Abstract
The notion of hypergraph cyclicity is important in numerous fields of application of hypergraph theory in computer science and relational database theory. The database scheme and query can be represented as a hypergraph. The database scheme (or query) has a cycle if the corresponding hypergraph has a cycle. An Acyclic database has several desired computational properties such as making query optimization easier and can be recognized in linear time. In this paper, we introduce a new type of cyclicity in hypergraphs via the notions of Quasi $\alpha $ -cycle(s) and the set of $\alpha $ -nodes in hypergraphs, which are based on the existence of an $\alpha $ –cycle(s). Then, it is proved that a hypergraph is acyclic if and only if it does not contain any $\alpha $ -nodes. Moreover, a polynomial-time algorithm is proposed to detect the set of $\alpha $ -nodes based on the existence of Quasi $\alpha $ -cycle(s), or otherwise claims the acyclicity of the hypergraph. Finally, a systematic discussion is given to show how to use the detected set of $\alpha $ -nodes to convert the cyclic hypergraph into acyclic one if the conversion is possible. The acyclic database and acyclic query enjoy time and/or space-efficient access paths for answering a query.
Highlights
In the last decades, a class of ‘‘acyclic’’ database scheme and different degrees of acyclicity has been introduced [1]
Each attribute of a database scheme R corresponds to a node in a hypergraph H and each relation scheme R in R corresponds to an edge in H [3], [4]
It is enough to detect the set of α–nodes to convert the cyclic hypergraph into acyclic one instead of detecting the α-cycle(s), that requires checking all permutations which is an NP problem
Summary
A class of ‘‘acyclic’’ database scheme and different degrees of acyclicity has been introduced [1]. Codd [2] has defined a relational database scheme as a collection of table skeletons (a set of subsets of the attributes, which are the column names of the database tables) These tables can be represented as hypergraphs. Due to the great importance of the acyclic database scheme, Graham [18] and Yu and Ozsoyoglu [19] have introduced a polynomial-time algorithm for detecting the acyclicity of hypergraphs that is known in the literature as Graham or the GYO algorithm. A polynomial-time algorithm is proposed to detect the set of α-nodes based on the existence of Quasi αcycle(s) or otherwise claims that the input hypergraph is an α-acyclic.
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