scholarly journals Novel Degree Constrained Minimum Spanning Tree Algorithm Based on an Improved Multicolony Ant Algorithm

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Xuemei Sun ◽  
Cheng Chang ◽  
Hua Su ◽  
Chuitian Rong
Keyword(s):  
Experimental Data ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Optimal Algorithm ◽  
Minimum Weight ◽  
Ant Algorithm ◽  
Random Perturbations ◽  
Optimal Solutions ◽  
Ant System ◽  
Multiple Groups

Degree constrained minimum spanning tree (DCMST) refers to constructing a spanning tree of minimum weight in a complete graph with weights on edges while the degree of each node in the spanning tree is no more thand(d≥ 2). The paper proposes an improved multicolony ant algorithm for degree constrained minimum spanning tree searching which enables independent search for optimal solutions among various colonies and achieving information exchanges between different colonies by information entropy. Local optimal algorithm is introduced to improve constructed spanning tree. Meanwhile, algorithm strategies in dynamic ant, random perturbations ant colony, and max-min ant system are adapted in this paper to optimize the proposed algorithm. Finally, multiple groups of experimental data show the superiority of the improved algorithm in solving the problems of degree constrained minimum spanning tree.

scholarly journals On the Rank of a Random Binary Matrix

10.37236/8092 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Applied Mathematics ◽  
Minimum Weight ◽  
Binary Matrix ◽  
The Matrix ◽  
Expected Length ◽  
Minimum Weight Spanning Tree ◽  
Random Hypergraph ◽  
Correct Estimate

We study the rank of a random $n \times m$ matrix $\mathbf{A}_{n,m;k}$ with entries from $GF(2)$, and exactly $k$ unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all ${n \choose k}$ such columns. We obtain an asymptotically correct estimate for the rank as a function of the number of columns $m$ in terms of $c,n,k$, and where $m=cn/k$. The matrix $\mathbf{A}_{n,m;k}$ forms the vertex-edge incidence matrix of a $k$-uniform random hypergraph $H$. The rank of $\mathbf{A}_{n,m;k}$ can be expressed as follows. Let $|C_2|$ be the number of vertices of the 2-core of $H$, and $|E(C_2)|$ the number of edges. Let $m^*$ be the value of $m$ for which $|C_2|= |E(C_2)|$. Then w.h.p. for $m<m^*$ the rank of $\mathbf{A}_{n,m;k}$ is asymptotic to $m$, and for $m \ge m^*$ the rank is asymptotic to $m-|E(C_2)|+|C_2|$. In addition, assign i.i.d. $U[0,1]$ weights $X_i, i \in {1,2,...m}$ to the columns, and define the weight of a set of columns $S$ as $X(S)=\sum_{j \in S} X_j$. Define a basis as a set of $n-𝟙 (k\text{ even})$ linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of Frieze [On the value of a random minimum spanning tree problem, Discrete Applied Mathematics, (1985)] that, for $k=2$,   the expected length of a minimum weight spanning tree tends to $\zeta(3)\sim 1.202$.

scholarly journals The Diameter of the Minimum Spanning Tree of a Complete Graph

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Louigi Addario-Berry ◽  
Nicolas Broutin ◽  
Bruce Reed
Keyword(s):  
Complete Graph ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Minimum Weight ◽  
Unique Minimum ◽  
International Audience ◽  
Minimum Weight Spanning Tree ◽  
Independent Identically Distributed

International audience Let $X_1,\ldots,X_{n\choose 2}$ be independent identically distributed weights for the edges of $K_n$. If $X_i \neq X_j$ for$ i \neq j$, then there exists a unique minimum weight spanning tree $T$ of $K_n$ with these edge weights. We show that the expected diameter of $T$ is $Θ (n^{1/3})$. This settles a question of [Frieze97].

A LEARNING AUTOMATA-BASED ALGORITHM TO THE STOCHASTIC MIN-DEGREE CONSTRAINED MINIMUM SPANNING TREE PROBLEM

2013 ◽  
Vol 24 (03) ◽  
pp. 329-348 ◽  
Author(s):  
JAVAD AKBARI TORKESTANI
Keyword(s):  
Spanning Tree ◽  
Sampling Method ◽  
Minimum Spanning Tree ◽  
Minimum Weight ◽  
Learning Automata ◽  
Optimal Solution ◽  
Simulation Experiments ◽  
Stochastic Graph ◽  
Standard Sampling ◽  
Martingale Theorem

Min-degree constrained minimum spanning tree (md-MST) problem is an NP-hard combinatorial optimization problem seeking for the minimum weight spanning tree in which the vertices are either of degree one (leaf) or at least degree d ≥ 2. md-MST problem is new to the literature and very few studies have been conducted on this problem in deterministic graph. md-MST problem has several appealing real-world applications. Though in realistic applications the graph conditions and parameters are stochastic and vary with time, to the best of our knowledge no work has been done on solving md-MST problem in stochastic graph. This paper proposes a decentralized learning automata-based algorithm for finding a near optimal solution to the md-MST problem in stochastic graph. In this work, it is assumed that the weight associated with the graph edge is random variable with a priori unknown probability distribution. This assumption makes the md-MST problem incredibly harder to solve. The proposed algorithm exploits an intelligent sampling technique avoiding the unnecessary samples by focusing on the edges of the min-degree spanning tree with the minimum expected weight. On the basis of the Martingale theorem, the convergence of the proposed algorithm to the optimal solution is theoretically proven. Extensive simulation experiments are performed on the stochastic graph instances to show the performance of the proposed algorithm. The obtained results are compared with those of the standard sampling method in terms of the sampling rate and solution optimality. Simulation experiments show that the proposed method outperforms the standard sampling method.

scholarly journals Algorithms for Euclidean Degree Bounded Spanning Tree Problems

2019 ◽  
Vol 29 (02) ◽  
pp. 121-160 ◽  
Author(s):  
Patrick J. Andersen ◽  
Charl J. Ras
Keyword(s):  
Approximation Algorithms ◽  
Spanning Tree ◽  
Maximum Degree ◽  
Euclidean Plane ◽  
Minimum Spanning Tree ◽  
Computational Experiments ◽  
Optimal Solutions ◽  
New Type ◽  
Disjoint Sets ◽  
Set Of Points

Given a set of points in the Euclidean plane, the Euclidean [Formula: see text]-minimum spanning tree ([Formula: see text]-MST) problem is the problem of finding a spanning tree with maximum degree no more than [Formula: see text] for the set of points such the sum of the total length of its edges is minimum. Similarly, the Euclidean [Formula: see text]-minimum bottleneck spanning tree ([Formula: see text]-MBST) problem, is the problem of finding a degree-bounded spanning tree for a set of points in the plane such that the length of the longest edge is minimum. When [Formula: see text], these two problems may yield disjoint sets of optimal solutions for the same set of points. In this paper, we perform computational experiments to compare the accuracies of a variety of heuristic and approximation algorithms for both these problems. We develop heuristics for these problems and compare them with existing algorithms. We also describe a new type of edge swap algorithm for these problems that outperforms all the algorithms we tested.

GA-BASED ALTERNATIVE APPROACHES FOR THE DEGREE-CONSTRAINED SPANNING TREE PROBLEM

2009 ◽  
Vol 23 (01) ◽  
pp. 129-144
Author(s):  
GENGUI ZHOU ◽  
ZHENYU CAO ◽  
ZHIQING MENG ◽  
JIAN CAO
Keyword(s):  
Genetic Algorithm ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Practical Importance ◽  
Numerical Results ◽  
Tree Structure ◽  
Np Hard ◽  
Optimal Solutions ◽  
Alternative Approaches ◽  
Prüfer Number

The degree-constrained minimum spanning tree (dc-MST) problem is of high practical importance. Up to now there are few effective algorithms to solve this problem because of its NP-hard complexity. More recently, a genetic algorithm (GA) approach for this problem was tried by using Prüfer number to encode a spanning tree. The Prüfer number is a skillful encoding for tree but not efficient enough to deal with the dc-MST problem. In this paper, a new tree-based encoding is developed directly based on the tree structure. We denote it as tree-based permutation encoding and apply it to the dc-MST problem by using the GA approach. Compared with the numerical results and CPU runtimes between two encodings, the new tree-based permutation is effective to deal with the dc-MST problem and even more efficient than the Prüfer number to evolve to the optimal or near-optimal solutions.

scholarly journals Parallel strategies for a multi-criteria GRASP algorithm

Production ◽  
2007 ◽  
Vol 17 (1) ◽  
pp. 84-93 ◽  
Author(s):  
Dalessandro Soares Vianna ◽  
José Elias Claudio Arroyo ◽  
Pedro Sampaio Vieira ◽  
Thiago Ribeiro de Azeredo
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Efficient Solutions ◽  
Search Problem ◽  
Pareto Optimal ◽  
Complete Graphs ◽  
Pareto Optimal Solutions ◽  
Adaptive Search ◽  
Optimal Solutions

This paper proposes different strategies of parallelizing a multi-criteria GRASP (Greedy Randomized Adaptive Search Problem) algorithm. The parallel GRASP algorithm is applied to the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem, a vector of costs is defined for each edge of the graph and the goal is to find all the efficient or Pareto optimal spanning trees (Pareto-optimal solutions). Each process finds a subset of efficient solutions. These subsets are joined using different strategies to obtain the final set of efficient solutions. The multi-criteria GRASP algorithm with the different parallel strategies are tested on complete graphs with n = 20, 30 and 50 nodes and r = 2 and 3 criteria. The computational results show that the proposed parallel algorithms reduce the execution time and the results obtained by the sequential version were improved.

scholarly journals Motif Discovery of Protein-Protein Interaction using Minimum Spanning Tree

2020 ◽  
Vol 9 (3) ◽  
pp. 990-994
Keyword(s):  
Protein Interaction ◽  
Spanning Tree ◽  
Motif Discovery ◽  
Minimum Spanning Tree ◽  
Minimum Weight ◽  
Interaction Network ◽  
Protein Protein Interaction ◽  
Minimum Path ◽  
Weight Value ◽  
Tree Approach

In protein Interaction Networks, counting subgraph is a tedious task. From the list of non induced occurrence of the subgraph, motif topology calculated by using Combi Motif and Slider techniques. But, this approach was taken more time to execute. To reduce the execution time, the minimum weight value between the nodes, the Minimum spanning tree concept proposed. Prim’s method implemented with the greedy technique (as Kruskal’s algorithm) to calculate the minimum path between the nodes in the Protein interaction network. This technique uses to compare the similarity of the minimum spanning tree approach. Initially, this algorithm has discovered the path then calculated the weight matrix and found the minimum weight value. From the computational experiments, the proposed approach of MST providing better results in terms of time consumption and accuracy to count the motif pattern in the network of the interacted proteins.

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