Efficient Top-k Indexing via General Reductions

Abstract

Let D be a set of n elements each associated with a real-valued weight, and Q be the set of all possible predicates allowed on those elements. Given a predicate in Q and integer k, a top-k query returns the k elements with the largest weights among the elements of D satisfying q. The corresponding data structure problem aims to store D in small space to allow every query to be answered efficiently. It is already known that, before settling the problem, one must be able to solve two degenerated accompanying problems: (i) prioritized reporting: given a predicate q ∈ Q and a real value τ, return all the elements of D satisfying q and having weights at least τ (ii) max reporting: top-k queries with k fixed to 1. In this paper we prove general reductions in external memory that explore the opposite direction. Our first reduction shows that, (under mild conditions) any prioritized reporting structure yields a static top-$k$ structure with only a slow-down in query time by a factor of O(logB n), where B is the block size. Our second reduction shows that if one additionally has a max reporting structure, then combining the two structures yields a top-k structure with no performance slow down (in space, query, and update) in expectation. These reductions significantly simplify the design of top-k structures, as we showcase on numerous problems including halfspace reporting, circular reporting, interval stabbing, point enclosure, and 3d dominance. All the techniques proposed work directly in the RAM model as well.