A General Technique for Top-$k$ Geometric Intersection Query Problems

In a top-k Geometric Intersection Query (top-k GIQ) problem, a set of n weighted, geometric objects in Rd is to be pre-processed into a compact data structure so that for any query geometric object, q, and integer k > 0, the k largest-weight objects intersected by q can be reported efficiently. While the top-k problem has been studied extensively for non-geometric problems (e.g., recommender systems), the geometric version has received little attention. This paper gives a general technique to solve any top-k GIQ problem efficiently. The technique relies only on the availability of an efficient solution for the underlying (non-top-k) GIQ problem, which is often the case. Using this, asymptotically efficient solutions are derived for several top-k GIQ problems, including top-k orthogonal and circular range search, point enclosure search, halfspace range search, etc. Implementations of some of these solutions, using practical data structures, show that they are quite efficient in practice. This paper also does a formal investigation of the hardness of the top-k GIQ problem, which reveals interesting connections between the top-k GIQ problem and the underlying (non-top-k) GIQ problem.