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Then any maximum-weight spanning tree of the clique graph is a junction tree. So, to construct a junction tree we just have to extract a maximum weight spanning tree out of the clique graph. This can be efficiently done by, for example, modifying Kruskal's algorithm. The last step is to apply belief propagation to the obtained junction tree. [10]
The tree decomposition of a graph is far from unique; for example, a trivial tree decomposition contains all vertices of the graph in its single root node. A tree decomposition in which the underlying tree is a path graph is called a path decomposition, and the width parameter derived from these special types of tree decompositions is known as ...
An example constraint satisfaction problem; this problem is binary, and the constraints are represented by edges of this graph. A decomposition tree; for every edge of the original graph, there is a node that contains both its endpoints; all nodes containing a variable are connected Solving a subproblem.
1 Relation to junction trees. 4 comments. 2 The recursion for Independent Set. 1 comment. 3 Books? 1 comment. 4 Treewidth of trees. 2 comments. 5 junction tree. 3 ...
It is a key step of the junction tree algorithm, used in belief propagation on graphical models. A directed acyclic graph The corresponding moral graph, with newly added arcs shown in red
Chow and Liu provide a simple algorithm for constructing the optimal tree; at each stage of the procedure the algorithm simply adds the maximum mutual information pair to the tree. See the original paper, Chow & Liu (1968), for full details. A more efficient tree construction algorithm for the common case of sparse data was outlined in Meilă ...
A simple alternative to the above algorithm uses chain decompositions, which are special ear decompositions depending on DFS-trees. [3] Chain decompositions can be computed in linear time by this traversing rule. Let C be a chain decomposition of G. Then G is 2-vertex-connected if and only if G has minimum degree 2 and C 1 is the only cycle in C.
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