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(nα(n) log n)-time algorithm to preprocess P into a data structure of size O(nα(n) log n), such that, given a query point q, we can find, in O(log 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 et al. [4], which uses slightly superquadratic preprocessing and storage. Work by Haim Kaplan was partially supported by Grant 2006/204 from the U.S.–Israel Binational Science Foundation, and by Grant 822/10 from the Israel Science Fund. Work by Micha Sharir was partially supported by NSF Grant CCR-08-30272, by Grant 2006/194 from the U.S.–Israel Binational Science Foundation, by Grant 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: [email protected] School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. E-mail: [email protected]