Group Nearest Neighbor Queries in the L 1 Plane

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 L1 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 L1 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 log n + k(log log n + logm)) time after preprocessing P in O(mn log n) time and space, or (2) a query can be handled in O(m logn + (k +m) log n) time after preprocessing P in O(mn log n) time using O(mn) space. We also show that our approach can be applied to the group k-farthest neighbor query problem.