Aggregate-MAX Top-k Nearest Neighbor Searching in the L1 Plane

Abstract

We study the aggregate/group top-k nearest neighbor searching for the Max operator in the plane, where the distances are measured by the L1 metric. Let P be a set of n points in the plane. Given a query set Q of m points, for each point p ∈ P, the aggregate-max distance from p to Q is defined to be the maximum distance from p to all points in Q. Given Q and an integer k with 1 ≤ k ≤ n, the query asks for the k points of P that have the smallest aggregate-max distances to Q. We build a data structure of O(n) size in O(n log n) time, such that each query can be answered in O(m+k log n) time and the k points are reported in sorted order by their aggregate-max distances to Q. Alternatively, we build a data structure of O(n log n) size in O(n log2 n) time that can answer each query in O(m + k + log3 n) time.