Generic Techniques for Building Top-k Structures

Abstract

A reporting query returns the objects satisfying a predicate q from an input set. In prioritized reporting, each object carries a real-valued weight (which can be query dependent), and a query returns the objects that satisfy q and have weights at least a threshold τ. A top-k query finds, among all the objects satisfying q, the k ones of the largest weights; a max query is a special instance with k = 1. We want to design data structures of small space to support queries (and possibly updates) efficiently. Previous work has shown that a top-k structure can also support max and prioritized queries with no performance deterioration. This article explores the opposite direction: do prioritized queries, possibly combined with max queries, imply top-k search? Subject to mild conditions, we provide affirmative answers with two reduction techniques. The first converts a prioritized structure into a static top-k structure with the same space complexity and only a logarithmic blowup in query time. If a max structure is available in addition, our second reduction yields a top-k structure with no degradation in expected performance (this holds for the space, query, and update complexities). Our techniques significantly simplify the design of top-k structures because structures for max and prioritized queries are often easier to obtain. We demonstrate this by developing top-k structures for interval stabbing, 3D dominance, halfspace reporting, linear ranking, and L∞ nearest neighbor search in the RAM and the external memory computation models.