scholarly journals The Spanning Tree of a Divisible Multiple Graph

2018 ◽  
Vol 25 (4) ◽  
pp. 388-401
Author(s):  
Alexander V. Smirnov
Keyword(s):  
Heuristic Algorithm ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Basic Properties ◽  
The Common ◽  
Divisible Graph ◽  
Multiple Edges ◽  
Definition Of ◽  
Multiple Graph ◽  
Multiple Tree

In this paper, we study undirected multiple graphs of any natural multiplicity k > 1. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of k linked edges, which connect 2 or k + 1 vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges, and it can be the common ending vertex to k linked edges of a multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of any other multi-edge. Special attention is paid to the class of divisible multiple graphs. The main peculiarity of them is a possibility to divide the graph into k parts, which are adjusted on the linked edges and which have no common edges. Each part is an ordinary graph. The definition of a multiple tree is stated and the basic properties of such trees are studied. Unlike ordinary trees, the number of edges in a multiple tree is not fixed. In the article, the evaluation of the minimum and maximum number of edges in the divisible tree is stated and proved. Next, the definitions of the spanning tree and the complete spanning tree of a multiple graph are given. The criterion of completeness of the spanning tree is proved for divisible graphs. It is also proved that a complete spanning tree exists in any divisible graph. If the multiple graph is weighted, the minimum spanning tree problem and the minimum complete spanning tree problem can be set. In the article, we suggest a heuristic algorithm for the minimum complete spanning tree problem for a divisible graph.

scholarly journals NP-completeness of the Minimum Spanning Tree Problem of a Multiple Graph of Multiplicity k ≥ 3

2021 ◽  
Vol 28 (1) ◽  
pp. 22-37
Author(s):  
Alexander Valeryevich Smirnov
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Optimization Problems ◽  
Initial Graph ◽  
The Common ◽  
Minimum Spanning Tree Problem ◽  
Multiple Edges ◽  
Multiple Graph ◽  
Multiple Tree

In this paper, we study undirected multiple graphs of any natural multiplicity k > 1. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of k linked edges, which connect 2 or (k + 1) vertices correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of k linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. A multiple tree is a connected multiple graph with no cycles. Unlike ordinary trees, the number of edges in a multiple tree is not fixed. The problem of finding the spanning tree can be set for a multiple graph. Complete spanning trees form a special class of spanning trees of a multiple graph. Their peculiarity is that a multiple path joining any two selected vertices exists in the tree if and only if such a path exists in the initial graph. If the multiple graph is weighted, the minimum spanning tree problem and the minimum complete spanning tree problem can be set. Also we can formulate the problems of recognition of the spanning tree and complete spanning tree of the limited weight. The main result of this article is the proof of NPcompleteness of such recognition problems for arbitrary multiple graphs as well as for divisible multiple graphs in the case when multiplicity k ≥ 3. The corresponding optimization problems are NP-hard.

Power Networks Coordinated Planning Based on Improved Steiner Minimum Spanning Tree Theory

2012 ◽  
Vol 433-440 ◽  
pp. 1903-1909
Author(s):  
Han Lin Liu ◽  
Jin Liang Jiang ◽  
Yong Jun Zhang
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Distribution Network ◽  
Weight Coefficient ◽  
Power Networks ◽  
Basic Properties ◽  
Definition Of ◽  
Load Weight ◽  
Better Than

110kV substation’s location and supply area is the crux of the coordination planning for main and distribution network. In this paper, an approach to the coordinated planning of the main grid and distribution network is proposed, which is based on improved weighted Steiner minimum spanning tree theory. This paper will analysis the basic properties of substation firstly, make the definition of Load Weight Coefficient, then use a bound circle to delineate the optimization area. At last, make Load Weight Coefficient as weight, use improved weighted Steiner minimum spanning tree theory to determine the final location of substation. Actual examples show that, in favor of coordinated planning, this method result is obviously better than traditional model’s.

Uncertain Distribution-Minimum Spanning Tree Problem

2016 ◽  
Vol 24 (04) ◽  
pp. 537-560 ◽  
Author(s):  
Jian Zhou ◽  
Xiajie Yi ◽  
Ke Wang ◽  
Jing Liu
Keyword(s):  
Risk Management ◽  
Value At Risk ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Graph Transformation ◽  
Numerical Examples ◽  
Uncertain Variables ◽  
Minimum Spanning Tree Problem ◽  
Definition Of ◽  
Uncertain Distribution

This paper studies the minimum spanning tree problem on a graph with uncertain edge weights, which are formulated as uncertain variables. The concept of ideal uncertain minimum spanning tree (ideal UMST) is initiated by extending the definition of the uncertain [Formula: see text]-minimum spanning tree to reect the overall properties of the α-minimum spanning tree weights at any confidence level [Formula: see text]. On the basis of this new concept, the definition of uncertain distribution-minimum spanning tree is proposed in three ways. Particularly, by considering the tail value at risk from the perspective of risk management, the notion of uncertain [Formula: see text]-distribution-minimum spanning tree ([Formula: see text]-distribution-UMST) is suggested. It is shown that the [Formula: see text]-distribution-UMST is just the uncertain expected minimum spanning tree when [Formula: see text] = 0. For any [Formula: see text], this problem can be effectively solved via the proposed deterministic graph transformation-based approach with the aid of the [Formula: see text]-distribution-path optimality condition. Furthermore, the proposed definitions and solutions are illustrated by some numerical examples.

A New Algorithm of Online Handwritten Mathematical Formula Structure Recognition

2011 ◽  
Vol 48-49 ◽  
pp. 64-70
Author(s):  
Liu Rong Hong
Keyword(s):  
Structural Analysis ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Recognition Rate ◽  
Mathematical Formula ◽  
Structure Recognition ◽  
The Common ◽  
New Algorithms

As a natural, fast mathematical formula input, online handwritten mathematical formula input have great prospects. The paper described the algorithm of recognizing the common online handwritten mathematical formula structure. Firstly, the forms were defined, the convention of formulas were expanded. A method of calculating weight for the minimum spanning tree(MST) was proposed according to the characteristics of hand-written formula. On the basis of them, MST algorithm was applied to formula structural analysis. Then two new algorithms were proposed to recognize the mix form and table structure respectively. Compared with other classical algorithms, the algorithm proposed in this paper can recognized the expansive structure, while recognition rate was improved.

Cellular Competitive Decision Algorithm for Minimum Ratio Spanning Tree

2012 ◽  
Vol 490-495 ◽  
pp. 365-369 ◽  
Author(s):  
Xiao Hua Xiong ◽  
Ai Bing Ning
Keyword(s):  
Combinatorial Optimization ◽  
Objective Function ◽  
Heuristic Algorithm ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Optimization Problems ◽  
Decision Algorithm ◽  
Combinatorial Optimization Problems ◽  
Linear Cost ◽  
Minimum Ratio

Minimum ratio spanning tree(MRST) is the problem of finding a minimum spanning tree when the objective function is the ratio of two linear cost functions(such as the ratio of cost to weight). MRST problem is NP-hard when the denominator is unrestricted in sign. Cellular Competitive decision algorithm(CCDA) is a newly proposed meta-heuristic algorithm for solving combinatorial optimization problems. In this paper, a cellular competitive decision algorithm for MRST is presented. To verify the efficiency of the algorithm, it is been coded in Delphi 7, by which series of instances are tested and results result in good performance.

scholarly journals An Approach of Community Search with Minimum Spanning Tree Based on Node Embedding

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jinglian Liu ◽  
Daling Wang ◽  
Shi Feng ◽  
Yifei Zhang
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Search Algorithm ◽  
Feature Representation ◽  
Graph Structure ◽  
Detection Problem ◽  
Second Stage ◽  
Community Search ◽  
Definition Of ◽  
Low Dimensional

Community search is a query-oriented variant of community detection problem, and the goal is to retrieve a single community from a given set of nodes. Most of the existing community search methods adopt handcrafted features, so there are some limitations in applications. Our idea is motivated by the recent advances of node embedding. Node embedding uses deep learning method to obtain feature representation of nodes directly from graph structure automatically and offers a new method to measure the distance between two nodes. In this paper, we propose a two-stage community search algorithm with a minimum spanning tree strategy based on node embedding. At the first stage, we propose a node embedding model NEBRW and map nodes to the points in a low-dimensional vector space. At the second stage, we propose a new definition of community from the distance viewpoint, transform the problem of community search to a variant of minimum spanning tree problem, and uncover the target community with an improved Prim algorithm. We test our algorithm on both synthetic and real-world network datasets. The experimental results show that our algorithm is more effective for community search than baselines.

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