Offline algorithms for dynamic minimum spanning tree problems

In view of today's unprecedented diverse and discrete mass text data processing, this paper presents a distributed MST (minimum spanning tree) algorithm based on MapReduce programming model. And with this MST algorithm, a distributed MST text clustering algorithm is designed and implemented. In this paper, this clustering algorithm is analyzed in three aspects: text feature vector extraction, graph construction and MST construction. And some data was used to experimentally compare this algorithm with ordinary MST clustering algorithm and MapReduce-based K-means clustering algorithm.

The minimum spanning tree problem with non-terminal set

This study’s objective is to assess the performance of routing processing between Shortest Paths Tree (SPT) and Minimum Spanning Tree (MST). In this paper we propose and create software to calculate the efficiency path of routing processing in the network. Extensive simulations are being carried out with two different algorithms specific for Dijkistra’s algorithm and Kruskal’s algorithm. The network loads in the performance plots are characterized by two important parameters (a number of vertices) - 7 nodes and 10 nodes, more vertices could show good performance in the simulation results and showed that the performance of Kruskal’s algorithm is better than Dijkistra’s algorithm when the network loads are increased from 7 nodes to 10 nodes. In this simulation, Because of the total cost of distance matrix for Kruskal is less than Dijkistra when the number of routers or nodes is increased from 7 nodes to 10 nodes. This software namely Network Aids Design (NAD), is useful for calculation about the performance of routing processing in the network.

The concept of the minimum spanning tree (MST) plays an important role in topological network design, because it models a cheapest connected network. In a tree, however, the failure of a vertex can disconnect the network. In order to tolerate such a failure, we generalize the MST to the concept of a cheapest biconnected network. For a set of points in the Euclidean plane, we show that it is NP-hard to find a cheapest biconnected spanning graph, where edge costs are the Euclidean distances of the respective points. We propose a different type of subgraph, based on forbidding (due to failure) the use of a vertex. A minimum spanning multi-tree is a spanning graph that contains for each possible forbidden vertex a spanning tree that is minimum among the spanning trees that do not use the forbidden vertex. We propose a worst-case time optimal algorithm for computing a minimum spanning multi-tree for a planar Euclidean point set. A minimum spanning multi-tree is cheap, even though it embeds a linear number of MSTs: Its cost is more than the MST cost only by a constant factor. Furthermore, we propose a linear time algorithm for computing a cheap vertex failure tolerant graph, given the Delaunay triangulation. This graph bounds the cost of the minimum spanning multi-tree from above.

The Minimum Spanning Tree problem aims to create a subset of a graph where all the vertices are connected with the minimum edge weights and with no cycle. In this field, an innovative method entitled Dhouib-Matrix-MSTP (DM-MSTP) is designed in this research work with a time complexity independently of the number of edges O(n*log(n)) where n is the number of vertices. DM-MSTP is a constructive algorithm based on a matrix navigation with two new lists (Min-Columns and MST-Path) in order to organize the steering flow. DM-MSTP is composed of four simple steps where the first and the fourth steps are repeated only once, whereas the second and third are reiterated (n-1). For more clarification, a step-by-step application of the proposed method is presented in details. Besides, the performance of DM-MSTP is proved through different examples from the literature including a graph with negative weighted edges and complete graphs from TSP-LIB. Moreover, with a simple modification (Min by Max) the DM-MSTP is tested on the Maximum (Largest) Spanning Tree Problem. Also, DM-MSTP is applied on eight case studies and compared to six methods developed in the literature. All these experimental results in the above different environments show that DM-MSTP can rapidly plan the shortest spanning tree with a stable performance and convivial representation of the optimal tree. Hence, DM-MSTP is developed under Python programming language using Matplotlib and Numpy standard libraries.

The minimum spanning tree (MST), a graph constructed from a distribution of points, draws lines between pairs of points so that all points are linked in a single skeletal structure that contains no loops and has minimal total edge length. The MST has been used in a broad range of scientific fields such as particle physics (to distinguish classes of events in collider collisions), in astronomy (to detect mass segregation in star clusters) and cosmology (to search for filaments in the cosmic web). Its success in these fields has been driven by its sensitivity to the spatial distribution of points and the patterns within. MiSTree, a public Python package, allows a user to construct the MST in a variety of coordinates systems, including Celestial coordinates used in astronomy. The package enables the MST to be constructed quickly by initially using a k-nearest neighbour graph (kNN, rather than a matrix of pairwise distances) which is then fed to Kruskal's algorithm to construct the MST. MiSTree enables a user to measure the statistics of the MST and provides classes for binning the MST statistics (into histograms) and plotting the distributions. Applying the MST will enable the inclusion of high-order statistics information from the cosmic web which can provide additional information to improve cosmological parameter constraints. This information has not been fully exploited due to the computational cost of calculating N-point statistics. MiSTree was designed to be used in cosmology but could be used in any field which requires extracting non-Gaussian information from point distributions. The source code for MiSTree is available on GitHub at https://github.com/knaidoo29/mistree

A super-stabilizing [formula omitted]-approximation algorithm for dynamic Steiner trees

N points are given in the Euclidean plane, and the minimum spanning tree problem seeks for a minimum spanning tree interconnecting the n points so that there is only one path between any two points. One of the classic and frequently-used algorithms for minimum spanning tree problem is Prim's algorithm, but it consumes large time and space complexity for the plane minimum spanning tree problem is of O(n2) numbers of edges. Luckily, it was proved that the plane minimum spanning tree is a sub-graph of Delaunay triangulation for the given points in the plane, and the number of edges in the triangulation is O(n). This motivates us to efficiently compute the Delaunay triangulation of the given points and then find the minimum spanning tree in the triangulation. This paper presents an algorithm based on the divide and conquer for Delaunay triangulation together with the Prim's algorithm to produce an O(nlogn) algorithm for minimum spanning tree problem in the plane, implements the visual graphic interface with various selected algorithms for plane minimum spanning tree and compares their running time.

The addition of new houses in the Kota Mas Cimahi housing complex has triggered an increase in the need for electricity distribution installations. To optimize the electricity distribution network, in this case minimizing the installation of electrical cables, an application is needed that can optimize the installation of electrical cables, namely the Kruskal algorithm which is an algorithm for finding the minimum spanning tree value on a weighted graph. In this study, two methods will be used, namely the manual method and the POM-QM for Windows software to help solve the minimum spanning tree problem. This study's goal is to identify the Kota Mas Cimahi housing complex's energy distribution network model's graph structure and determine a minimum spanning tree solution with POM-QM for Windows software and the Kruskal technique done by hand. In order to gather data for this study, PLN was the primary and secondary source Cimahi UP3. The distribution data for the electricity cable network is arranged in the form of a network image. By representing the electricity distribution network in the form of a graph, the minimum spanning tree is then obtained, and the solution to the MST (Minimum Spanning Tree) problem uses the Kruskal algorithm and TORA software. The results of the study showed that the minimum spanning tree cable length which was originally 1424.61 meters, after using the Kruskal algorithm manually became 1258.05 meters and also by using the POM-QM for Windows software obtained a minimum result of 1258.05 meters, so it can be concluded that there is a cable length efficiency of 11.69%.

In this paper, we consider an interesting variant of the inverse minimum traveling salesman problem. Given an in-stance ( G , w ) of the minimum traveling salesman problem defined on a metric space, we fix a specified Hamiltonian cycle HC 0 . The task is then to adjust the edge cost vector w to w ' so that the new cost vector w ' satisfies the triangle inequality condition and HC 0 can be returned by the minimum spanning tree algorithm in the TSP-instance defined with w '. The objective is to minimize the total deviation between the original and the new cost vectors with respect to the L 1 -norm. We call this problem the inverse metric traveling salesman problem against the minimum spanning tree algorithm and show that it is closely related to the inverse metric spanning tree problem. Keywords: Combinatorial Optimization, Inverse Optimization, TSP, Minimum Spanning Tree Algorithm * Corresponding Author, E-mail: yerimchung@yonsei.ac.kr 1. INTRODUCTION In (Chung, 2010; Chung and Demange, 2012), some variant of the inverse optimization problem is defined with respect to a specific algorithm. For a combinatorial optimization problem, we are given an instance

This study addresses constructing Minimum Spanning Trees (MST) in stochastic weighted distribution networks, where edge costs have inherent uncertainties with known means and variances. Traditional deterministic methods often fail, and existing stochastic approaches are frequently unstable or computationally complex under high uncertainty. A novel variance-based deterministic transformation algorithm is proposed. Its core feature is transforming stochastic edge costs into robust deterministic equivalents by computing an aggregate variance term from the largest (n – 1) edge variances, enabling MST construction via classical algorithms. This method fundamentally enhances stability and ensures feasibility, particularly in high-variance scenarios, improving upon traditional confidence interval-based techniques. The algorithm’s efficacy was rigorously validated. Its performance was compared against a probabilistic Qij-based method under moderate variance, demonstrating consistent and accurate MSTs. It was then applied to a complex 21-edge distribution network with high variance parameters. Results confirm the algorithm’s broad applicability, precision, and capability to construct reliable spanning trees under both moderate and substantial uncertainty. The algorithm demonstrates significant computational efficiency (O(r log r)), ensuring practicality and scalability across varying uncertainty levels. Unlike iterative or constraint-heavy models, this algorithm simplifies optimization while preserving uncertainty representation. This makes it well-suited for large-scale networks and real-world systems where cost variability is critical. Future research includes expanding this approach to multi-objective optimization or dynamic networks

An effective genetic algorithm approach to the quadratic minimum spanning tree problem

The Minimum Spanning Tree (MST) problem is one of the most popular and important problems in the research area of distributed computing and networks. Contrary to the theoretical models where we usually have a global knowledge of all nodes and the corresponding distances for MST construction, in a realistic network (e.g., Internet) a node always has to rely on local knowledge only, that is it neither knows all other nodes nor exact distances between these nodes. In this paper we propose an approach for MST approximation based on local knowledge of a small subset of existing nodes by using the CARMA metric as a distance substitute. According to the evaluation results, our approach achieves a good MST approximation with respect to a communication cost and avoids extraneous communication needed for latency measurements.

Based on a recently proposed q-dependent detrended cross-correlation coefficient, ρ_{q} [J. Kwapień, P. Oświęcimka, and S. Drożdż, Phys. Rev. E 92, 052815 (2015)PLEEE81539-375510.1103/PhysRevE.92.052815], we generalize the concept of the minimum spanning tree (MST) by introducing a family of q-dependent minimum spanning trees (qMSTs) that are selective to cross-correlations between different fluctuation amplitudes and different time scales of multivariate data. They inherit this ability directly from the coefficients ρ_{q}, which are processed here to construct a distance matrix being the input to the MST-constructing Kruskal's algorithm. The conventional MST with detrending corresponds in this context to q=2. In order to illustrate their performance, we apply the qMSTs to sample empirical data from the American stock market and discuss the results. We show that the qMST graphs can complement ρ_{q} in disentangling "hidden" correlations that cannot be observed in the MST graphs based on ρ_{DCCA}, and therefore, they can be useful in many areas where the multivariate cross-correlations are of interest. As an example, we apply this method to empirical data from the stock market and show that by constructing the qMSTs for a spectrum of q values we obtain more information about the correlation structure of the data than by using q=2 only. More specifically, we show that two sets of signals that differ from each other statistically can give comparable trees for q=2, while only by using the trees for q≠2 do we become able to distinguish between these sets. We also show that a family of qMSTs for a range of q expresses the diversity of correlations in a manner resembling the multifractal analysis, where one computes a spectrum of the generalized fractal dimensions, the generalized Hurst exponents, or the multifractal singularity spectra: the more diverse the correlations are, the more variable the tree topology is for different q's. As regards the correlation structure of the stock market, our analysis exhibits that the stocks belonging to the same or similar industrial sectors are correlated via the fluctuations of moderate amplitudes, while the largest fluctuations often happen to synchronize in those stocks that do not necessarily belong to the same industry.

Trees, including minimum spanning trees (MSTs), are commonly used in phylogenetic studies. But, for the research community, it may be unclear that the presented tree is just a hypothesis, chosen from among many possible alternatives. In this scenario, it is important to quantify our confidence in both the trees and the branches/edges included in such trees. In this paper, we address this problem for MSTs by introducing a new edge betweenness metric for undirected and weighted graphs. This spanning edge betweenness metric is defined as the fraction of equivalent MSTs where a given edge is present. The metric provides a per edge statistic that is similar to that of the bootstrap approach frequently used in phylogenetics to support the grouping of taxa. We provide methods for the exact computation of this metric based on the well known Kirchhoff’s matrix tree theorem. Moreover, we implement and make available a module for the PHYLOViZ software and evaluate the proposed metric concerning both effectiveness and computational performance. Analysis of trees generated using multilocus sequence typing data (MLST) and the goeBURST algorithm revealed that the space of possible MSTs in real data sets is extremely large. Selection of the edge to be represented using bootstrap could lead to unreliable results since alternative edges are present in the same fraction of equivalent MSTs. The choice of the MST to be presented, results from criteria implemented in the algorithm that must be based in biologically plausible models.