global influence evaluation, respectively. Indirect reputation is determined based on users' importance and similarity with interacting peers. Considering the comprehensive reputation value derived from the combined assessment of direct and indirect reputations, and integrating the collaborative capabilities among users, we have formulated a feature function for contribution. This function is applied within an enhanced Shapley value method to assess the relative contributions of each user, achieving a more equitable distribution of earnings. Finally, experiments conducted on real datasets validate the fairness of this mechanism.

The Integration of Dual Evaluation and Minimum Spanning Tree Clustering to Support Decision-Making in Territorial Spatial Planning

Spatial planning, a policy instrument for creating sustainable environments that meet the needs of the current and future generations, has been implemented extensively worldwide. However, it is difficult for urban planners to thoroughly determine the spatial value of a territory and make informed decisions regarding the efficient utilization of regional resources in the real world. This study proposes a novel methodological framework for spatial pattern optimization that can guide future land use by integrating Minimum Spanning Tree (MST) clustering with a comprehensive evaluation system (dual evaluation). Furthermore, the validity of this framework is demonstrated through a case study of territorial spatial planning in Deyang, China. The findings indicate that (1) the methodological framework presented in this study offers valuable guidance for the spatial arrangement of territorial resources, especially in practical projects; and (2) the combination of dual evaluation and MST clustering can facilitate automatic regionalization to identify spatial clusters exhibiting functional similarity in terms of land use. By focusing on methodological advancements, this study concludes that the integration of dual evaluation (DE) and MST clustering not only simplifies the identification of optimal land-use patterns but also promotes a more systematic and efficient approach to support spatial planning.

Minimum spanning tree analysis of EEG resting-state functional networks in schizophrenia

Schizophrenia is a serious and complex mental disease, known to be associated with various subtle structural and functional deviations in the brain. Recently, increased attention is given to the analysis of brain-wide, global mechanisms, strongly altering the communication of long-distance brain areas in schizophrenia. Data of 32 patients with schizophrenia and 28 matched healthy control subjects were analyzed. Two minutes long 64-channel EEG recordings were registered during resting, eyes closed condition. Average connectivity strength was estimated with Weighted Phase Lag Index (wPLI) in lower frequencies: delta and theta, and Amplitude Envelope Correlation with leakage correction (AEC-c) in higher frequencies: alpha, beta, lower gamma and higher gamma. To analyze functional network topology Minimum Spanning Tree (MST) algorithms were applied. Results show that patients have weaker functional connectivity in delta and alpha frequency bands. Concerning network differences, the result of lower diameter, higher leaf number, and also higher maximum degree and maximum betweenness centrality in patients suggest a star-like, and more random network topology in patients with schizophrenia. Our findings are in accordance with some previous findings based on resting-state EEG (and fMRI) data, suggesting that MST network structure in schizophrenia is biased towards a less optimal, more centralized organization.

Analysing Malaysian Technology Sector during COVID-19: A Minimum Spanning Tree Approach

This paper examines the structural changes of the companies listed in the technology sector in Bursa Malaysia during the COVID-19 pandemic. The closing price of each stock is used as the basis for calculating the cross-correlation between the companies' returns. Minimum spanning trees are employed to analyse the relationships and visualize the correlation changes for each period. Additionally, degree centrality is utilized to analyse the network's structure further. This study covers the period from 2020 to 2021 and involves 39 companies within the technology sector. The findings reveal significant fluctuations in the companies' correlations each year. Moreover, the most influential companies vary according to the period, demonstrating position changes. The fact that Inari Amertron (INAR), Revenue Group (REVE), and My EG Services (MYEG) consistently become leading players in different years shows how strong and influential they are in the technology sector. This suggests that these companies have the ability to shape market trends and possibly even control the direction of the industry. The results provide valuable insights for investors and policymakers, aiding in visualizing company relationships and facilitating strategic portfolio optimization.

An Amalgamated Kruskal’s Algorithm to ascertain the shortest route in the futuristic smart world

As the globe enters a new era, web applications will become indispensable to managing business. Businesses can easily grow, become simpler, and accomplish their objective much faster by employing web applications. Creating a web application in cloud computing allows for the more affordable leveraging of cloud-based services. This makes it easier to avoid setting up and maintaining several servers. To get around cloud computing’s built-in restrictions such as scalability, security, and bandwidth limitations, the future smart world of cloud computing will be coupled with LiFi connectivity. Beyond creating the web application, it is important to promote this web application among the network of users as quickly and effectively as possible. This manuscript proposes a strategy to address these challenges. There are two primary components to this MCDM technique. The first step is to model the problem as a graph and weigh the edges by employing the Hamacher aggregation operator. The second step involves using a fresh iteration of Kruskal’s technique in conjunction with this approach to discover a Minimum Spanning Tree as a resolution. This manuscript adds to the literature by solving real-world Minimum Spanning Tree problems by combining existing algorithms with MCDM techniques. This technique is demonstrated for marketing a web application(created via cloud service) in a future smart world using LiFi technology.

Characterization of Endothelial to Mesenchymal Transition in Air Pollution Mediated Atherosclerosis: Insights from Trajectory Inference and Functional Analyses in Single cell RNA Sequencing Data

Background: Air pollution is the world’s leading environmental risk factor for atherosclerotic cardiovascular disease (ASCVD). Regressive transcriptional reprogramming and transition of cells to primordial states has been identified as a driver of atherosclerosis. Endothelial cell to mesenchymal cell transition (EndoMT) whereby an endothelial cell transforms to a mesenchymal-like cellular phenotype is one such mechanism. However its relevance to air pollution exposure remains uncharacterized. Methods: Male ApoE−/− mice were inhalationally exposed to concentrated particulate matter <2.5μ component (PM2.5) or filtered air (FA, controls) using a Versatile Aerosol Concentration Enrichment System (VACES, 5–6 h/day, 5 d/week for 36 weeks, n = 7/group. PM2.5 concentrations were 70-80 μg/m3 or ~10X ambient levels. Following euthanasia, single-cell RNA-sequencing of aortic cells was performed using the 10-X platform (CA). For all subsequent analyses, aortic cell samples were integrated. Unsupervised clustering (k-NN) and low-dimensional space visualization (UMAP) were performed with a supervised graph method (Minimum Spanning Tree and Principal Curves) to infer and visualize trajectory analysis of novel cell lineages (branching and cell ordering in pseudotime/pseudospace). Functional analyses were carried out by statistical enrichment analysis using Over Representation Analysis (ORA) and Gene Set Enrichment Analysis (GSEA) against databases of Gene Ontologies (GO), Biological Pathways, putative regulatory motifs, proteins, or disease annotations to find over-representation of terms and functions. Pseudotime distribution assessed along exposure groups and cell-types lineages were statistically compared using GLM models. Differential Expression Analysis within lineages and between groups were tested by GLM and GAM models and FDR correction, followed by functional analyses. Results: Comparison of the pseudotime lineage with lineage root node at the endothelial cells revealed a similarity between observed endothelial (EC) to vascular smooth muscle (VSMC) cell transition with the differential transition noted with PM2.5 from FA. This overlap suggests a potential exaggeration of EC to VSMC transition in response to PM2.5 exposure. Differential gene expression profiling upon PM2.5-exposure in EC revealed significant changes in GO terms related to cell differentiation, cell migration, and regulation of cell population proliferation. Specific genes with variable patterns of expression (4481) were identified along the EC to VSMC lineage. KEGG pathway analysis along the lineage revealed enriched pathways including PI3K/Akt signaling pathway, MAPK signaling pathway, and Rap1 signaling pathway which are all implicated in the development and progression of atherosclerosis. Conclusion: Single-cell analysis reveals that chronic exposure to PM2.5 results in the exaggeration of EC to VSMC transition consistent with the EndoMT paradigm that may help explain the acceleration of ASCVD in response to air pollution exposure. Additional analysis of cell specific pathways may provide improved understanding of mechanisms progression of atherosclerosis in response to environmental exposures. NIH: 1R35ES031702-01, R01 HL155450, 3R35ES031702-02S1; Burroughs Wellcome Fund: 1060485. This is the full abstract presented at the American Physiology Summit 2024 meeting and is only available in HTML format. There are no additional versions or additional content available for this abstract. Physiology was not involved in the peer review process.

Some Algorithms of Graph Theory in Cryptology

The inventive use of concepts from Graph Theory plays a significant role in hiding the original Plain-text for resulting in a significantly safe data transfer. In this work, the tree traversal algorithms like Inorder, Preorder, Postorder, Kruskal’s algorithm for making minimal spanning tree and the modified graph labelling scheme of graceful labelling allowing repetition of exactly one vertex label for certain graphs, have been employed to create highly hidden Cipher-texts. Encryption and decryption algorithms for all these methods are being presented in this work.

Transmission characteristics in Tuberculosis by WGS: nationwide cross-sectional surveillance in China

ABSTRACT China, with the third largest share of global tuberculosis cases, faces a substantial challenge in its healthcare system as a result of the high burden of multidrug-resistant and rifampicin-resistant tuberculosis (MDR/RR-TB). This study employs a genomic epidemiological approach to assess recent tuberculosis transmissions between individuals, identifying potential risk factors and discerning the role of transmitted resistant isolates in the emergence of drug-resistant tuberculosis in China. We conducted a population-based retrospective study on 5052 Mycobacterium tuberculosis (MTB) isolates from 70 surveillance sites using whole genome sequencing (WGS). Minimum spanning tree analysis identified resistance mutations, while epidemiological data analysis pinpointed transmission risk factors. Of the 5052 isolates, 23% (1160) formed 452 genomic clusters, with 85.6% (387) of the transmissions occurring within the same counties. Individuals with younger age, larger family size, new cases, smear positive, and MDR/RR were at higher odds for recent transmission, while higher education (university and above) and occupation as a non-physical workers emerged as protective factors. At least 61.4% (251/409) of MDR/RR-TB were likely a result of recent transmission of MDR/RR isolates, with previous treatment (crude OR = 2.77), smear-positive (cOR = 2.07) and larger family population (cOR = 1.13) established as risk factors. Our findings highlight that local transmission remains the predominant form of TB transmission in China. Correspondingly, drug-resistant tuberculosis is primarily driven by the transmission of resistant tuberculosis isolates. Targeted interventions for high-risk populations to interrupt transmission within the country will likely provide an opportunity to reduce the prevalence of both tuberculosis and drug-resistant tuberculosis.

Fuzzy Tree t -Spanners of Ratio Labelled Butterfly Network

Background/Objectives: The reliability and optimality of a communication system depend on the network that has a minimum distance. The minimum distance is the shortest path between the vertices. Even in error correction codes, the minimum Hamming distance ensures data integrity. Hence to have efficient routing protocols, the network is preferred to be a spanning tree that connects all the nodes of the communication system. As tree t- spanner is the optimization technique in determining minimum spanning tree, finding tree t-spanners of a ratio labelled fuzzy graph for the Butterfly network BF(n) is notion of this study. Methods: This study has introduced a new labelling called Ratio Labelling (RL) for examining the fuzziness in a crisp graph. Using the RL method, the crisp graphs were examined for the admittance of fuzziness and were thereby found, to have a tree t-spanners of ratio labelled fuzzy graphs. Findings: The quality of a network can be measured using the stretch factor of the network. The issues of network survival or link failures in a communication networks can be rectified by approximating the best flow spanner. As the optimal connectivity between the nodes is ensured by interconnection networks, the fuzzy tree t- spanners of undirected fuzzy Butterfly Network BF(n) are examined by labelling the vertices and edges using ratio labelling. The Breadth First Search (BFS) algorithm enhances the process of minimizing the spanning tree for an unweighted graph. Since the edge weightage is unique for all the edges of BF(n) under RL, searching for an edge with minimum weight by Kruskal’s algorithm or Prim’s algorithm are less meaningful. Hence, the BFS algorithm is adopted in finding the spanning tree T of Ratio Labelled BF(n). The bounds for the stretch factor t of a fuzzy tree t- spanner of ratio labelled BF(n) were obtained. Novelty: The literature survey shows that various classes of graphs were examined for the admissibility of tree t-spanners. Finding a tree t-spanner for a fuzzy graph was not found in literature, as per our knowledge. Extending the problem of finding a tree t -spanner of a ratio labelled fuzzy graph is an advancement in the field of fuzzy graphs. Keywords: Fuzzy Graph, Fuzzy distance, Ratio Labelling, Tree t- spanner, Butterfly Network

Cluster formation due to repulsive spanning trees in attractively coupled networks.

Ensembles of coupled nonlinear oscillators are a popular paradigm and an ideal benchmark for analyzing complex collective behaviors. The onset of cluster synchronization is found to be at the core of various technological and biological processes. The current literature has investigated cluster synchronization by focusing mostly on the case of attractive coupling among the oscillators. However, the case of two coexisting competing interactions is of practical interest due to their relevance in diverse natural settings, including neuronal networks consisting of excitatory and inhibitory neurons, the coevolving social model with voters of opposite opinions, and ecological plant communities with both facilitation and competition, to name a few. In the present article, we investigate the impact of repulsive spanning trees on cluster formation within a connected network of attractively coupled limit-cycle oscillators. We successfully predict which nodes belong to each cluster and the emergent frustration of the connected networks independent of the particular local dynamics at the network nodes. We also determine local asymptotic stability of the cluster states using an approach based on the formulation of a master stability function. We additionally validate the emergence of solitary states and antisynchronization for some specific choices of spanning trees and networks.

Voronoi diagrams and tree structures in HRP-EE: Enhancing IoT network lifespan with WSNs

Within the domain of the Internet of Things (IoT), wireless sensor networks (WSNs) play a pivotal role, facilitating advancements in sectors such as smart urban infrastructures, intelligent healthcare systems, and residential automation. Despite their versatility, WSNs grapple with challenges like limited energy reserves and constrained computational capacities, making energy conservation a paramount concern for IoT deployments underpinned by WSNs. This study introduces the Hybrid Routing Protocol for Efficient Energy (HRP-EE) designed to enhance the operational efficiency of WSNs. The HRP-EE protocol unfolds over three meticulous stages: the selection of cluster heads, the formation of clusters, and the establishment of routing pathways. Initially, the protocol evaluates nodes based on their residual energy and their Euclidean distance to the central sink (or gateway) device to designate apt cluster heads. Subsequently, these chosen cluster heads are integrated into the Voronoi diagram as central nodes to orchestrate cluster architectures. In the terminal stage, an innovative hybrid algorithm is instituted. This algorithm amalgamates the principles of the Minimum Spanning Tree for structuring intra-cluster communication trees and Dijkstra’s algorithm to ascertain the most efficient paths for inter-cluster data transmission from cluster heads to the sink device. The primary objective of this protocol is the judicious utilization of the sensor nodes’ energy, thereby optimizing the overall network longevity. To assess the proficiency of HRP-EE, we executed a series of simulations using NS2. Comparative analyses reveal that HRP-EE outperforms existing protocols such as LEACH-VA, PEGCP, and TBC, delivering superior energy efficiency, extended network lifespan, and enhanced throughput across both homogeneous and heterogeneous network architectures.

The concept of Minimal Spanning Tree (MST) for Symmetric Encryption

As the digital communication has become a vital part of the human life nowadays, cybercrimes are ever expanding. One major tool to protect the message from hacking and tampering is concealing the original information. Modern cryptography not only concerns the encryption of the message but also focuses on the protection of the encrypted information from cyber-attacks.Apart from secret communication modern cryptography focuses on authentication and integrity of the message. The present paper explains a symmetric block cipher basing on graph structures. In this technique the message is encrypted twice. The first stage encryption using adjacency matrix of undirected graph; the second stage using exclusive OR (XOR) operation on the output of first stage. Encryption/Decryption Keys are distinct for distinct blocks and distinct rounds; derived from Minimum Span Tree (MST) technique of an undirected graph. Here multiple peers can communicate messages simultaneously. One of the entries selects some random ASCII characters whose number is same as the block length and uploads those characters in the cloud. All the members of the group establish block keys using key generation technique which is secret among them.

Low Functional network integrity in cognitively unimpaired and MCI subjects with depressive symptoms: results from a multi-center fMRI study

Evidence suggests that depressive symptomatology is a consequence of network dysfunction rather than lesion pathology. We studied whole-brain functional connectivity using a Minimum Spanning Tree as a graph-theoretical approach. Furthermore, we examined functional connectivity in the Default Mode Network, the Frontolimbic Network (FLN), the Salience Network, and the Cognitive Control Network. All 183 elderly subjects underwent a comprehensive neuropsychological evaluation and a 3 Tesla brain MRI scan. To assess the potential presence of depressive symptoms, the 13-item version of the Beck Depression Inventory (BDI) or the Geriatric Depression Scale (GDS) was utilized. Participants were assigned into three groups based on their cognitive status: amnestic mild cognitive impairment (MCI), non-amnestic MCI, and healthy controls. Regarding affective symptoms, subjects were categorized into depressed and non-depressed groups. An increased mean eccentricity and network diameter were found in patients with depressive symptoms relative to non-depressed ones, and both measures showed correlations with depressive symptom severity. In patients with depressive symptoms, a functional hypoconnectivity was detected between the Anterior Cingulate Cortex (ACC) and the right amygdala in the FLN, which impairment correlated with depressive symptom severity. While no structural difference was found in subjects with depressive symptoms, the volume of the hippocampus and the thickness of the precuneus and the entorhinal cortex were decreased in subjects with MCI, especially in amnestic MCI. The increase in eccentricity and diameter indicates a more path-like functional network configuration that may lead to an impaired functional integration in depression, a possible cause of depressive symptomatology in the elderly.

Efficient Maintenance of Minimum Spanning Trees in Dynamic Weighted Undirected Graphs

This paper presents an algorithm for effectively maintaining the minimum spanning tree in dynamic weighted undirected graphs. The algorithm efficiently updates the minimum spanning tree when the underlying graph structure changes. By identifying the portion of the original tree that can be preserved in the updated tree, our algorithm avoids recalculating the minimum spanning tree from scratch. We provide proof of correctness for the proposed algorithm and analyze its time complexity. In general scenarios, the time complexity of our algorithm is comparable to that of Kruskal’s algorithm. However, the experimental results demonstrate that our algorithm outperforms the approach of recomputing the minimum spanning tree by using Kruskal’s algorithm, especially in medium- and large-scale dynamic graphs where the graph undergoes iterative changes.

A reliable method for data aggregation on the industrial internet of things using a hybrid optimization algorithm and density correlation degree

The Internet of Things (IoT) is a new information technology sector in which each device may receive and distribute data across a network. Industrial IoT (IIoT) and related areas, such as Industrial Wireless Networks (IWNs), big data, and cloud computing, have made significant strides recently. Using IIoT requires a reliable and effective data collection system, such as a spanning tree. Many previous spanning tree algorithms ignore failure and mobility. In such cases, the spanning tree is broken, making data delivery to the base station difficult. This study proposes an algorithm to construct an optimal spanning tree by combining an artificial bee colony, genetic operators, and density correlation degree to make suitable trees. The trees’ fitness is measured using hop count distances of the devices from the base station, residual energy of the devices, and their mobility probabilities in this technique. The simulation outcomes highlight the enhanced data collection reliability achieved by the suggested algorithm when compared to established methods like the Reliable Spanning Tree (RST) construction algorithm in IIoT and the Hop Count Distance (HCD) based construction algorithm. This proposed algorithm shows improved reliability across diverse node numbers, considering key parameters including reliability, energy consumption, displacement probability, and distance.

Negative emotions mediate the association between the topology of the complex brain network and smartphone use disorder: A resting-state EEG study.

Increasing research has examined the factors related to smartphone use disorder. However, limited research has explored its neural basis. We aimed to examine the relationship between the topology of the resting-state electroencephalography (rs-EEG) brain network and smartphone use disorder using minimum spanning tree analysis. Furthermore, we examined how negative emotions mediate this relationship. This study included 113 young, healthy adults (mean age = 20.87 years, 46.9% males). The results showed that the alpha- and delta-band kappas and delta-band leaf fraction were positively correlated with smartphone use disorder. In contrast, the alpha-band diameter was negatively correlated with smartphone use disorder. Negative emotions fully mediated the relationship between alpha-band kappa and alpha-band diameter and smartphone use disorder. Furthermore, negative emotions partially mediated the relationship between delta-band kappa and smartphone use disorder. The findings suggest that excessive scale-free alpha- and delta-band brain networks contribute to the emergence of smartphone use disorder. In addition, the findings also demonstrate that negative emotions and smartphone use disorder share the same neural basis. Negative emotions play a mediating role in the association between topological deviations and smartphone use disorder. To the best of our knowledge, this is the first study to examine the neural basis of smartphone use disorder from the perspective of the topology of the rs-EEG brain network. Therefore, neuromodulation may be a potential intervention for smartphone use disorder.

The Impact of the Measure Used to Calculate the Distance between Exchange Rate Time Series on the Topological Structure of the Currency Network.

Structural properties of the currency market were examined with the use of topological networks. Relationships between currencies were analyzed by constructing minimal spanning trees (MSTs). The dissimilarities between time series of currency returns were measured in various ways: by applying Euclidean distance, Pearson's linear correlation coefficient, Spearman's rank correlation coefficient, Kendall's coefficient, partial correlation, dynamic time warping measure, and Kullback-Leibler relative entropy. For the constructed MSTs, their topological characteristics were analyzed and conclusions were drawn regarding the influence of the dissimilarity measure used. It turned out that the strength of most types of correlations was highly dependent on the choice of the numeraire currency, while partial correlations were invariant in this respect. It can be stated that a network built on the basis of partial correlations provides a more adequate illustration of pairwise relationships in the foreign exchange market. The data for quotations of 37 of the most important world currencies and four precious metals in the period from 1 January 2019 to 31 December 2022 were used. The outbreak of the COVID-19 pandemic in 2020 and Russia's invasion of Ukraine in 2022 triggered changes in the topology of the currency network. As a result of these crises, the average distances between tree nodes decreased and the centralization of graphs increased. Our results confirm that currencies are often pegged to other currencies due to countries' geographic locations and economic ties. The detected structures can be useful in descriptions of the currency market, can help in constructing a stable portfolio of the foreign exchange rates, and can be a valuable tool in searching for economic factors influencing specific groups of countries.