Finding the Maximal Empty Rectangle Containing a Query Point

Let $P$ be a set of $n$ points in an axis-parallel rectangle $B$ in the plane. We present an $O(n\alpha (n){\log }^{4}n)$-time algorithm to preprocess $P$ into a data structure of size $O(n\alpha (n){\log }^{3}n)$, such that, given a query point $q$, we can find, in $O({\log }^{4}n)$ time, the largest-area axis-parallel rectangle that is contained in $B$, contains $q$, and its interior contains no point of $P$. This is a significant improvement over the previous solution of Augustine {\em et al.} \cite{qmex}, which uses slightly superquadratic preprocessing and storage.