Clustering Vertices in Weighted Graphs

2011 ◽  
pp. 285-298
Author(s):  
Derry Tanti Wijaya ◽  
Stephane Bressan
Keyword(s):  
Graph Algorithms ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Weighted Graph ◽  
Graph Clustering ◽  
Weighted Graphs ◽  
Similarity Relations ◽  
Natural Notion ◽  
Cluster Density ◽  
Markov Clustering

Clustering is the unsupervised process of discovering natural clusters so that objects within the same cluster are similar and objects from different clusters are dissimilar. In clustering, if similarity relations between objects are represented as a simple, weighted graph where objects are vertices and similarities between objects are weights of edges; clustering reduces to the problem of graph clustering. A natural notion of graph clustering is the separation of sparsely connected dense sub graphs from each other based on the notion of intra-cluster density vs. inter-cluster sparseness. In this chapter, we overview existing graph algorithms for clustering vertices in weighted graphs: Minimum Spanning Tree (MST) clustering, Markov clustering, and Star clustering. This includes the variants of Star clustering, MST clustering and Ricochet.

scholarly journals A Constant-Factor Approximation for the Generalized Cable-Trench Problem

2019 ◽  
Author(s):  
Marcelo Benedito ◽  
Lehilton Pedrosa ◽  
Hugo Rosado
Keyword(s):  
Shortest Path ◽  
Steiner Tree ◽  
Spanning Tree ◽  
Optimization Problem ◽  
Minimum Spanning Tree ◽  
Weighted Graph ◽  
Constant Factor ◽  
Steiner Tree Problem ◽  
Negative Factor ◽  
Path Lengths

In the Cable-Trench Problem (CTP), the objective is to find a rooted spanning tree of a weighted graph that minimizes the length of the tree, scaled by a non-negative factor , plus the sum of all shortest-path lengths from the root, scaled by another non-negative factor. This is an intermediate optimization problem between the Single-Destination Shortest Path Problem and the Minimum Spanning Tree Problem. In this extended abstract, we consider the Generalized CTP (GCTP), in which some vertices need not be connected to the root, but may serve as cost-saving merging points; this variant also generalizes the Steiner Tree Problem. We present an 8.599-approximation algorithm for GCTP. Before this paper, no constant approximation for the standard CTP was known.

scholarly journals IMPLEMENTASI ALGORITMA KRUSKAL DAN ALGORITMA PRIM SUATU GRAPH DENGAN APLIKASI BERBASIS DESKTOP

2020 ◽  
Vol 3 (2) ◽  
pp. 89-93
Author(s):  
Siti Alvi Sholikhatin ◽  
Adi Budi Prasetyo ◽  
Ade Nurhopipah
Keyword(s):  
Spanning Tree ◽  
Research Study ◽  
Minimum Spanning Tree ◽  
Greedy Algorithms ◽  
Relevant Literature ◽  
Weighted Graph ◽  
Self Assessment ◽  
Technological Readiness ◽  
Actual Environment ◽  
Application Design

A graph has several algorithms in its solution, including the Kruskal algorithm and Prim algorithm, both of which are greedy algorithms for determining the minimum spanning tree. Completion of graphs is useful in various fields of life, so an accurate graph calculation is important. Making an application to solve a graph, especially the Kruskal algorithm and Prim algorithm aims to facilitate the work of the graph so as to produce an accurate final result. The flow of research carried out are: a background review of research, study of literature and relevant literature, application design, building desktop-based applications, as well as implementation and application tests. The level of technological readiness or TKT in this research is based on self-assessment which is at level 7, meaning the prototype demonstration system in the actual environment, with details of the TKT indicators as follows: TKT indicator 1 is met, TKT indicator 2 is met, TKT indicator 3 is not met, TKT indicator 4, TKT indicator 5 are met, TKT indicator 6 are met, TKT indicator 7 is met, TKT indicator 8 and 9 are not met. The application that has been built is useful for completing a graph with the Kruskal algorithm and Prim algorithm. This research was conducted to answer the crucial needs of a weighted graph settlement application.

On the Euclidean Minimum Spanning Tree Problem

2005 ◽  
Vol 1 (1) ◽  
pp. 11-14 ◽  
Author(s):  
Sanguthevar Rajasekaran
Keyword(s):  
Spanning Tree ◽  
Euclidean Distance ◽  
Minimum Spanning Tree ◽  
Linear Time ◽  
Randomized Algorithm ◽  
Weighted Graph ◽  
Deterministic Algorithm ◽  
Probabilistic Algorithms ◽  
Tree Algorithms ◽  
Minimum Spanning Tree Problem

Given a weighted graph G(V;E), a minimum spanning tree for G can be obtained in linear time using a randomized algorithm or nearly linear time using a deterministic algorithm. Given n points in the plane, we can construct a graph with these points as nodes and an edge between every pair of nodes. The weight on any edge is the Euclidean distance between the two points. Finding a minimum spanning tree for this graph is known as the Euclidean minimum spanning tree problem (EMSTP). The minimum spanning tree algorithms alluded to before will run in time O(n2) (or nearly O(n2)) on this graph. In this note we point out that it is possible to devise simple algorithms for EMSTP in k- dimensions (for any constant k) whose expected run time is O(n), under the assumption that the points are uniformly distributed in the space of interest.CR Categories: F2.2 Nonnumerical Algorithms and Problems; G.3 Probabilistic Algorithms

Mathematical Modelling for Transportation With Application to Airline Transportation Network

2021 ◽  
pp. 28-51
Author(s):  
Sadiqah Almarzooq ◽  
Njwd Albishi
Keyword(s):  
Graph Theory ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Transportation Networks ◽  
Transportation Network ◽  
Weighted Graph ◽  
Search Method ◽  
Nearest Neighbour ◽  
Water Pipelines ◽  
Regional Airports

Graph theory is a basic tool to solve real-world problems such as communication between people, water pipelines, and transportation networks. A transportation network can be modeled as connected weighted graph. This chapter starts by introducing some fundamental concepts of graph theory to be applied to three main problems: the minimum spanning tree, the shortest path, and the travel salesperson. The authors discuss some appropriated algorithms such as depth first algorithm, Prim's algorithm, Kruskal's algorithm, Dijkstra's algorithm, the nearest neighbour algorithm, the minimum spanning tree depth first search method (MST-DFS) algorithm, and the Christofides' algorithm to solve these problems and apply them the airlines network between international and regional airports in Saudi Arabia.

The Application of Minimum Spanning Tree Clustering in the Ontology Mapping Pretreatment

2014 ◽  
Vol 644-650 ◽  
pp. 2151-2154
Author(s):  
Jin Hui Cheng ◽  
Wan Long Li ◽  
Shan Hong Zheng ◽  
Dong Han
Keyword(s):  
Objective Function ◽  
Information Entropy ◽  
Spanning Tree ◽  
Clustering Algorithm ◽  
Minimum Spanning Tree ◽  
Graph Clustering ◽  
Ontology Mapping ◽  
System Information

Screening candidates is the key step to improve the efficiency of ontology mapping. Minimum spanning tree clustering is one of the important ways of graph clustering algorithm. Defining the related concepts and methods first, according to the characteristics of the ontology file itself, Select graph clustering of minimum spanning tree clustering algorithm, To screening candidates of participate in the concept of mapping, Aiming at the deficiency and improvement of objective function in the algorithm, based on the system information entropy instead of the complicated calculation of similarity to supervise the clustering. To reduce the computational scale and improve the efficiency.

scholarly journals Undergraduate Students’ Solutions of Modeling Problems in Algorithmic Graph Theory

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 572
Author(s):  
Medová ◽  
Páleníková ◽  
Rybanský ◽  
Naštická
Keyword(s):  
Undergraduate Students ◽  
Graph Algorithms ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Hierarchical Cluster ◽  
Discrete Mathematics ◽  
Chinese Postman Problem ◽  
Item Response Theory Analysis ◽  
Chinese Postman ◽  
Minimum Spanning Tree Problem

Graphs can be considered as useful mathematical models. Graph algorithms are a common part of undergraduate courses in discrete mathematics. Even though they have been successfully implemented in secondary curricula, little research has been dedicated to the analysis of students’ work. Within a discrete mathematics course for university students, several graph algorithms were introduced via their applications. At the end of the course, the students took a test focused, inter alia, on applications of the algorithms. The mistakes that occurred in 127 students’ solutions of three problems (the Chinese postman problem, the shortest path problem, and the minimum spanning tree problem) were categorized and compared. Surprisingly, no mistakes were identified in the mathematization of situations or in the interpretation of results with respect to the wording of the problem. The categories of errors varied regardless of the problem types. Hierarchical cluster analysis grouped together the students’ solutions for the Chinese postman problem and the minimum spanning tree problem. By means of nonparametric item response theory analysis, the Chinese postman problem was identified as the most problematic for students. Possible sources of this difficulty are discussed in more detail herein.

NEW METAHEURISTIC APPROACHES FOR THE LEAF-CONSTRAINED MINIMUM SPANNING TREE PROBLEM

2008 ◽  
Vol 25 (04) ◽  
pp. 575-589 ◽  
Author(s):  
ALOK SINGH ◽  
ANURAG SINGH BAGHEL
Keyword(s):  
Genetic Algorithm ◽  
Tabu Search ◽  
Ant Colony Optimization ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Weighted Graph ◽  
The Other ◽  
Greedy Heuristic ◽  
New Approaches

Given an undirected, connected, weighted graph, the leaf-constrained minimum spanning tree (LCMST) problem seeks a spanning tree of the graph with smallest weight among all spanning trees of the graph, which contains at least l leaves. In this paper we have proposed two new metaheuristic approaches for the LCMST problem. One is an ant-colony optimization (ACO) algorithm, whereas the other is a tabu search based algorithm. Similar to a previously proposed genetic algorithm, these metaheuristic approaches also use the subset coding that represents a leaf-constrained spanning tree by the set of its interior vertices. Our new approaches perform well in comparison with two best heuristics reported in the literature for the problem — the subset-coded genetic algorithm and a greedy heuristic.

scholarly journals ALGORITMA “HANCURKAN SEMUA SIKEL” UNTUK MENENTUKAN POHON PERENTANG MINIMUM DARI SUATU GRAF BERBOBOT

2019 ◽  
Vol 21 (2) ◽  
pp. 91-98
Author(s):  
Ricky Aditya
Keyword(s):  
Graph Theory ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Weighted Graph ◽  
Alternative Algorithm

The minimum spanning tree is one of the applications of graph theory in various fields. There are several algorithms for determining the minimum spanning tree of a weighted graph, such as Kruskal's algorithm and Prim's algorithm. These two algorithms are not really easy to teach to students in general. Therefore in this paper presented an alternative algorithm called the algorithm "Destroy All Sikel", which is more intuitive and easier to understand. Furthermore, there are also examples of implementation and comparison with two other algorithms.

Graph Theoretic Approaches for Image Analysis

2017 ◽  
pp. 193-224
Author(s):  
Biplab Banerjee ◽  
Sudipan Saha ◽  
Krishna Mohan Buddhiraju
Keyword(s):  
Image Analysis ◽  
Image Segmentation ◽  
Spanning Tree ◽  
Object Representation ◽  
Minimum Spanning Tree ◽  
Graph Clustering ◽  
Clustering Techniques ◽  
Graph Theoretic ◽  
Pairwise Interactions ◽  
Image Pixels

Different graph theoretic approaches are prevalent in the field of image analysis. Graphs provide a natural representation of image pixels exploring their pairwise interactions among themselves. Graph theoretic approaches have been used for problem like image segmentation, object representation, matching for different kinds of data. In this chapter, we mainly aim at highlighting the applicability of graph clustering techniques for the purpose of image segmentation. We describe different spectral clustering techniques, minimum spanning tree based data clustering, Markov Random Field (MRF) model for image segmentation in this respect.

Finding the most vital edge with respect to minimum spanning tree in weighted graphs

1991 ◽  
Vol 39 (5) ◽  
pp. 277-281 ◽  
Author(s):  
Lih-Hsing Hsu ◽  
Rong-Hong Jan ◽  
Yu-Che Lee ◽  
Chun-Nan Hung ◽  
Maw-Sheng Chern
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Weighted Graphs
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