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Tree Decomposition
In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph. Tree decompositions are also called junction trees, clique trees, or join trees. They play an important role in problems like probabilistic inference, constraint satisfaction, query optimization, and matrix decomposition. The concept of tree decomposition was originally introduced by . Later it was rediscovered by and has since been studied by many other authors. Definition Intuitively, a tree decomposition represents the vertices of a given graph as subtrees of a tree, in such a way that vertices in are adjacent only when the corresponding subtrees intersect. Thus, forms a subgraph of the intersection graph of the subtrees. The full intersection graph is a chordal graph. Each subtree associates a graph vertex with a set of tree nodes. To define this formally, we represent each t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree Decomposition
In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph. Tree decompositions are also called junction trees, clique trees, or join trees. They play an important role in problems like probabilistic inference, constraint satisfaction, query optimization, and matrix decomposition. The concept of tree decomposition was originally introduced by . Later it was rediscovered by and has since been studied by many other authors. Definition Intuitively, a tree decomposition represents the vertices of a given graph as subtrees of a tree, in such a way that vertices in are adjacent only when the corresponding subtrees intersect. Thus, forms a subgraph of the intersection graph of the subtrees. The full intersection graph is a chordal graph. Each subtree associates a graph vertex with a set of tree nodes. To define this formally, we represent each t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haven (graph Theory)
In graph theory, a haven is a certain type of function on sets of vertices in an undirected graph. If a haven exists, it can be used by an evader to win a pursuit–evasion game on the graph, by consulting the function at each step of the game to determine a safe set of vertices to move into. Havens were first introduced by as a tool for characterizing the treewidth of graphs. Their other applications include proving the existence of small separators on minor-closed families of graphs, and characterizing the ends and clique minors of infinite graphs... Definition If is an undirected graph, and is a set of vertices, then an -flap is a nonempty connected component of the subgraph of formed by deleting . A haven of order in is a function that assigns an -flap to every set of fewer than vertices. This function must also satisfy additional constraints which are given differently by different authors. The number is called the ''order'' of the haven.. In the original defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trees (graph Theory)
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are usable as lumber or plants above a specified height. In wider definitions, the taller palms, tree ferns, bananas, and bamboos are also trees. Trees are not a taxonomic group but include a variety of plant species that have independently evolved a trunk and branches as a way to tower above other plants to compete for sunlight. The majority of tree species are angiosperms or hardwoods; of the rest, many are gymnosperms or softwoods. Trees tend to be long-lived, some reaching several thousand years old. Trees have been in existence for 370 million years. It is estimated that there are some three trillion mature trees in the world. A tree typically has many secondary branches supported clear of the ground by the trunk. This trunk typically ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decomposition Method (constraint Satisfaction)
In constraint satisfaction, a decomposition method translates a constraint satisfaction problem into another constraint satisfaction problem that is binary and acyclic. Decomposition methods work by grouping variables into sets, and solving a subproblem for each set. These translations are done because solving binary acyclic problems is a tractable problem. Each structural restriction defined a measure of complexity of solving the problem after conversion; this measure is called ''width''. Fixing a maximal allowed width is a way for identifying a subclass of constraint satisfaction problems. Solving problems in this class is polynomial for most decompositions; if this holds for a decomposition, the class of fixed-width problems form a tractable subclass of constraint satisfaction problems. Overview Decomposition methods translate a problem into a new one that is easier to solve. The new problem only contains binary constraints; their scopes form a directed acyclic graph. The va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch-decomposition
In graph theory, a branch-decomposition of an undirected graph ''G'' is a hierarchical clustering of the edges of ''G'', represented by an unrooted binary tree ''T'' with the edges of ''G'' as its leaves. Removing any edge from ''T'' partitions the edges of ''G'' into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of any pair of subgraphs formed in this way. The branchwidth of ''G'' is the minimum width of any branch-decomposition of ''G''. Branchwidth is closely related to tree-width: for all graphs, both of these numbers are within a constant factor of each other, and both quantities may be characterized by forbidden minors. And as with treewidth, many graph optimization problems may be solved efficiently for graphs of small branchwidth. However, unlike treewidth, the branchwidth of planar graphs may be computed exactly, in polynomial time. Branch-decompositions and branchwidth may also be generalized from graphs to matroids. Defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haven (graph Theory)
In graph theory, a haven is a certain type of function on sets of vertices in an undirected graph. If a haven exists, it can be used by an evader to win a pursuit–evasion game on the graph, by consulting the function at each step of the game to determine a safe set of vertices to move into. Havens were first introduced by as a tool for characterizing the treewidth of graphs. Their other applications include proving the existence of small separators on minor-closed families of graphs, and characterizing the ends and clique minors of infinite graphs... Definition If is an undirected graph, and is a set of vertices, then an -flap is a nonempty connected component of the subgraph of formed by deleting . A haven of order in is a function that assigns an -flap to every set of fewer than vertices. This function must also satisfy additional constraints which are given differently by different authors. The number is called the ''order'' of the haven.. In the original defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bramble (graph Theory)
In graph theory, a bramble for an undirected graph is a family of connected subgraphs of that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in that has one endpoint in each subgraph. The ''order'' of a bramble is the smallest size of a hitting set, a set of vertices of that has a nonempty intersection with each of the subgraphs. Brambles may be used to characterize the treewidth of .. In this reference, brambles are called "screens" and their order is called "thickness". Treewidth and havens A haven of order ''k'' in a graph ''G'' is a function ''β'' that maps each set ''X'' of fewer than ''k'' vertices to a connected component of ''G'' − ''X'', in such a way that every two subsets ''β''(''X'') and ''β''(''Y'') touch each other. Thus, the set of images of ''β'' forms a bramble in ''G'', with order ''k''. Conversely, every bramble may be used to determine a haven: for each set ''X'' of size smaller ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Junction Tree Algorithm
The junction tree algorithm (also known as 'Clique Tree') is a method used in machine learning to extract marginalization in general graphs. In essence, it entails performing belief propagation on a modified graph called a junction tree. The graph is called a tree because it branches into different sections of data; nodes of variables are the branches. The basic premise is to eliminate cycles by clustering them into single nodes. Multiple extensive classes of queries can be compiled at the same time into larger structures of data. There are different algorithms to meet specific needs and for what needs to be calculated. Inference algorithms gather new developments in the data and calculate it based on the new information provided. Junction tree algorithm Hugin algorithm * If the graph is directed then moralize it to make it un-directed. *Introduce the evidence. *Triangulate the graph to make it chordal. *Construct a junction tree from the triangulated graph (we will call t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Machine Learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data, known as training data, in order to make predictions or decisions without being explicitly programmed to do so. Machine learning algorithms are used in a wide variety of applications, such as in medicine, email filtering, speech recognition, agriculture, and computer vision, where it is difficult or unfeasible to develop conventional algorithms to perform the needed tasks.Hu, J.; Niu, H.; Carrasco, J.; Lennox, B.; Arvin, F.,Voronoi-Based Multi-Robot Autonomous Exploration in Unknown Environments via Deep Reinforcement Learning IEEE Transactions on Vehicular Technology, 2020. A subset of machine learning is closely related to computational statistics, which focuses on making predicti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
