Geometric clustering
- Finding minimum area \(k\)-gons.
D. Eppstein, M. Overmars, G. Rote, and G. Woeginger.
Disc. Comp. Geom. 7 (1): 45–58, 1992.Uses dynamic programming to choose sets of \(k\) points optimizing various criteria on the quality of their convex hull (in particular area). The time complexity (cubic in \(n\)) was later improved to quadratic in my paper "New algorithms for minimum area \(k\)-gons", which however continues to use the same dynamic program as a subroutine.
- Dynamic three-dimensional linear programming.
D. Eppstein.
Tech. Rep. 91-53, ICS, UCI, 1991.
32nd IEEE Symp. Foundations of Comp. Sci., San Juan, Puerto Rico, 1991, pp. 488–494.
ORSA J. Computing 4: 360–368, 1992 (special issue on computational geometry).Uses Dobkin-Kirkpatrick hierarchies to perform linear programming queries in the intersection of several convex polyhedra. By maintaining a collection of halfspaces as several subsets, represented by polyhedra, this leads to algorithms for a dynamic linear program in which updates change the set of constraints. The fully dynamic results have largely been subsumed by Agarwal and Matoušek, but this paper also includes polylog time results for semi-online problems, and uses them to give a fast randomized algorithm for the planar 2-center problem (later improved by various authors, most recently in "Faster Construction of Planar Two-Centers", which re-uses the data structures described here).
- New algorithms for minimum area k-gons.
D. Eppstein.
Tech. Rep. 91-59, ICS, UCI, 1991.
3rd ACM-SIAM Symp. Discrete Algorithms, Orlando, 1992, pp. 83–86.Shows that the minimum area polygon containing k of n points must be near a line determined by two points, and uses this observation to find the polygon quickly. Merged with "Iterated nearest neighbors and finding minimal polytopes" in the journal version.
- New algorithms for minimum measure simplices and one-dimensional
weighted Voronoi diagrams.
D. Eppstein and J. Erickson.
Tech. Rep. 92-55, ICS, UCI, 1992.Looks at space complexity of finding minimum simplices – solves the problem in O(n2) space and O(nd) time (matching the best known time bounds) or in linear space at the expense of an additional log in time. Also finds one-dimensional multiplicatively weighted Voronoi diagrams in linear time for sorted inputs (O(n log n) was known).
- Iterated nearest neighbors and finding minimal polytopes.
D. Eppstein and J. Erickson.
Tech. Rep. 92-71, ICS, UCI, 1992.
4th ACM-SIAM Symp. Discrete Algorithms, Austin, 1993, pp. 64–73.
Disc. Comp. Geom. 11: 321–350, 1994.Showed that for various optimization criteria, the optimal polygon containing k of n points must be near one of the points, hence one can transform time bounds involving several factors of n to bounds linear in n but polynomial in k. Used as a subroutine are data structures for finding several nearest neighbors in rectilinear metric spaces, and algorithms for finding the deepest point in an arrangement of cubes or spheres.
- 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.
- Faster construction of planar two-centers.
D. Eppstein.
Tech. Rep. 96-12, ICS, UCI, 1996.
8th ACM-SIAM Symp. Discrete Algorithms, New Orleans, 1997, pp. 131–138.Improving on a recent breakthrough of Sharir, we use data structures from "Dynamic three-dimensional linear programming" to find two circular disks of minimum radius covering a set of points in the Euclidean plane, in randomized expected time O(n log2 n).