Dynamic Smooth Compressed Quadtrees

We introduce dynamic smooth (a.k.a. balanced) compressed quadtrees with worst-case constant time updates in constant dimensions. We distinguish two versions of the problem. First, we show that quadtrees as a space-division data structure can be made smooth and dynamic subject to split and merge operations on the quadtree cells. Second, we show that quadtrees used to store a set of points in R^d can be made smooth and dynamic subject to insertions and deletions of points. The second version uses the first but must additionally deal with compression and alignment of quadtree components. In both cases our updates take 2^{O(d log d)} time, except for the point location part in the second version which has a lower bound of Omega(log n); but if a pointer (finger) to the correct quadtree cell is given, the rest of the updates take worst-case constant time. Our result implies that several classic and recent results (ranging from ray tracing to planar point location) in computational geometry which use quadtrees can deal with arbitrary point sets on a real RAM pointer machine.

[1]  Maarten Löffler,et al.  Dynamic Planar Point Location with Sub-logarithmic Local Updates , 2012, WADS.

[2]  I. Jntroductjon.  Neighbor Finding Techniques for Images Represented by Quadtrees * , 1980 .

[3]  Jon Louis Bentley,et al.  Quad trees a data structure for retrieval on composite keys , 1974, Acta Informatica.

[4]  Mihai Pop,et al.  De-amortization of Algorithms , 1998, COCOON.

[5]  Maarten Löffler,et al.  Dynamic Stabbing Queries with Sub-logarithmic Local Updates for Overlapping Intervals , 2017, CSR.

[6]  Olivier Devillers,et al.  Fully Dynamic Delaunay Triangulation in Logarithmic Expected Time Per Operation , 1992, Comput. Geom..

[7]  David Eppstein,et al.  Provably good mesh generation , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[8]  Stefan Kimmerle,et al.  Hierarchical Techniques in Collision Detection for Cloth Animation , 2003, WSCG.

[9]  Pravin M. Vaidya,et al.  Minimum Spanning Trees in k-Dimensional Space , 1988, SIAM J. Comput..

[10]  Wolfgang Mulzer,et al.  Delaunay Triangulations in O(sort(n)) Time and More , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[11]  David Eppstein,et al.  Parallel Construction of Quadtrees and Quality Triangulations , 1993, WADS.

[12]  Hanan Samet,et al.  The Design and Analysis of Spatial Data Structures , 1989 .

[13]  Maarten Löffler,et al.  Preprocessing Imprecise Points for Delaunay Triangulation: Simplified and Extended , 2010, Algorithmica.

[14]  Y. V. Venkatesh,et al.  Threaded Linear Hierarchical Quadtree for Computation of Geometric Properties of Binary Images , 1988, IEEE Trans. Software Eng..

[15]  Chee-Keng Yap,et al.  Amortized analysis of smooth quadtrees in all dimensions , 2014, Comput. Geom..

[16]  S. Rao Kosaraju,et al.  A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields , 1995, JACM.

[17]  Bettina Speckmann,et al.  Kinetic Compressed Quadtrees in the Black-Box Model with Applications to Collision Detection for Low-Density Scenes , 2012, ESA.

[18]  Christos Levcopoulos,et al.  Computing a Threaded Quadtree from the Delaunay Triangulation in linear Time , 1998, Nord. J. Comput..

[19]  David M. Mount,et al.  A Self-adjusting Data Structure for Multidimensional Point Sets , 2012, ESA.

[20]  Kellogg S. Booth,et al.  Heuristics for ray tracing using space subdivision , 1990, The Visual Computer.

[21]  Boris Aronov,et al.  Cost prediction for ray shooting in octrees , 2006, Comput. Geom..

[22]  Chee-Keng Yap,et al.  Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams , 2016, Comput. Graph. Forum.

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

[24]  David Eppstein,et al.  The skip quadtree: a simple dynamic data structure for multidimensional data , 2005, SCG.

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

[26]  Maarten Löffler,et al.  Triangulating the square and squaring the triangle: quadtrees and Delaunay triangulations are equivalent , 2011, SODA '11.