Tree Problem Research Articles

DEGREE-CONSTRAINED MINIMUM SPANNING TREE PROBLEM IN STOCHASTIC GRAPH

A degree-constrained minimum spanning tree (DCMST) problem is an NP-hard combinatorial optimization problem in graph theory seeking the minimum cost spanning tree with the additional constraint on the vertex degree. Several different approaches have been proposed in the literature to solve this problem using a deterministic graph. However, to the best of the author's knowlege, no work has been performed on solving the problem using stochastic edge-weighted graphs. In this article, a learning automata–based algorithm is proposed to find a near optimal solution of the DCMST problem using a stochastic graph, where the cost associated with the graph edge is a random variable with a priori unknown probability distribution. The convergence of the proposed algorithm to the optimal solution is theoretically proved based on the Martingale theorem. To show the performance of the proposed algorithm, several simulation experiments are conducted on stochastic Euclidean graph instances. Numerical results are compared with those of the standard sampling method (SSM). The numerical results confirm the superiority of the proposed sampling technique over the SSM both in terms of the sampling rate and solution optimality.

A Scheduling Algorithm for Connected Target Coverage in Rotatable Directional Sensor Networks

A key challenge in developing energy-efficient sensor networks is to extend network lifetime in resource-limited environments. As sensors are often densely distributed, they can be scheduled on alternative duty cycles to conserve energy while satisfying the system requirements. Directional sensor networks composed of a large number of directional sensors equipped with a limited battery and with a limited angle of sensing have recently attracted attention. Many types of directional sensors can rotate to face a given direction. Maximizing network lifetime while covering all of the targets in a given area and forwarding sensor data to the sink is a challenge in developing such rotatable directional sensor networks. In this paper, we address the maximum directional cover tree (MDCT) problem of organizing directional sensors into a group of non-disjoint subsets to extend network lifetime. One subset, in which the directional sensors cover all of the targets and forward the data to the sink, is activated at a time, while the others sleep to conserve energy. For the MDCT problem, we first present an energy-consumption model that mainly takes into account the energy expenditure for sensor rotation as well as for the sensing and relaying of data. We also develop a heuristic scheduling algorithm called directional coverage and connectivity (DCC)-greedy to solve the MDCT problem. To verify and evaluate the algorithm, we conduct extensive simulations and show that it extends network lifetime to a reasonable degree.

An efficient method to solve least-cost minimum spanning tree (LC-MST) problem

In this paper, least-cost minimum spanning tree (LC-MST) problem is defined as a method to construct a minimum cost spanning tree that has the least-cost edges in the network by using the distance (cost) matrix. The paper presents a new algorithm based on the distance matrix to solve the LC-MST problem. The studied cases show that the presented algorithm is efficient to solve the LC-MST problem in less time. Also, the presented algorithm can be modified to solve the DC-MST (Delay Constrained-Minimum Spanning Tree) problem presented by Lee and Atiquzzaman (2007) and the MRCT (Minimum Routing Cost Tree) problem presented by Cambos and Ricardo (2008), given as the applications of the presented algorithm.

Genomics-based breeding in forest trees: are we there yet?

Efforts to develop genetic marker based approaches to breeding forest trees began in the late 1980s. Approaches based on first generation of markers, allozymes, were not feasible due to the very limited number of markers (<50). The first DNA-based markers, RFLPs, brought more hope as moderately dense genetic maps could be constructed to scan the genome and map quantitative trait loci (QTLs). This approach was quite effective toward mapping QTLs in many forest tree species but the approach could not be brought to application in tree breeding due to low levels of linkage disequilibrium (LD) in forest tree breeding populations and recombination between flanking markers and QTLs with each generation. The next generation of DNA markers based of the polymerase chain reaction (PCR), RAPD, AFLP and SSR, did not solve the LD and recombination problem, even though more markers were available and throughput increased. The situation began to change in the early 2000s with the availability of automated DNA sequencing technology and single nucleotide polymorphism (SNP) genetic markers. Now association studies could be performed where SNPs within candidate genes controlling complex traits could be identified and thus “solving” or minimizing the LD and recombination limitation. This approach to complex trait dissection has been widely applied in forest trees and the early approach of QTL mapping in segregating populations has been mostly abandoned. The association genetic approach has been used to find candidate gene SNPs associated to a broad array of quantitative traits of interest (wood properties, growth, abiotic stresses and disease resistance). However, like QTL mapping before, individual SNP x trait associations only account for a small proportion of the variation (generally less than 2-3% of the total phenotypic variance) and the total variation explained by all markers is generally less than 50%. In human genetics, this situation is called the “missing heritability” and there has been great debate over whether this problem can ever be solved in complex trait dissection in humans. In a few agricultural systems however, notably dairy cattle, researchers are now accounting for nearly all the heritable variation. Efforts to realize genomics-based breeding in conifer tree improvement in the US have culminated under a collaborative research project funded by the USDA. The Conifer Translational Genomics Network Coordinated Agricultural Project (CTGN CAP) has successfully brought genomics-based breeding to application in the four major cooperative forest tree breeding programs in the US. The project has completed the basic research on allele discovery of economic traits in conifers, conducted the translational research, and provided education and training to tree breeders to realize this goal during the four-year project period (2007 to 2011). Genomics-based breeding can now be applied to the production of over 1.3 billion loblolly pine, slash pine and Douglas-fir seedlings planted annually in the US. The CTGN CAP obtained high-density single nucleotide polymorphic (SNP) genotypes (over 7000 SNPs) on nearly 10,000 loblolly pine, slash pine, and Douglas-fir trees from the breeding cooperatives. These data were used to estimate molecular breeding values that can now be used to make selections as an alternative to breeding values based on mature traits that can take many years to the time of evaluation and can be expensive to measure in field tests. The CTGN CAP was the first to apply the advanced Illumina Infininium SNP-genotyping technology that is now being used in the breeding of most major crops such as corn, wheat, tomato, potato, and many more. The CTGN CAP culminates a long-term investment by the USDA in this team of researchers and their work on the technological advancement of tree breeding in the US. USDA’s efforts began more than 20 years ago with funding of a series of grants under the National Research Initiative Plant Genome Program and continued under the CSREES IFAFS and AFRI programs. Recently, NIFA has awarded a grant to members of this team to conduct full genome sequencing in loblolly pine, sugar pine, and Douglas-fir. The full genome sequence will soon allow tree breeders to practice genomics-based breeding using genetic variation from the entire genome (genomic selection), and thus minimizing or even eliminating the missing heritability problem in forest trees. The partnership between USDA research programs and this team of researchers has resulted in a modernization of tree breeding technology in the US that will secure US competitiveness in the production of forest products.

Convex Recoloring of Paths

Abstract Let ( T , C ) be a pair consisting of a tree T and a coloring C of its vertices. We say that C is a convex coloring if, for each color, the vertices in T with the same color induces a subtree of T. The convex recoloring problem (of trees) is defined as follows. Given a pair ( T , C ), find a recoloring of a minimum number of vertices of T such that the resulting coloring is convex. This problem is known to be NP-hard [Shlomo Moran and Sagi Snir. Convex recolorings of strings and trees: definitions, hardness results and algorithms. J. Comput. System Sci., 74(5):850–869, 2008]. It was motivated by problems on phylogenetic trees [Shlomo Moran and Sagi Snir. Convex recolorings of strings and trees: definitions, hardness results and algorithms. J. Comput. System Sci., 74(5):850–869, 2008]. We investigate here the convex recoloring problem on paths, denoted here CRP. We present an integer programming formulation for CRP and show some computational results obtained by exploring this formulation. We also present some preliminary results of a heuristic that is based on this formulation. Furthermore, we investigate a special case of CRP, denoted here 2-CRP, restricted to paths in which the number of vertices of each color is at most 2, a problem known to be NP-hard [Iyad Kanj and Dieter Kratsch. Convex recoloring revisited: Complexity and exact algorithms. In Algorithms and data structures, volume 5609 of Lect. Notes in Comput. Sci., pages 388–397. Springer, Berlin, 2009]. The best approximation result for CRP was obtained by Moran and Snir [Shlomo Moran and Sagi Snir. Efficient approximation of convex recolorings. J. Comput. System Sci., 73(7):1078–1089, 2007], who showed a 2-approximation algorithm. We show in this paper a 3/2-approximation algorithm for 2-CRP.

The algorithmic complexity of mixed domination in graphs

Let G = ( V , E ) be a simple graph with vertex set V and edge set E . A subset W ⊆ V ∪ E is a mixed dominating set if every element x ∈ ( V ∪ E ) ∖ W is either adjacent or incident to an element of W . The mixed domination problem is to find a minimum mixed dominating set of G . In this paper we first prove that a connected graph is a tree if and only if its total graph is strongly chordal, and thus we obtain a polynomial-time algorithm for this problem in trees. Further we design another linear-time labeling algorithm for this problem in trees. At the end of the paper, we show that the mixed domination problem is NP-complete even when restricted to split graphs, a subclass of chordal graphs.

Applying the DISCOVER Model to Nurture Creativity in Pre-Service Early Childhood Teachers

The purpose of this study was to illustrate how professors can motivate pre-service early childhood teachers to be more creative using the Discovering Intellectual Strengths and Capabilities while Observing Varied Ethnic Responses (DISCOVER) Model (Maker, 2005; Maker &amp; Schiever, 2005; Schiever &amp; Maker, 1991, 1997). The participants were 50 pre-service early childhood teachers from a private rural college in MiaoLi County, Taiwan. They were enrolled in a Children's Literature course in the Department of Child Care and Education. The picture book was used to increase the engagement and motivation of the participants through the DISCOVER Model's six problem-solving types. Tree choice, authenticity, problem solving, and creative ability were factors included in every task. The participants' creative work demonstrated the effectiveness of the method. Pedagogic ideas and methods for integrating the model into higher education are included in the recommendations.

Dominating tree construction in wireless networks using all-pair shortest paths in graph

The dominating tree (DT) problem has engineering value in wireless network design and network routing. A novel algorithm for the DT problem is presented. Simulation results confirm the effectiveness of the proposed method.

An inverse approach to convex ordered median problems in trees

The convex ordered median problem is a generalization of the median, the k-centrum or the center problem. The task of the associated inverse problem is to change edge lengths at minimum cost such that a given vertex becomes an optimal solution of the location problem, i.e., an ordered median. It is shown that the problem is NP-hard even if the underlying network is a tree and the ordered median problem is convex and either the vertex weights are all equal to 1 or the underlying problem is the k-centrum problem. For the special case of the inverse unit weight k-centrum problem a polynomial time algorithm is developed.

A Distributed Algorithm for Min-Max Tree and Max-Min Cut Problems in Communication Networks

We consider the problem of finding a multicast tree rooted at the source node and including all the destination nodes such that the maximum weight of the tree arcs is minimized. It is of paramount importance for many optimization problems, e.g., the maximum-lifetime multicast problem in multihop wireless networks, in the data networking community. We explore some important properties of this problem from a graph theory perspective and obtain a min-max-tree max-min-cut theorem, which provides a unified explanation for some important while separated results in the recent literature. We also apply the theorem to derive an algorithm that can construct a global optimal min-max multicast tree in a distributed fashion. In random networks with n nodes and m arcs, our theoretical analysis shows that the expected communication complexity of our distributed algorithm is in the order of O(m). Specifically, the average number of messages is 2(n - 1 - γ) -2 ln(n-1) + m at most, in which γ is the Euler constant. To our best knowledge, this is the first contribution that possesses the distributed and scalable properties for the min-max multicast problem and is especially desirable to the large-scale resource-limited multihop wireless networks, like sensor networks.

New common ancestor problems in trees and directed acyclic graphs

We derive a new generalization of lowest common ancestors (LCAs) in dags, called the lowest single common ancestor (LSCA). We show how to preprocess a static dag in linear time such that subsequent LSCA-queries can be answered in constant time. The size is linear in the number of nodes. We also consider a “fuzzy” variant of LSCA that allows to compute a node that is only an LSCA of a given percentage of the query nodes. The space and construction time of our scheme for fuzzy LSCAs is linear, whereas the query time has a sub-logarithmic slow-down. This “fuzzy” algorithm is also applicable to LCAs in trees, with the same complexities.

Review

132 Michigan Historical Review John Gaertner. The Duluth, South Shore & Atlantic Railway: A History of the Lake Superior District’s Pioneer Iron Ore Hauler. Bloomington: Indiana University Press, 2009. Pp. 349. Appendix. Bibliography. Illustrations. Index. Maps. Notes. Cloth, $49.95. Formed in 1887 from earlier short-line railroads, the Duluth, South Shore & Atlantic Railway (DSS&A) served Michigan’s Upper Peninsula and northern Wisconsin until it was merged into the Soo Line in 1961. At its most extensive, it ran from Duluth, Minnesota, through northern Wisconsin to Sault Ste. Marie and St. Ignace, Michigan. The latter towns were as close as it ever got to the Atlantic seaboard. The road was controlled through nearly all of its history by the Canadian Pacific Railroad, which treated it as something of a stepchild. As the book’s subtitle indicates, the DSS&A served the Upper Peninsula’s iron fields but it also carried the region’s timber products and passengers as well. Competing with other roads for the traffic of a thinly populated territory, it seldom made money. When the area’s iron and timber resources became depleted and highways increasingly cut into passenger and freight traffic, the road frequently trimmed its services and trackage. It never became part of an intercontinental trunk line as its promoters occasionally hoped. The Duluth, South Shore & Atlantic Railway is a business history written from management’s point of view. Labor and community relations are seldom mentioned. The text, moreover, is exceedingly dense. John Gaertner has done a more than commendable job of research into the road’s records and related sources. He has produced a reference book that no one should ever need to duplicate. But there is a serious forest vs. trees problem: seventy-four years of relentless operational detail are so rarely leavened by summary or evaluation that the book is a burden to read. Of great help, on the other hand, are the maps amply distributed throughout the book, something that cannot be said of every railroad history. There are many illustrations as well, primarily of rolling stock. And finally there is an appendix containing for the years 1886 to 1960 a complete locomotive roster; a fifteen-page list of all stations with the years in which they operated; and a year-by-year list of operating statistics—including miles covered, earnings, expenses, passengers carried, and freight tonnage moved. John Gaertner has done this part of his work very well indeed. Allen W. Trelease University of North Carolina at Greensboro, Emeritus ...

An evolutionary creative design approach for optimising the broadcasting trees in MANET

Finding the optimal broadcasting tree is a challenging problem in networks like mobile add-hoc network (MANET). Finding the optimal tree faces engineering problems such as the broadcasting storm problem and the complexity of computations. Therefore, several heuristic approaches have been introduced in order to find a sub-optimal solution. In this work, the creative design approach was adopted in order to provide means to tackle this problem by searching for a structure of a suitable spanning tree. This structure can then be optimised in order to meet the performance indexes related to the broadcasting problem. The proposed model was compared with three main algorithms: simple flooding (SF), counter base scheme (CBS) and minimum spanning tree (MST). The results showed that the proposed approach could create a near optimal solution for the minimum spanning tree (MST) problem at reasonable and reliable computation time. Additionally, it has lower redundancy and the broadcasting overhead than the simple flooding and the counter base scheme.

The weight-constrained maximum-density subtree problem and related problems in trees

Given a tree T =( V , E ) of n nodes such that each node v is associated with a value-weight pair ( val v , w v ), where value val v is a real number and weight w v is a non-negative integer, the d...

Disjoint paths in sparse graphs

We generalize all the results obtained for maximum integer multiflow and minimum multicut problems in trees by Garg, Vazirani and Yannakakis [N. Garg, V.V. Vazirani, M. Yannakakis, Primal-dual approximation algorithms for integral flow and multicut in trees, Algorithmica 18 (1997) 3–20] to graphs with a fixed cyclomatic number, while this cannot be achieved for other classical generalizations of trees. We also introduce the k -edge-outerplanar graphs, a class of planar graphs with arbitrary (but bounded) tree-width that generalizes the cacti, and show that the integrality gap of the maximum edge-disjoint paths problem is bounded in these graphs.

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

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.

Efficient list cost coloring of vertices and/or edges of bounded cyclicity graphs

We consider a list cost coloring of vertices and edges in the model of vertex, edge, total and pseudototal coloring of graphs. We use a dynamic programming approach to derive polynomial-time algorithms for solving the above problems for trees. Then we generalize this approach to arbitrary graphs with bounded cyclomatic numbers and to their multicolorings.

A local search heuristic for bounded-degree minimum spanning trees

The bounded-degree minimum spanning tree (BDMST) problem has many practical applications. Unlike the unbounded case, the BDMST problem is NP-hard, and many attempts have been made to devise good approximation methods, including evolutionary algorithms. Inspired by recent applications to wireless communications, the present article focuses on the geometric version of the problem, i.e. the weights assigned to links (u, v) are equal to the Euclidean distance between u and v, but no grid geometry is used as an underlying structure. The proposed genetic local search procedure for BDMST-approximations utilizes a specific edge crossover operation, and the local search in-between applications of crossover performs alternating sequences of descending and ascending steps for each individual of the population. The length of a sequence with uniform direction is controlled by the estimated value of the maximum depth of local minima of the associated fitness landscape. The computational experiments were executed on ten synthetic networks, and a comparison to two recently published BDMST algorithms is presented.

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

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.

A SOLUTION OF REAL-WORLD OCST PROBLEMS WITH A NEW TREE ENCODING-BASED GENETIC ALGORITHM

For embodying and implementing the ubiquitous computing environments, the recent increase in concern for communication systems has been leading the evolution of the theory and technology of the related fields. Among these network-related problems, the minimum spanning tree (MST) problems have many applications in the communication systems field, because they are solved as one of the conventional combinatorial optimization problems. In particular, the optimal communication spanning tree (OCST) problem is a famous application of the MST problem. The OCST problem is defined by finding a spanning tree that connects all nodes and satisfies their communication requirements for connecting all nodes. This OCST problem can be applied to many network optimization fields, such as network topology design, multicast tree configuration, and ad-hoc network or ubiquitous sensor network routing, etc. This paper presents a genetic algorithm with a new encoding method, which is based on the Prüfer number (PN) and a clustering string, for solving the OCST problems. We will also demonstrate that the efficiency and effectiveness of our proposed method can be shown by several experimental results, employing the proposed method as the solution method of the OCST problems.