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Let $P$ be a set of $n$ points in an axis-parallel rectangle $B$ in theplane. 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 querypoint $q$, we can find, in $O(\log^4 n)$ time, the largest-area axis-parallelrectangle that is contained in $B$, contains $q$, and its interior contains nopoint of $P$. This is a significant improvement over the previous solution ofAugustine {\em et al.} \cite{qmex}, which uses slightly superquadraticpreprocessing and storage.

Finding the Maximal Empty Rectangle Containing a Query Point