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In this paper, we study top-$k$ aggregate (or group) nearest neighbor queriesusing the weighted SUM operator under the $L_1$ metric in the plane. Given aset $P$ of $n$ points, for any query consisting of a set $Q$ of $m$ weightedpoints and an integer $k$, $ 1 \le k \le n$, the top-$k$ aggregate nearestneighbor query asks for the $k$ points of $P$ whose aggregate distances to $Q$are the smallest, where the aggregate distance of each point $p$ of $P$ to $Q$is the sum of the weighted distances from $p$ to all points of $Q$. We build an$O(n\log n\log\log n)$-size data structure in $O(n\log n \log\log n)$ time,such that each top-$k$ query can be answered in $O(m\log m+(k+m)\log^2 n)$time. We also obtain other results with trade-off between preprocessing andquery. Even for the special case where $k=1$, our results are better than thepreviously best method (in PODS 2012), which requires $O(n\log^2 n)$preprocessing time, $O(n\log^2 n)$ space, and $O(m^2\log^3 n)$ query time. Inaddition, for the one-dimensional version of this problem, our approach canbuild an $O(n)$-size data structure in $O(n\log n)$ time that can support$O(\min\{k,\log m\}\cdot m+k+\log n)$ time queries. Further, we extend ourtechniques to the top-$k$ aggregate farthest neighbor queries, with the samebounds.

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