Deterministic Rectangle Enclosure and Offline Dominance Reporting on the RAM

We revisit a classical problem in computational geometry that has been studied since the 1980s: in the rectangle enclosure problem we want to report all k enclosing pairs of n input rectangles in 2D. We present the first deterministic algorithm that takes O(nlogn + k) worst-case time and O(n) space in the word-RAM model. This improves previous deterministic algorithms with O((nlogn + k)loglogn) running time. We achieve the result by derandomizing the algorithm of Chan, Larsen and Pătrascu [SoCG’11] that attains the same time complexity but in expectation.

[1]  Timothy M. Chan.  Persistent predecessor search and orthogonal point location on the word RAM , 2011, SODA '11.

[2]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[3]  Yijie Han,et al.  Deterministic sorting in O(nloglogn) time and linear space , 2004, J. Algorithms.

[4]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[5]  Christos Makris,et al.  An Improved Algorithm for Static 3D Dominance Reporting in the Pointer Machine , 2012, ISAAC.

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

[7]  Michael T. Goodrich,et al.  Planar Separators and Parallel Polygon Triangulation , 1995, J. Comput. Syst. Sci..

[8]  D. Wood,et al.  Data structures for the rectangle containment and enclosure problems , 1980 .

[9]  Timothy M. Chan,et al.  Orthogonal range searching on the RAM, revisited , 2011, SoCG '11.

[10]  Peyman Afshani.  On Dominance Reporting in 3D , 2008, ESA.

[11]  Peter van Emde Boas,et al.  Design and implementation of an efficient priority queue , 1976, Mathematical systems theory.

[12]  HanYijie.  Deterministic sorting in O(nlog logn) time and linear space , 2004 .

[13]  Derick Wood,et al.  An Optimal Worst Case Algorithm for Reporting Intersections of Rectangles , 1980, IEEE Transactions on Computers.

[14]  D. T. Lee,et al.  An Improved Algorithm for the Rectangle Enclosure Problem , 1982, J. Algorithms.

[15]  Christos Makris,et al.  A New Algorithm for Rectangle Enclosure Reporting , 1999, Inf. Process. Lett..

[16]  Jirí Matousek,et al.  Reporting Points in Halfspaces , 1992, Comput. Geom..

[17]  Edgar A. Ramos,et al.  On range reporting, ray shooting and k-level construction , 1999, SCG '99.

[18]  Timothy M. Chan.  All-Pairs Shortest Paths with Real Weights in O(n3/log n) Time , 2008, Algorithmica.

[19]  Timothy M. Chan,et al.  Counting inversions, offline orthogonal range counting, and related problems , 2010, SODA '10.

[20]  Timothy M. Chan,et al.  Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time , 2009, SIAM J. Comput..

[21]  Konstantinos Tsakalidis,et al.  Optimal Deterministic Shallow Cuttings for 3D Dominance Ranges , 2014, SODA.

[22]  Michael L. Fredman,et al.  Trans-Dichotomous Algorithms for Minimum Spanning Trees and Shortest Paths , 1994, J. Comput. Syst. Sci..

[23]  Michiel H. M. Smid,et al.  The Rectangle Enclosure and Point-Dominance Problems Revisited , 1997, Int. J. Comput. Geom. Appl..