Group Nearest Neighbor Queries in the L ₁ Plane

Abstract

Let P be a set of n points in the plane. The k-nearest neighbor (k-NN) query problem is to preprocess P into a data structure that quickly reports k closest points in P for a query point q. This paper addresses a generalization of the k-NN query problem to a query set Q of points, namely, the group nearest neighbor problem, in the L ₁ plane. More precisely, a query is assigned with a set Q of at most m points and a positive integer k with k ≤ n, and the distance between a point p and a query set Q is determined as the sum of L ₁ distances from p to all q ∈ Q. The maximum number m of query points Q is assumed to be known in advance and to be at most n; that is, m ≤ n. In this paper, we propose two methods, one based on the range tree and the other based on the segment dragging query, obtaining the following complexity bounds: (1) a group k-NN query can be handled in O(m ²logn + k(loglogn + logm)) time after preprocessing P in O(m ² n log² n) time and space, or (2) a query can be handled in O(m ²logn + (k + m)log² n) time after preprocessing P in O(m ² nlogn) time using O(m ² n) space. We also show that our approach can be applied to the group k-farthest neighbor query problem.

This research was supported by NRF grant 2011-0030044 (SRC-GAIA) funded by the government of Korea.