scholarly journals A Note on Random Minimum Length Spanning Trees

10.37236/1519 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Alan Frieze ◽  
Miklós Ruszinkó ◽  
Lubos Thoma
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Minimum Length ◽  
Edge Connectivity ◽  
Vertex Graph ◽  
Expected Length ◽  
Independent Edge

Consider a connected $r$-regular $n$-vertex graph $G$ with random independent edge lengths, each uniformly distributed on $[0,1]$. Let $mst(G)$ be the expected length of a minimum spanning tree. We show in this paper that if $G$ is sufficiently highly edge connected then the expected length of a minimum spanning tree is $\sim {n\over r}\zeta(3)$. If we omit the edge connectivity condition, then it is at most $\sim {n\over r}(\zeta(3)+1)$.

scholarly journals FUZZY SHORTEST ROUTE ALGORITHM FOR TELEPHONE LINE CONNECTION USING THE LC-MST ALGORITHM

2015 ◽  
Vol 2 (2) ◽  
pp. 37-39
Author(s):  
Vijayalakshmi D ◽  
Kalaivani R
Keyword(s):  
Transmission Lines ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Minimum Cost ◽  
Distribution Networks ◽  
Minimum Length ◽  
Fuzzy Graph ◽  
Telephone Line ◽  
Water Pipelines ◽  
Communication Links

In computer science, there are many algorithms that finds a minimum spanning tree for a connected weighted undirected fuzzy graph. The minimum length (or cost) spanning tree problem is one of the nicest and simplest problems in network optimization, and it has a wide variety of applications. The problem is tofind a minimum cost (or length) spanning tree in G. Applications include the design of various types of distribution networks in which the nodes represent cities, centers etc.; and edges represent communication links (fiber glass phone lines, data transmission lines, cable TV lines, etc.), high voltage power transmissionlines, natural gas or crude oil pipelines, water pipelines, highways, etc. The objective is to design a network that connects all the nodes using the minimum length of cable or pipe or other resource in this paper we find the solution to the problem is to minimize the amount of new telephone line connection using matrixalgorithm with fuzzy graph.

scholarly journals A Minimum-Spanning-Tree-Inspired Algorithm for Channel Assignment in 802.11 Networks

2016 ◽  
Vol 62 (4) ◽  
pp. 379-388 ◽  
Author(s):  
Iwona Dolińska ◽  
Mariusz Jakubowski ◽  
Antoni Masiukiewicz ◽  
Grzegorz Rządkowski ◽  
Kamil Piórczyński
Keyword(s):  
Greedy Algorithm ◽  
Channel Assignment ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Minimum Spanning Trees ◽  
Undirected Graphs ◽  
Interference Level ◽  
802.11 Networks ◽  
Crowded Environment

Abstract Channel assignment in 2.4 GHz band of 802.11 standard is still important issue as a lot of 2.4 GHz devices are in use. This band offers only three non-overlapping channels, so in crowded environment users can suffer from high interference level. In this paper, a greedy algorithm inspired by the Prim’s algorithm for finding minimum spanning trees (MSTs) in undirected graphs is considered for channel assignment in this type of networks. The proposed solution tested for example network distributions achieves results close to the exhaustive approach and is, in many cases, several orders of magnitude faster.

ON THE STEINER RATIO IN $\mathcal{R}_{n}$

2011 ◽  
Vol 03 (04) ◽  
pp. 473-489
Author(s):  
HAI DU ◽  
WEILI WU ◽  
ZAIXIN LU ◽  
YINFENG XU
Keyword(s):  
Euclidean Space ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Dimensional Euclidean Space ◽  
Minimum Length ◽  
Steiner Ratio ◽  
Line Segments ◽  
Finite Set ◽  
Steiner Minimum Tree ◽  
Set Of Points

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let [Formula: see text] denote the n-dimensional Euclidean space. The Steiner ratio in [Formula: see text] is defined to be [Formula: see text], where Ls(P) and Lm(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for [Formula: see text] in the literature is 0.615. In this paper, we show that [Formula: see text] for any n ≥ 2.

scholarly journals Meminimalisir Biaya Pembuatan Saluran Air PDAM Di Wilayah KLU Menggunakan Algoritma Kruskal

2019 ◽  
Vol 1 (1) ◽  
pp. 27
Author(s):  
Devi Lastri ◽  
Masriani ◽  
Nadia W ◽  
Parizal Hidayatullah ◽  
Wahyu Ulfayandhie Misuki ◽  
...  
Keyword(s):  
Spanning Tree ◽  
Water Distribution ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Clean Water ◽  
Minimum Spanning Trees ◽  
Tree Algorithm ◽  
Shortest Route ◽  
Minimize Costs

The kruskal algorithm is an algorithm to search for minimum spanning trees directly based on the general MST (Minimum Spanning Tree) algorithm. In the kruskal algorithm, the sides in the graph are sorted first based on their weight from small to large. The kruskal algorithm in the search for a minimum spanning tree can be applied to the distribution of clean water of PDAM in North Lombok district. This problem is intended to get the shortest route for PDAM water distribution in North Lombok district in order to minimize costs.

scholarly journals Development of Gis Tool for the Solution of Minimum Spanning Tree Problem using Prim's Algorithm

2014 ◽  
Vol XL-8 ◽  
pp. 1105-1114 ◽  
Author(s):  
S. Dutta ◽  
D. Patra ◽  
H. Shankar ◽  
P. Alok Verma
Keyword(s):  
Spanning Tree ◽  
Road Network ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Transportation Network ◽  
Travelling Salesman Problem ◽  
Optimal Path ◽  
Road Transportation ◽  
Gis Technology ◽  
Weighted Network

minimum spanning tree (MST) of a connected, undirected and weighted network is a tree of that network consisting of all its nodes and the sum of weights of all its edges is minimum among all such possible spanning trees of the same network. In this study, we have developed a new GIS tool using most commonly known rudimentary algorithm called Prim’s algorithm to construct the minimum spanning tree of a connected, undirected and weighted road network. This algorithm is based on the weight (adjacency) matrix of a weighted network and helps to solve complex network MST problem easily, efficiently and effectively. The selection of the appropriate algorithm is very essential otherwise it will be very hard to get an optimal result. In case of Road Transportation Network, it is very essential to find the optimal results by considering all the necessary points based on cost factor (time or distance). This paper is based on solving the Minimum Spanning Tree (MST) problem of a road network by finding it’s minimum span by considering all the important network junction point. GIS technology is usually used to solve the network related problems like the optimal path problem, travelling salesman problem, vehicle routing problems, location-allocation problems etc. Therefore, in this study we have developed a customized GIS tool using Python script in ArcGIS software for the solution of MST problem for a Road Transportation Network of Dehradun city by considering distance and time as the impedance (cost) factors. It has a number of advantages like the users do not need a greater knowledge of the subject as the tool is user-friendly and that allows to access information varied and adapted the needs of the users. This GIS tool for MST can be applied for a nationwide plan called Prime Minister Gram Sadak Yojana in India to provide optimal all weather road connectivity to unconnected villages (points). This tool is also useful for constructing highways or railways spanning several cities optimally or connecting all cities with minimum total road length.

scholarly journals Zastosowanie metody minimalnego drzewa rozpinającego (najkrótszego dendrytu) w ocenie efektywności i spójności sieci osadniczej województwa mazowieckiego = Application of the minimum spanning tree method in assessing the effectiveness and cohesion of the settlement network of Mazowieckie voivodeship

2019 ◽  
Vol 91 (2) ◽  
pp. 61-80 ◽  
Author(s):  
Przemysław Śleszyński ◽  
Paweł Sudra
Keyword(s):  
Urban Areas ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Research Area ◽  
Rural Settlements ◽  
Single Family ◽  
Network Effectiveness ◽  
Spatial Cohesion ◽  
The 19Th Century

Contemporary settlement systems observed in Poland bear numerous traces of historical transformations of rural settlements which took place in the 19th century, at the time of foreign partitioning of Polish territory, in different ways in particular regions. The result of processes occurring from the second half of the 20th century is the extensive development of urban areas, and – after 1990 – chaotic, spontaneous processes of transformation in suburban zones. Research methods using graph theory have been applied for years in investigating settlement networks on various scales. One of the more useful graphs is the minimum spanning tree (MST), which connects all vertices in such a way that the sum of the distances between them is the shortest. This article presents the application of the minimum spanning tree (or shortest dendrite) method with a view to its suitability for determining the degree of dispersion and spatial cohesion of urbanised structures being assessed. Two indicators have been proposed thanks to alignment of the shortest dendrite length to other variables. The settlement network effectiveness indicator is the ratio of MST length to the population in an area. The settlement network cohesion indicator is in turn the ratio of the MST length to population density. Mazowieckie voivodeship has been chosen as the research area, while address points obtained from the central official database collecting data from municipal records have been chosen as the source dataset. Over 1 million address points were considered, in line with their status as at the end of 2016. Minimum spanning trees were plotted for each of the 314 gminas (local-authority areas) aking up the voivodeship, using ArcGIS software. Subsequently, the proposed indicators were calculated by reference to the MSTs. The results were then mapped. The proposed indicators may be helpful in studies on the origin of settlements, allowing areas with varying degrees of uniformity or isolation of building locations to be indicated. They can be made use of in comparative studies, especially concerning rural settlements, in which single-family housing predominates, and hamlets and uildings standing in isolation are present. The effectiveness indicator can be used in the assessment of infrastructural coverage, i.a. in the ontext of the costs of spatial chaos and demographic capacity.

scholarly journals Does Determination of Initial Cluster Centroids Improve the Performance of K-Means Clustering Algorithm? Comparison of Three Hybrid Methods by Genetic Algorithm, Minimum Spanning Tree, and Hierarchical Clustering in an Applied Study

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Saeedeh Pourahmad ◽  
Atefeh Basirat ◽  
Amir Rahimi ◽  
Marziyeh Doostfatemeh
Keyword(s):  
Genetic Algorithm ◽  
Hierarchical Clustering ◽  
Spanning Tree ◽  
Clustering Algorithm ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Hybrid Methods ◽  
Optimal Solution ◽  
Initial Cluster ◽  
Algorithm Comparison

Random selection of initial centroids (centers) for clusters is a fundamental defect in K-means clustering algorithm as the algorithm’s performance depends on initial centroids and may end up in local optimizations. Various hybrid methods have been introduced to resolve this defect in K-means clustering algorithm. As regards, there are no comparative studies comparing these methods in various aspects, the present paper compared three hybrid methods with K-means clustering algorithm using concepts of genetic algorithm, minimum spanning tree, and hierarchical clustering method. Although these three hybrid methods have received more attention in previous researches, fewer studies have compared their results. Hence, seven quantitative datasets with different characteristics in terms of sample size, number of features, and number of different classes are utilized in present study. Eleven indices of external and internal evaluating index were also considered for comparing the methods. Data indicated that the hybrid methods resulted in higher convergence rate in obtaining the final solution than the ordinary K-means method. Furthermore, the hybrid method with hierarchical clustering algorithm converges to the optimal solution with less iteration than the other two hybrid methods. However, hybrid methods with minimal spanning trees and genetic algorithms may not always or often be more effective than the ordinary K-means method. Therefore, despite the computational complexity, these three hybrid methods have not led to much improvement in the K-means method. However, a simulation study is required to compare the methods and complete the conclusion.

Board Game Supporting Learning Prim’s Algorithm and Dijkstra’s Algorithm

2012 ◽  
pp. 256-270
Author(s):  
Wen-Chih Chang ◽  
Te-Hua Wang ◽  
Yan-Da Chiu
Keyword(s):  
Data Structure ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Experimental Results ◽  
Dijkstra’S Algorithm ◽  
Board Game ◽  
Tree Algorithm ◽  
Dijkstra's Algorithm ◽  
Tree Algorithms

The concept of minimum spanning tree algorithms in data structure is difficult for students to learn and to imagine without practice. Usually, learners need to diagram the spanning trees with pen to realize how the minimum spanning tree algorithm works. In this paper, the authors introduce a competitive board game to motivate students to learn the concept of minimum spanning tree algorithms. They discuss the reasons why it is beneficial to combine graph theories and board game for the Dijkstra and Prim minimum spanning tree theories. In the experimental results, this paper demonstrates the board game and examines the learning feedback for the mentioned two graph theories. Advantages summarizing the benefits of combining the graph theories with board game are discussed.

CLUSTERING WITH MINIMUM SPANNING TREES

2011 ◽  
Vol 20 (01) ◽  
pp. 139-177 ◽  
Author(s):  
YAN ZHOU ◽  
OLEKSANDR GRYGORASH ◽  
THOMAS F. HAIN
Keyword(s):  
Spanning Tree ◽  
Clustering Algorithm ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Clustering Algorithms ◽  
Predefined Criterion ◽  
Point Set ◽  
Color Clustering ◽  
Representative Points ◽  
Set Of Points

We propose two Euclidean minimum spanning tree based clustering algorithms — one a k-constrained, and the other an unconstrained algorithm. Our k-constrained clustering algorithm produces a k-partition of a set of points for any given k. The algorithm constructs a minimum spanning tree of a set of representative points and removes edges that satisfy a predefined criterion. The process is repeated until k clusters are produced. Our unconstrained clustering algorithm partitions a point set into a group of clusters by maximally reducing the overall standard deviation of the edges in the Euclidean minimum spanning tree constructed from a given point set, without prescribing the number of clusters. We present our experimental results comparing our proposed algorithms with k-means, X-means, CURE, Chameleon, and the Expectation-Maximization (EM) algorithm on both artificial data and benchmark data from the UCI repository. We also apply our algorithms to image color clustering and compare them with the standard minimum spanning tree clustering algorithm as well as CURE, Chameleon, and X-means.

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