In probability theory, Bayesian statistics, artificial intelligence and database theory the minimum cross-entropy principle is often used to estimate a distribution with a given set P of marginal distributions under the proportionality assumption with respect to a given "prior'' distribution q. Such an estimation problem admits a solution if and only if there exists an extension of P that is dominated by q. In this paper we consider the case that q is not given explicitly, but is specified as the maximum-entropy extension of an auxiliary set Q of distributions. There are three problems that naturally arise: (1) the existence of an extension of a distribution set (such as P and Q), (2) the existence of an extension of P that is dominated by the maximum entropy extension of Q, (3) the numeric computation of the minimum cross-entropy extension of P with respect to the maximum entropy extension of Q. In the spirit of a divide-and-conquer approach, we prove that, for each of the three above-mentioned problems, the global solution can be easily obtained by combining the solutions to subproblems defined at node level of a suitable tree.
A new kind of a deterministic pushdown automaton, called a \emph{Tree Compression Automaton}, is presented. The tree compression automaton represents a complete compressed index of a set of trees for subtrees and accepts all subtrees of given trees. The algorithm for constructing our pushdown automaton is incremental. For a single tree with n nodes, the automaton has at most n+1 states, its transition function cardinality is at most 4n and there are 2n+1 pushdown store symbols. If hashing is used for storing automaton's transitions, thus removing a factor of logn, the construction of the automaton takes linear time and space with respect to the length n of the input tree(s). Our pushdown automaton construction can also be used for finding all subtree repeats without augmenting the overall complexity.
In this work we deal with tree pattern matching over ranked trees, where the pattern set to be matched against is defined by a regular tree expression. We present a new method that uses a tree automaton constructed inductively from a regular tree expression. First we construct a special tree automaton for the regular tree expression of the pattern E, which is somehow a generalization of Thompson automaton for strings. Then we run the constructed automaton on the subject tree t. The pattern matching algorithm requires an O(|t||E|) time complexity, where |t| is the number of nodes of t and |E| is the size of the regular tree expression E. The novelty of this contribution besides the low time complexity is that the set of patterns can be infinite, since we use regular tree expressions to represent patterns.