The Implementation of Kruskal’s Algorithm for Minimum Spanning Tree in a Graph

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.

This paper proposes a parallel shortest path-searching algorithm and implements it on a newly structured parallel reconfigurable processor, DAPDNA-2 (IPFlex Inc). Routing determines the shortest paths from the source to the ultimate destination through intermediate nodes. In open shortest path first (OSPF), Dijkstra's shortest path algorithm, which is the conventional one, finds the shortest paths from the source on a program counter-based processor. The calculation time for Dijkstra's algorithm is O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) when the number of nodes is N. When the network scale is large, calculation time required by Dijkstra's algorithm increases rapidly. It's very difficult to compute Dijkstra's algorithm in parallel because of the need for previous calculation results, so Dijkstra's algorithm is unsuitable for parallel processors. Our proposed scheme finds the shortest paths using a simultaneous multi-path search method. In contrast with Dijkstra's algorithm, several nodes can be determined at one time. Moreover, we partition the network into different groups (network groups) and find the all-node pair's shortest path in each group using a pipeline operation. Networks can be abstracted, and the shortest paths in very large networks can be found easily. The proposed scheme can decrease calculation time from O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) to O(N) using a pipeline operation on DAPDNA-2. Our simulations show that the proposed algorithm uses 99.6% less calculation time than Dijkstra's algorithm. The proposed algorithm can be applied to the very large Internet network designs of the future.

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

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

Texture synthesis employs neighbourhood matching to generate appropriate new content. Terrain synthesis has the added constraint that new content must be geographically plausible. The profile recognition and polygon breaking algorithm (PPA) [Chang et al. 1998] provides a robust mechanism for characterizing terrain as systems of valley and ridge lines in digital elevation maps. We exploit this to create a terrain characterization metric that is robust, efficient to compute and is sensitive to terrain properties.

In wireless ad hoc and sensor networks, energy is a scarce resource and a considerable amount of energy is dissipated due to interference. Therefore, interference is one of the major challenges in wireless ad hoc and sensor networks. It alters or disrupts a message as it is being transmitted along a channel between source and destination. Since the messages are disrupted when the interference occurs, they have to be detected and the interfered messages have to be retransmitted. In this paper, we propose central and distributed heuristic algorithms for reducing average interference in a receiver-centric interference model. In the literature, the minimum spanning tree (MST) algorithm is generally used through the interference coverage graph directly or indirectly in order to generate minimum average interference topology. Our algorithm, dynamic average interference (DAI), however, generates lower average interference as well as more sparse topology than MST. We realized that if the transmission ranges of nodes are taken into consideration at each stage of the topology control algorithm, the interference of links are changed dynamically. This interference changing enables up to 22% more energy saving than the MST algorithm. Thus, DAI provides energy saving by reducing the interference as far as possible in generated topology.

We study the problem of privately releasing an approximate minimum spanning tree (MST). Given a graph G = ( V , E , W ) where V is a set of n vertices, E is a set of m undirected edges, and W ∈ ℝ |E| is an edge-weight vector, our goal is to publish an approximate MST under edge-weight differential privacy, as introduced by Sealfon in PODS 2016, where V and E are considered public and the weight vector is private. Our neighboring relation is 𝓁 ∞ -distance on weights: for a sensitivity parameter Δ ∞ , graphs G = ( V , E , W ) and G' = ( V , E , W ') are neighboring if || W - W '|| ∞ ≤ Δ ∞ ). Existing private MST algorithms face a trade-off, sacrificing either computational efficiency or accuracy. We show that it is possible to get the best of both worlds: With a suitable random perturbation of the input that does not suffice to make the weight vector private, the result of any non-private MST algorithm will be private and achieves a state-of-the-art error guarantee. Furthermore, by establishing a connection to Private Top-k Selection [Steinke and Ullman, FOCS '17], we give the first privacy-utility trade-off lower bound for MST under approximate differential privacy, demonstrating that the error magnitude, ~O(n 3/2 ), is optimal up to logarithmic factors. That is, our approach matches the time complexity of any non-private MST algorithm and at the same time achieves optimal error. We complement our theoretical treatment with experiments that confirm the practicality of our approach.

In a complete graphwith independent uniform(or exponential) edge weights, letbe the minimum‐weight spanning tree (MST), andthe MST after deleting the edges of all previous trees. We show that each tree's weightconverges in probability to a constant, with, and we conjecture that. The problem is distinct from Frieze and Johansson's minimum combined weightofkedge‐disjoint spanning trees; indeed,. With an edge of weightw“arriving” at time, Kruskal's algorithm defines forests, initially empty and eventually equal to, each edge added to the first possible. Using tools of inhomogeneous random graphs we obtain structural results including that the fraction of vertices in the largest component ofconverges to some. We conjecture that the functionstend to time translations of a single function.

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

Wireless Sensor Network (WSN) has many applications in day to day life. One of the applications of WSN is target tracking that uses sensors to track and localize the targets. This makes sensor node deployment in a network necessary and highly important. But the main challenge faced by sensor networks is the power consumption by sensor nodes. Hence an energy optimization technique must be in place that ultimately reduces the actual power consumption of the sensor nodes. Here, the shortest path algorithms are used in routing and it helps in reducing the energy consumption. The shortest path algorithms used in wireless sensor network are Dijkstra's algorithm and Floyd-Warshall's algorithm. They are also used for calculating throughput, packet delivery ratio and energy consumption for given number of nodes. The Floyd-Warshall's algorithm is not scalable and efficient for routing in WSN. Hence, we use modified Floyd-Warshall's algorithm that obtains better throughput, packet delivery ratio and minimized energy consumption. The new Modified Floyd-Warshall's algorithm calculates the shortest path from node to node by adding an acknowledgment mechanism. We have done a comparative analysis of Dijkstra's algorithm, Floyd-Warshall's algorithm, and modified Floyd-Warshall's algorithm for parameters like packet deliver ratio, throughput, and energy optimization for multiple nodes. From simulation results it is observed that the modified Floyd-Warshall's algorithm works better than Dijkstra's and Floyd Warshall's algorithm in terms of Packet Delivery ratio, throughput and energy consumption.

Social networks are platforms where users share their experiences and interact with each other. Unlike traditional web pages, users are not only passive consumers but also content producers and spreaders. Thus, it is necessary to benefit from the interactions between users in order to spread the information in the shortest and most effective way in social media networks. This research explores the idea of prioritization of social network connections by representing a social media network as undirected intuitionistic fuzzy weighted graph. The link weights are calculated by the intuitionistic fuzzy MCDM with the Hamacher aggregation operator. Then, the minimum spanning tree (or the highest weighted link for the most effective spreading) between all users is determined by the Kruskal algorithm. The proposed methodology contributes the literature by integrating Kruskal's algorithm with an intuitionistic fuzzy MCDM solution to solve MST problems in real life scenarios. To illustrate this hybrid approach, it is applied for spreading content on social media problem.

With technology advancement, Vehicular Ad Hoc Networks (VANETs) have been requested to provide one-to-many group communications (i.e. multicasting) for business and entertainment applications such as video conferences and file sharings. For these group communication applications, the existing broadcasting-based and unicasting-based approaches designed for safety applications however cause high packet generations and redundant data transmissions, respectively. Although the well-known Shortest Path Tree (SPT) and Minimum Spanning Tree (MST) algorithms for multicasting have been proposed, they achieve only a specific goal such as the reduction of delay or tree cost. Thus, for reducing end-to-end delay, delay variation, and transmission number requested for delay-sensitive and cost-efficient group communications and supporting high mobility of vehicles, this paper proposes a multicast protocol which uses the Farthest destination Selection and Shortest path Connection (FSSC) strategy to form a multicast tree. FSSC relies on the road map information and the location information of vehicles and intersections. Simulation results are provided to verify the effectiveness of FSSC.

Context. Young stellar clusters harbour complex spatial structures emerging from the star formation process. Identifying stellar over-densities is a key step in better constraining how these structures are formed. The high accuracy of distances derived from Gaia DR2 parallaxes still do not allow us to locate individual stars within clusters of ≈1 pc in size with certainty. Aims. In this work, we explore how such uncertainty on distance estimates can lead to the misidentification of membership of sub-clusters selected by the minimum spanning tree (MST) algorithm. Our goal is to assess how this impacts their estimated properties. Methods. Using N-body simulations, we build gravity-driven fragmentation models that self-consistently reproduce the early stellar configurations of a star forming region. Stellar groups are then identified both in two and three dimensions by the MST algorithm, representing respectively an inaccurate and an ideal identification. We compare the properties derived for these resulting groups in order to assess the systematic bias introduced by projection and incompleteness. Results. We show that in such fragmented configurations, the dynamical mass of groups identified in projection is systematically underestimated compared to those of groups identified in 3D. This systematic error is statistically of 50% for more than half of the groups and reaches 100% in a quarter of them. Adding incompleteness further increases this bias. Conclusions. These results challenge our ability to accurately identify sub-clusters in most nearby star forming regions where distance estimate uncertainties are comparable to the size of the region. New clump-finding methods need to tackle this issue in order to better define the dynamical state of these substructures.

This paper presents new parallel algorithms for generating Euclidean minimum spanning trees and spatial clustering hierarchies (known as HDBSCAN*). Our approach is based on generating a well-separated pair decomposition followed by using Kruskal's minimum spanning tree algorithm and bichromatic closest pair computations. We introduce a new notion of well-separation to reduce the work and space of our algorithm for HDBSCAN*. We also give a new parallel divide-and-conquer algorithm for computing the dendrogram and reachability plots, which are used in visualizing clusters of different scale that arise for both EMST and HDBSCAN*. We show that our algorithms are theoretically efficient: they have work (number of operations) matching their sequential counterparts, and polylogarithmic depth (parallel time). We implement our algorithms and propose a memory optimization that requires only a subset of well-separated pairs to be computed and materialized, leading to savings in both space (up to 10x) and time (up to 8x). Our experiments on large real-world and synthetic data sets using a 48-core machine show that our fastest algorithms outperform the best serial algorithms for the problems by 11.13--55.89x, and existing parallel algorithms by at least an order of magnitude.