Quadtrees and related hierarchical decompositions
Provably good mesh generation.
M. Bern, D. Eppstein, and J. Gilbert.
31st IEEE Symp. Foundations of Comp. Sci., St. Louis, Missouri, 1990, pp. 231–241.
J. Comp. Sys. Sci. 48: 384–409, 1994 (special issue for 31st FOCS).In this paper, we construct triangulations of point sets and polygons by using quadtrees to add extra vertices to the input. As a result we can guarantee that all triangles have angles bounded away from zero, using a number of triangles within a constant of optimal; this was the first paper to provide simultaneous bounds on mesh element quality and mesh complexity of this form, and therefore the first to provide finite element mesh generation algorithms that guarantee both the robustness of the algorithm against unexpected input geometries and the quality of its output.
In the same paper we also use quadtrees to triangulate planar point sets so that all angles are non-obtuse, using linearly many triangles, and to triangulate higher dimensional point sets with no small solid angles and a number of simplices within a constant of optimal. Also, we can augment any higher dimensional point set so the Delaunay triangulation has linear complexity.
In later follow-up work, I showed that the same technique can also be used to find a triangulation whose edges have total length within a constant factor of optimal. Bern, Mitchell, and Ruppert showed that alternative methods can be used to triangulate any polygon without obtuse angles; see "Faster circle packing with application to nonobtuse triangulation" for an algorithmic improvement to their technique. Additionally, with Bern, Chew, and Ruppert, we showed that any point set in higher dimensions can be triangulated with nonobtuse simplices. Bern and I surveyed these and related results in our paper "Mesh generation and optimal triangulation".
Approximating the minimum weight Steiner triangulation.
D. Eppstein.
Tech. Rep. 91-55, ICS, UCI, 1991.
3rd ACM-SIAM Symp. Discrete Algorithms, Orlando, 1992, pp. 48–57.
Disc. Comp. Geom. 11: 163–191, 1994.Quadtree based triangulation gives a large but constant factor approximation to the minimum weight triangulation of a point set or convex polygon, allowing extra Steiner points to be added as vertices. Includes proofs of several bounds on triangulation weight relative to the minimum spanning tree or non-Steiner triangulation, and a conjecture that for convex polygons the only points that need to be added are on the polygon boundary.
Algorithms for proximity problems in higher dimensions.
M. T. Dickerson and D. Eppstein.
Comp. Geom. Theory & Applications 5: 277–291, 1996.Combines a method from "Provably good mesh generation" for finding sparse high-dimensional Delaunay triangulations, a method of Dickerson, Drysdale, and Sack ["Simple algorithms for enumerating interpoint distances", IJCGA 1992] for using Delaunay triangulations to search for nearest neighbors, and a method of Frederickson for speeding up tree-based searches. The results are fast algorithms for several proximity problems such as finding the k nearest neighbors to each point in a given point set.
Parallel construction of quadtrees and quality triangulations.
M. Bern, D. Eppstein, and S.-H. Teng.
3rd Worksh. Algorithms and Data Structures, Montreal, 1993.
Springer, Lecture Notes in Comp. Sci. 709, 1993, pp. 188–199.
Tech. Rep. 614, MIT Lab. for Comp. Sci., 1994.
Int. J. Comp. Geom. & Appl. 9 (6): 517–532, 1999.A parallelization of the quadtree constructions in "Provably good mesh generation", in an integer model of computation, based on a technique of sorting the input points using values formed by shuffling the binary representations of the coordinates. A side-effect is an efficient construction for the "fair split tree" hierarchical clustering method used by Callahan and Kosaraju for various nearest neighbor problems.
Faster geometric k-point MST approximation.
D. Eppstein.
Tech. Rep. 95-13, ICS, UCI, 1995.
Comp. Geom. Theory & Applications 8: 231–240, 1997.Various authors have looked at a variant of geometric clustering in which one must select k points that can be connected by a small spanning tree. The problem is NP-complete (for variable k); good approximations are known based on dynamic programming techniques but the time dependence on n is high. This paper describes a faster approximation algorithm based on dynamic programming in quadtrees, and a general technique based on that in "Iterated nearest neighbors" for reducing the dependence on n in any approximation algorithm.
Algorithms for coloring quadtrees.
M. Bern, D. Eppstein, and B. Hutchings.
arXiv:cs.CG/9907030.
Algorithmica 32 (1): 87–94, 2002.We consider several variations of the problem of coloring the squares of a quadtree so that no two adjacent squares are colored alike. We give simple linear time algorithms for 3-coloring balanced quadtrees with edge adjacency, 4-coloring unbalanced quadtrees with edge adjacency, and 6-coloring balanced or unbalanced quadtrees with corner adjacency. The number of colors used by the first two algorithms is optimal; for the third algorithm, 5 colors may sometimes be needed.
The skip quadtree: a simple dynamic data structure for multidimensional data.
D. Eppstein, M. T. Goodrich, and J. Z. Sun.
21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296–305.
arXiv:cs.CG/0507049.
Int. J. Comp. Geom. & Appl. 18(1-2): 131–160, 2008.We describe a data structure consisting of a sequence of compressed quadtrees for successively sparser samples of an input point set, with connections between the same squares in successive members of the sequence. Using this structure, we can insert or delete points and answer approximate range queries and approximate nearest neighbor queries in O(log n) time per operation.
Skip-webs: efficient distributed data structures for multi-dimensional data sets.
L. Arge, D. Eppstein, and M. T. Goodrich.
Proc. 24th ACM SIGACT-SIGOPS Symp. Principles of Distributed Computing (PODC 2005), Las Vegas, July 2005, pp. 69–76.
arXiv:cs.DC/0507050.Describes efficient distributed versions of skip quadtrees and related spatial searching structures.
Diamond-kite meshes: adaptive quadrilateral meshing and orthogonal circle packing.
D. Eppstein.
arXiv:1207.5082.
21st International Meshing Round Table, San Jose, California, 2012, pp. 261–277.
Engineering with Computers 30 (2): 223–235 (special issue for the 21st Int. Meshing Roundtable), 2014.We describe a recursive subdivision of the plane into quadrilaterals in the form of rhombi and kites with 60, 90, and 120 degree angles. The vertices of the resulting quadrilateral mesh form the centers of a set of circles that cross orthogonally for every two adjacent vertices, and it has many other properties that are important in finite element meshing.