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Kruskal's tree theorem then states: If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T 1, T 2, … of rooted trees labeled in X, there is some < so that .)
Kruskal's algorithm [1] finds a minimum spanning forest of an undirected edge-weighted graph.If the graph is connected, it finds a minimum spanning tree.It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle. [2]
The article as written is mostly about the TREE() function, and describes barely more of Kruskal's tree theorem than is needed to explain the TREE() function's terminology. Even the subsection describing the weak tree() function fails to explain who defined it or for what purpose; is it any more relevant to Kruskal's trees than the rest of this? G.
Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system ATR 0 codifying the principles acceptable based on a philosophy of mathematics called predicativism. [8]
Kruskal's tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on basis of a philosophy of mathematics called predicativism.
Kruskal's tree theorem states that, in every infinite set of finite trees, there exists a pair of trees one of which is homeomorphically embedded into the other; another way of stating the same fact is that the homeomorphisms of trees form a well-quasi-ordering.
The little Bernshtein theorem: A function that is absolutely monotonic on a closed interval [,] can be extended to an analytic function on the interval defined by | | <. A function that is absolutely monotonic on [ 0 , ∞ ) {\displaystyle [0,\infty )} can be extended to a function that is not only analytic on the real line but is even the ...
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]