Range-Aggregate Proximity Queries

In a range-aggegate query problem we wish to preprocess a set S of geometric objects such that given a query orthogonal range q, a certain intersection or proximity query on the objects of S intersected by q can be answered efficiently. Although range-aggregate queries have been widely investigated in the past for aggregation functions like average, count, min, max, sum etc. there is little work on proximity problems. In this paper, we solve two problems. We first consider the problem of determining if any pair of points in a query orthogonal rectangle are within a constant λ of each other and give a solution that takes O(n log n) space and O(log n) query time. Subsequently, we solve the problem of finding the closest pair in a query orthogonal rectangle which takes O(n log n) space and O(log n) query time.