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.

scholarly journals Algorithms for the power-p Steiner tree problem in the Euclidean plane

2018 ◽  
Vol 25 (4) ◽  
pp. 28
Author(s):  
Christina Burt ◽  
Alysson Costa ◽  
Charl Ras
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Valid Inequalities ◽  
Minimum Cost ◽  
Performance Ratio ◽  
Mixed Integer ◽  
Steiner Tree Problem ◽  
Steiner Points ◽  
Tree Construction ◽  
The Cost

We study the problem of constructing minimum power-$p$ Euclidean $k$-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most $k$ additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of $p$ (where $p\geq 1$), and the cost of a network is the sum of all edge costs. We propose two heuristics: a ``beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio $\kappa$ of the beaded-MST heuristic satisfies $\sqrt{3}^{p-1}(1+2^{1-p})\leq \kappa\leq 3(2^{p-1})$. We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the $p=2$ case.

scholarly journals Some Theoretical Links Between Shortest Path Filters and Minimum Spanning Tree Filters

2019 ◽  
Vol 61 (6) ◽  
pp. 745-762
Author(s):  
Sravan Danda ◽  
Aditya Challa ◽  
B. S. Daya Sagar ◽  
Laurent Najman
Keyword(s):  
Shortest Path ◽  
Spanning Tree ◽  
Minimum Spanning Tree

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

scholarly journals Hybrid greedy genetic algorithm for the Euclidean Steiner tree problem

2019 ◽  
Author(s):  
Andrey Oliveira ◽  
Danilo Sanches ◽  
Bruna Osti
Keyword(s):  
Genetic Algorithm ◽  
Steiner Tree ◽  
Optimization Problem ◽  
Spanning Trees ◽  
Evolutionary Strategies ◽  
Steiner Tree Problem ◽  
Minimum Length ◽  
Genetic Operators ◽  
Steiner Points ◽  
Euclidean Steiner Tree Problem

This paper presents a genetic algorithm for the Euclidean Steiner tree problem. This is an optimization problem whose objective is to obtain a minimum length tree to interconnect a set of fixed points, and for this purpose to be achieved, new auxiliary points, called Steiner points, can be added. The proposed heuristic uses a genetic algorithm to manipulate spanning trees, which are then transformed into Steiner trees by inserting and repositioning the Steiner points. Greedy genetic operators and evolutionary strategies are tested. Results of numerical experiments for benchmark library problem (OR-Library) are presented and discussed.This paper presents a genetic algorithm for the Euclidean Steiner tree problem. This is an optimization problem whose objective is to obtain a minimum length tree to interconnect a set of fixed points, and for this purpose to be achieved, new auxiliary points, called Steiner points, can be added. The proposed heuristic uses a genetic algorithm to manipulate spanning trees, which are then transformed into Steiner trees by inserting and repositioning the Steiner points. Greedy genetic operators and evolutionary strategies are tested. Results of numerical experiments for benchmark library problem (OR-Library) are presented and discussed.

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

2021 ◽  
Vol 348 ◽  
pp. 01001
Author(s):  
Paryati ◽  
Krit Salahddine
Keyword(s):  
Shortest Path ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Dijkstra’S Algorithm ◽  
Dijkstra Algorithm ◽  
Application System ◽  
Path Creation ◽  
Dijkstra's Algorithm

Kruskal’s Algorithm is an algorithm used to find the minimum spanning tree in graphical connectivity that provides the option to continue processing the least-weighted margins. In the Kruskal algorithm, ordering the weight of the ribs makes it easy to find the shortest path. This algorithm is independent in nature which will facilitate and improve path creation. Based on the results of the application system trials that have been carried out in testing and comparisons between the Kruskal algorithm and the Dijkstra algorithm, the following conclusions can be drawn: that a strength that is the existence of weight sorting will facilitate the search for the shortest path. And considering the characteristics of Kruskal’s independent algorithm, it will facilitate and improve the formation of the path. The weakness of the Kruskal algorithm is that if the number of nodes is very large, it will be slower than Dijkstra’s algorithm because it has to sort thousands of vertices first, then form a path.

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.

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