Aggregate-Max Nearest Neighbor Searching in the Plane

TL;DR

This paper introduces efficient data structures for aggregate maximum nearest neighbor queries in the plane under L1 and L2 distances, improving upon previous heuristics and approximations.

Contribution

It presents novel exact data structures for MAX nearest neighbor queries with optimal or near-optimal complexities for both L1 and L2 metrics.

Findings

L1 data structure answers queries in O(m+log n) time

L2 data structure achieves O(m√n log^O(1) n) query time

Top-k queries are answered efficiently with O(n) space and O(n log n) preprocessing

Abstract

We study the aggregate/group nearest neighbor searching for the MAX operator in the plane. For a set $P$ of $n$ points and a query set $Q$ of $m$ points, the query asks for a point of $P$ whose maximum distance to the points in $Q$ is minimized. We present data structures for answering such queries for both $L_1$ and $L_2$ distance measures. Previously, only heuristic and approximation algorithms were given for both versions. For the $L_1$ version, we build a data structure of O(n) size in $O(n\log n)$ time, such that each query can be answered in $O(m+\log n)$ time. For the $L_2$ version, we build a data structure in $O(n\log n)$ time and $O(n\log \log n)$ space, such that each query can be answered in $O(m\sqrt{n}\log^{O(1)} n)$ time, and alternatively, we build a data structure in $O(n^{2+\epsilon})$ time and space for any $\epsilon>0$, such that each query can be answered in $O(m\log n)$ time. Further, we extend our result for the $L_1$ version to the top-$k$ queries where each query asks for the $k$ points of $P$ whose maximum distances to $Q$ are the smallest for any $k$ with $1\leq k\leq n$: We build a data structure of O(n) size in $O(n\log n)$ time, such that each top-$k$ query can be answered in $O(m+k\log n)$ time.