On Top-$k$ Weighted SUM Aggregate Nearest and Farthest Neighbors in the $L_1$ Plane

In this paper, we present algorithms for the top-k nearest neighbor searching where the input points are exact and the query point is uncertain under the L1 metric in the plane. The uncertain query point is represented by a discrete probability distribution function, and the goal is to efficiently return the top-k expected nearest neighbors, which have the smallest expected distances to the query point. Given a set of n exact points in the plane, we build an O(n logn log log n)-size data structure in O(n logn log logn) time, such that for any uncertain query point with m possible locations and any integer k with 1 ≤ k ≤ n, the top-k expected nearest neighbors can be found in O(m logm + (k + m) log n) time. Even for the special case where k = 1, our result is better than the previously best method (in PODS 2012), which requires O(n log n) preprocessing time, O(n log n) space, and O(m log n) query time. In addition, for the one-dimensional version of this problem, our approach can build an O(n)-size data structure in O(n logn) time that can support O(min{k, logm} ·m+ k+ log n) time queries and the query time can be reduced to O(k+m+ logn) time if the locations of Q are given sorted. In fact, the problem is equivalent to the aggregate or group nearest neighbor searching with the weighted Sum as the aggregate distance function operator.