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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 ...
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 ...
Join follows the right spine of t 1 until a node c which is balanced with t 2. At this point a new node with left child c, root k and right child t 2 is created to replace c. The new node may invalidate the balancing invariant. This can be fixed with rotations. The following is the join algorithms on different balancing schemes. The join ...
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 ...
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In fact in order to answer a level ancestor query, the algorithm needs to jump from a path to another until it reaches the root and there could be Θ(√ n) of such paths on a leaf-to-root path. This leads us to an algorithm that can pre-process the tree in O(n) time and answers queries in O(√ n). In order to reach the optimal query time, we ...
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