Constructing Optimal Bushy Trees Possibly Containing Cross Products for Order Preserving Joins is in P

One of the main features of XQuery compared to traditional query languages like SQL, is that it preserves the input order—unless specified otherwise. As a consequence, order-preserving algebraic operators are needed to capture the semantics of XQuery correctly. One important algebraic operator is the orderpreserving join. The order-preserving join is associative but, in contrast to the traditional join operator, not commutative. Since join ordering (i.e. finding the optimal execution plan for a given set of join operators) has been an important topic of query optimization for SQL, it is expected that it will also play a major role in optimizing XQuery. The search space for ordering traditional joins is exponential in size. Although the lack of commutativity reduces the search space for ordering order-preserving joins, we show that it is still exponential. This raises the question whether the join ordering problem is also NP-hard, as in the traditional setting. We answer this question by introducing the first polynomial algorithm that produces optimal bushy trees possibly containing cross products.