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.

[1]  Leonidas J. Guibas,et al.  Fractional cascading: I. A data structuring technique , 1986, Algorithmica.

[2]  Mikhail J. Atallah,et al.  A note on finding a maximum empty rectangle , 1986, Discret. Appl. Math..

[3]  Philip N. Klein,et al.  Multiple-Source Multiple-Sink Maximum Flow in Directed Planar Graphs in Near-Linear Time , 2011, FOCS.

[4]  Robert L. Scot Drysdale,et al.  Voronoi diagrams based on convex distance functions , 1985, SCG '85.

[5]  Michael A. Bender,et al.  The LCA Problem Revisited , 2000, LATIN.

[6]  Bernard Chazelle,et al.  Computing the Largest Empty Rectangle , 1984, SIAM J. Comput..

[7]  John Augustine,et al.  Recognizing the Largest Empty Circle and Axis-Parallel Rectangle in a Desired Location , 2010, ArXiv.

[8]  Maria M. Klawe,et al.  Superlinear Bounds for Matrix Searching Problems , 1992, J. Algorithms.

[9]  Alok Aggarwal,et al.  Geometric Applications of a Matrix Searching Algorithm , 1986, Symposium on Computational Geometry.

[10]  Jeffrey D Ullma Computational Aspects of VLSI , 1984 .

[11]  Subhas C. Nandy,et al.  Largest Empty Rectangle among a Point Set , 1999, FSTTCS.

[12]  Jorge Urrutia,et al.  Finding the largest axis aligned rectangle in a polygon in o(n log n) time , 2001, CCCG.

[13]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[14]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[15]  Henry S. Baird,et al.  Image segmentation by shape-directed covers , 1990, [1990] Proceedings. 10th International Conference on Pattern Recognition.

[16]  Subhas C. Nandy,et al.  Location of the Largest Empty Rectangle among Arbitrary Obstacles , 1994, FSTTCS.

[17]  Daniel J. Kleitman,et al.  An Almost Linear Time Algorithm for Generalized Matrix Searching , 1990, SIAM J. Discret. Math..

[18]  John Hershberger,et al.  Finding the Upper Envelope of n Line Segments in O(n log n) Time , 1989, Inf. Process. Lett..

[19]  D. T. Lee,et al.  On the maximum empty rectangle problem , 1984, Discret. Appl. Math..

[20]  Micha Sharir,et al.  Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams , 2016, Discret. Comput. Geom..

[21]  Alok Aggarwal,et al.  Fast algorithms for computing the largest empty rectangle , 1987, SCG '87.