Abstract: We consider the problem of query optimization in the presence of limitations on access patterns to the data (i.e., when one must provide values for one of the attributes of a relation in order to obtain tuples). We show that in the presence of limited access patterns we must search a space of annotated query plans, where the annotations describe the inputs that must be given to the plan. We describe a theoretical and experimental analysis of the resulting search space and a novel query optimization algorithm that is designed to perform well under the different conditions that may arise. The algorithm searches the set of annotated query plans, pruning invalid and non-viable plans as early as possible in the search space, and it also uses a best-first search strategy in order to produce a first complete plan early in the search. We describe experiments to illustrate the performance of our algorithm.