Semistructured data comprises irregular structure and has no a-priori database schema. therefore we encounter several problems such as inefficient data retrieval and wasteful data storage. To cope with such problems. some schema extraction algorithms over semistructured data have been proposed, ill which data is modeled as all unordered, tree. However, the order of elements, is indispensable for document data, therefore we consider extracting an optimal database schema over an ordered free. We consider all optimization problem to extract a smallest database schema such that the density of each class is no less than a given threshold, where the density of a class represents a similarity between the type of the class and those of the objects in the class. We first prove that the corresponding decision problem is strongly NP-complete, and show that another version of the, problem is strongly NP-hard and belongs to Delta(2)P. Then we show that for any r < 3/2; there is no polynomial-time r-approximation algorithm that solves the optimization problem unless P = NP. Finally, we propose a kind of class called bounded class that can be constructed efficiently, then show a polynomial-time algorithm for constructing a database schema by using bounded classes.
