Incremental Planar Nearest Neighbor Queries with Optimal Query Time

In this paper we show that two-dimensional nearest neighbor queries can be answered in optimal $O(\log n)$ time while supporting insertions in $O({\log }^{1+\epsilon }n)$ time. No previous data structure was known that supports $O(\log n)$-time queries and polylog-time insertions. In order to achieve logarithmic queries our data structure uses a new technique related to fractional cascading that leverages the inherent geometry of this problem. Our method can be also used in other semi-dynamic scenarios.