Querying for the Largest Empty Geometric Object in a Desired Location

We study new types of geometric query problems defined as follows: given a geometric set $P$, preprocess it such that given a query point $q$, the location of the largest circle that does not contain any member of $P$, but contains $q$ can be reported efficiently. The geometric sets we consider for $P$ are boundaries of convex and simple polygons, and point sets. While we primarily focus on circles as the desired shape, we also briefly discuss empty rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the results have been improved in Section 5.Please note that the change in title and abstract indicate that we have expanded the scope of the problems we stud

Extensions of the Maximum Bichromatic Separating Rectangle Problem

In this paper, we study two extensions of the maximum bichromatic separating rectangle (MBSR) problem introduced in \cite{Armaselu-CCCG, Armaselu-arXiv}. One of the extensions, introduced in \cite{Armaselu-FWCG}, is called \textit{MBSR with outliers} or MBSR-O, and is a more general version of the MBSR problem in which the optimal rectangle is allowed to contain up to $k$ outliers, where $k$ is given as part of the input. For MBSR-O, we improve the previous known running time bounds of $O(k^7 m \log m + n)$ to $O(k^3 m + m \log m + n)$. The other extension is called \textit{MBSR among circles} or MBSR-C and asks for the largest axis-aligned rectangle separating red points from blue unit circles. For MBSR-C, we provide an algorithm that runs in $O(m^2 + n)$ time.Comment: 14 pages, 14 figures, full version of CCCG pape

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