Algorithms and applications for universal quantification in relational databases

Laws for Rewriting Queries Containing Division Operators

“…However, special applications like frequent itemset discovery could be processed efficiently and formulated more intuitively if division would be a first-class operator. Suppose that an RDBMS offers one or more efficient implementations of division, that is, physical division operators like hash-division or merge-sort division [16,35]. Since division is a derived operator, an optimizer could replace the division operator by an expression that simulates the operator and apply transformation rules on the basic operators in the expression.…”

“…We have recently implemented a collection of physical great divide operators into a Java query execution engine prototype based on the class library XXL [3]. A description of several great divide algorithms together with cost estimations based on input data characteristics (such as grouped or sorted columns in the dividend and divisor) was given in [35]. Future work will assess the effectiveness of the algebraic laws when implemented as transformation rules in a query optimizer.…”

“…We have recently suggested a generalization of division that we called set containment division, denoted by ÷ * 1 , because of its similarity to the set containment join [35]. Let R 1 (A ∪ B), R 2 (B ∪ C), and R 3 (A ∪ C) be relation schemas, where A = {a 1 , .…”

“…However, while the publications on generalized division [12] and great divide [11] solely focus on the relationship between the logical operator and the basic division operator, our previous work on the set containment division operator [33,35] put its emphasis on algorithms that implement physical operators and investigated applications for this operator. In the rest of the paper, we will use Demolombe's term generalized division and use the symbol ÷ * for the operator.…”

Processing frequent itemset discovery queries by division and set containment join operators

“…Since SCD is based on the division operator, as shown by the definition in Section 1.3, many implementations of division come into consideration for realizing SCD depending on the current data characteris- tics, e.g., the data may be grouped or even sorted on certain attributes, as discussed in [20]. The SCD operator that we used for the performance tests is based on the hash-division algorithm [6].…”