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 Degree-Constrained k -Minimum Spanning Tree Problem

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-25
Author(s):  
Pablo Adasme ◽  
Ali Dehghan Firoozabadi
Keyword(s):  
Learning Strategies ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Learning Strategy ◽  
Learning Algorithm ◽  
Hamiltonian Path ◽  
Minimum Cost ◽  
Mixed Integer ◽  
Neighborhood Search ◽  
Q Learning

Let G V , E be a simple undirected complete graph with vertex and edge sets V and E , respectively. In this paper, we consider the degree-constrained k -minimum spanning tree (DC k MST) problem which consists of finding a minimum cost subtree of G formed with at least k vertices of V where the degree of each vertex is less than or equal to an integer value d ≤ k − 2 . In particular, in this paper, we consider degree values of d ∈ 2,3 . Notice that DC k MST generalizes both the classical degree-constrained and k -minimum spanning tree problems simultaneously. In particular, when d = 2 , it reduces to a k -Hamiltonian path problem. Application domains where DC k MST can be adapted or directly utilized include backbone network structures in telecommunications, facility location, and transportation networks, to name a few. It is easy to see from the literature that the DC k MST problem has not been studied in depth so far. Thus, our main contributions in this paper can be highlighted as follows. We propose three mixed-integer linear programming (MILP) models for the DC k MST problem and derive for each one an equivalent counterpart by using the handshaking lemma. Then, we further propose ant colony optimization (ACO) and variable neighborhood search (VNS) algorithms. Each proposed ACO and VNS method is also compared with another variant of it which is obtained while embedding a Q-learning strategy. We also propose a pure Q-learning algorithm that is competitive with the ACO ones. Finally, we conduct substantial numerical experiments using benchmark input graph instances from TSPLIB and randomly generated ones with uniform and Euclidean distance costs with up to 400 nodes. Our numerical results indicate that the proposed models and algorithms allow obtaining optimal and near-optimal solutions, respectively. Moreover, we report better solutions than CPLEX for the large-size instances. Ultimately, the empirical evidence shows that the proposed Q-learning strategies can bring considerable improvements.

scholarly journals The Clustered Selected-Internal Steiner Tree Problem

2021 ◽  
pp. 1-12
Author(s):  
Yen Hung Chen
Keyword(s):  
Approximation Algorithm ◽  
Complete Graph ◽  
Steiner Tree ◽  
Minimum Cost ◽  
Performance Ratio ◽  
Steiner Tree Problem ◽  
The Cost

Given a complete graph [Formula: see text], with nonnegative edge costs, two subsets [Formula: see text] and [Formula: see text], a partition [Formula: see text] of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text], a clustered Steiner tree is a tree [Formula: see text] of [Formula: see text] that spans all vertices in [Formula: see text] such that [Formula: see text] can be cut into [Formula: see text] subtrees [Formula: see text] by removing [Formula: see text] edges and each subtree [Formula: see text] spans all vertices in [Formula: see text], [Formula: see text]. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of [Formula: see text] is a clustered Steiner tree for [Formula: see text] if all vertices in [Formula: see text] are internal vertices of [Formula: see text]. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree [Formula: see text] for [Formula: see text] and [Formula: see text] in [Formula: see text] with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio [Formula: see text] for the clustered selected-internal Steiner tree problem, where [Formula: see text] is the best-known performance ratio for the Steiner tree problem.

scholarly journals Dynamic minimum spanning tree construction and maintenance for Wireless Sensor Networks

2019 ◽  
pp. 57-69
Author(s):  
Sergio Diaz ◽  
Diego Mendez
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Dynamic Networks ◽  
Convergence Time ◽  
Wireless Sensor ◽  
Neighbor Discovery ◽  
Channel Quality ◽  
Tree Construction ◽  
Depth Analysis ◽  
The Cost

In a wireless sensor network (WSN), finding the optimal route from each node to the sink is not a straightforward task because of the distributed and dynamic characteristics of the network. For instance, the network suffers frequent changes because the channel quality varies over time and the nodes can leave or join the network at any moment. In order to deal with this variability, we propose the Dynamic Gallager-Humblet-Spira (DGHS) algorithm that builds and maintains a minimum spanning tree for distributed and dynamic networks, and we have found that DGHS reduces the number of control messages and the energy consumption, at the cost of a slight increase in the memory size and convergence time. This paper presents a detailed description of the different stages of the DGHS algorithm (neighbor discovery, tree construction and data collection), an in-depth analysis of extensive results that validates our proposal, and an exhaustive description of the GHS limitations.

A claims problem approach to the cost allocation of a minimum cost spanning tree

2021 ◽  
Author(s):  
José-Manuel Giménez-Gómez ◽  
Josep E. Peris ◽  
Begoña Subiza
Keyword(s):  
Spanning Tree ◽  
Cost Allocation ◽  
Minimum Cost ◽  
Claims Problem ◽  
The Cost

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 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 Fast Self-stabilizing Minimum Spanning Tree Construction

2010 ◽  
pp. 480-494 ◽  
Author(s):  
Lélia Blin ◽  
Shlomi Dolev ◽  
Maria Gradinariu Potop-Butucaru ◽  
Stephane Rovedakis
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Tree Construction

A Low Cost MST-FSM Obfuscation Method for Hardware IP Protection

2020 ◽  
Vol 29 (13) ◽  
pp. 2050208
Author(s):  
Yuejun Zhang ◽  
Zhao Pan ◽  
Pengjun Wang ◽  
Xiaowei Zhang
Keyword(s):  
Integrated Circuit ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Hamming Distance ◽  
Low Cost ◽  
The Arts ◽  
Finite State ◽  
Ip Cores ◽  
Building Program ◽  
The Cost

Effective resistance to intellectual property (IP) piracy, overproduction and reverse engineering are becoming more and more necessary in the integrated circuit (IC) supply chain. To protect the hardware, the obfuscation methodology hides the original function by adding a large number of redundant states. However, existing hardware obfuscation approaches have hardware overhead and efficiency of obfuscation limitations. This paper proposed a novel methodology for IP security using the minimum spanning tree finite state machine (MST-FSM) obfuscation. In the minimum spanning tree (MST) algorithm, the Hamming distance defines the cost of obfuscated states. The Kruskal algorithm optimizes the connection relationship of obfuscated states by computing the Hamming distance of the MST-FSM. The proposed MST-FSM is automatically generated and embedded in the hardware IP with the self-building program. Finally, the MST-FSM is applied on the itc99 benchmark circuits and encryption standard IP cores. Compared with other state-of-the-arts, the obfuscation potency is improved by 3.57%, and the average hardware cost is decreased by about 6.01%.

Sign in / Sign up

Export Citation Format

Share Document