Discrete & Computational Geometry
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.
Edge insertion for optimal triangulations.
M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T.S. Tan.
1st Latin Amer. Symp. Theoretical Informatics, Sao Paulo, 1992.
Springer, Lecture Notes in Comp. Sci. 583, 1992, pp. 46–60.
Tech. Rep. EDC UILU-ENG-92-1702, Univ. Illinois, Urbana-Champaign, 1992.
Disc. & Comp. Geom. 10: 47–65, 1993.One standard way of constructing Delaunay triangulations is by iterated local improvement, in which each step flips the diagonal of some quadrilateral. For many other optimal triangulation problems, flipping is insufficient, but the problems can instead be solved by a more general local improvement step in which a new edge is added to the triangulation, cutting through several triangles, and the region it cuts through is retriangulated on both sides.
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.
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.
On the number of minimal 1-Steiner trees.
B. Aronov, M. Bern, and D. Eppstein.
Disc. & Comp. Geom. 12: 29–34, 1994.Given a d-dimensional set of n points, the number of combinatorially different minimum spanning trees that can be formed by adding one more point is within a polylogarithmic factor of nd.
Dynamic Euclidean minimum spanning trees and extrema of binary functions.
D. Eppstein.
Tech. Rep. 92-05, ICS, UCI, 1992.
Tech. Rep. 92-88, ICS, UCI, 1992.
Disc. Comp. Geom. 13: 111–122, 1995.Shows that bichromatic nearest neighbors can be maintained under point insertion and deletion essentially as quickly as known solutions to the post office problem, and that the minimum spanning tree can be maintained in the same time except for an additive O(sqrt n) needed for solving the corresponding graph problem. TR 92-88's title was actually "Fully dynamic maintenance of Euclidean minimum spanning trees and maxima of decomposable functions". TR 92-05's title left out the part about maxima; that version gave a slower algorithm superseded by the result in 92-88.
Geometric lower bounds for parametric matroid optimization.
D. Eppstein.
Tech. Rep. 95-11, ICS, UCI, 1995.
27th ACM Symp. Theory of Computing, Las Vegas, 1995, pp. 662–671.
Disc. Comp. Geom. 20: 463–476, 1998.Considers graphs in which edge weights are linear functions of time. Shows nonlinear lower bounds on the number of different minimum spanning trees appearing over time by translation from geometric problem of lower envelopes of line segments. A matroid generalization has a better lower bound coming from many faces in line arrangements, and the uniform matroid problem is equivalent to the geometric k-set problem.
On nearest-neighbor graphs.
D. Eppstein, M. S. Paterson, and F. F. Yao.
Disc. Comp. Geom. 17: 263–282, 1997.Paterson and Yao presented a paper at ICALP showing among other things that any connected nearest neighbor forest with diameter D has O(D9) vertices. This paper is the journal version; my contribution consists of improving that bound to O(D5), which is tight.
Raising roofs, crashing cycles, and playing pool: applications of a data structure for finding pairwise interactions.
D. Eppstein and J. Erickson.
14th ACM Symp. Comp. Geom., Minneapolis, 1998, pp. 58–67.
Disc. Comp. Geom. 22 (4): 569–592, 1999 (special issue for SCG 1998).We use my dynamic closest pair data structure from "Dynamic Euclidean minimum spanning trees" to detect collisions among a collection of moving objects in sublinear time per collision. As one application, we can construct the straight skeleton of Aichholzer et al (and the mitered offset curves from which it is defined) in subquadratic time.
Regression depth and center points.
N. Amenta, M. Bern, D. Eppstein, and S.-H. Teng.
arXiv:cs.CG/9809037.
3rd CGC Worksh. Computational Geometry, Brown Univ., 1998.
Disc. Comp. Geom. 23 (3): 305–323, 2000.We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least ceiling(n/(d+1)). as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d dimensions there exists a point that cannot escape to infinity without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our approach to related questions on the existence of partitions of the data into subsets such that a common plane has nonzero regression depth in each subset, and to the computational complexity of regression depth problems.
Multivariate regression depth.
M. Bern and D. Eppstein.
arXiv:cs.CG/9912013.
16th ACM Symp. Comp. Geom., Hong Kong, 2000, pp. 315–321.
Disc. Comp. Geom. 28 (1): 1–17, 2002.We generalize regression depth to k-flats. The k=0 case gives the classical notion of center points. We prove that for any set of n points in Rd there always exists a k-flat with depth at least a constant fraction of n. As a consequence, we derive a linear-time (1+epsilon)-approximation algorithm for the deepest flat. The full version of this paper also includes results from "Computing the Depth of a Flat".
Guest editor's forward.
D. Eppstein.
Disc. Comp. Geom. 30: 1–2, 2003 (special issue for 17th Symp. Comp. Geom.)
Ramified rectilinear polygons: coordinatization by dendrons.
H.-J. Bandelt, V. Chepoi, and D. Eppstein.
arXiv:1005.1721.
Disc. Comp. Geom. 54 (4): 771–797, 2015.We characterize the graphs that can be isometrically embedded into the Cartesian product of two trees (partial double dendrons), and the metric spaces obtained as the median complexes of these graphs. These spaces include the space of geodesic distance in axis-parallel polygons in the L1 plane, hence the title. An algorithm based on lexicographic breadth-first search can be used to recognize partial double dendrons in linear time.
Drawing trees with perfect angular resolution and polynomial area.
C. Duncan, D. Eppstein, M. T. Goodrich, S. Kobourov, and M. Nöllenburg.
Proc. 18th Int. Symp. Graph Drawing, Konstanz, Germany, 2010.
Springer, Lecture Notes in Comp. Sci. 6502, 2011, pp. 183–194.
arXiv:1009.0581.
Discrete Comput. Geom. 49 (2): 157–182, 2013.We consider balloon drawings of trees, in which each subtree of the root is drawn recursively within a disk, and these disks are arranged radially around the root, with the edges at each node spaced equally around the node so as to achieve the best possible angular resolution. If we are allowed to permute the children of each node, then we show that a drawing of this type can be made in which all edges are straight line segments and the area of the drawing is a polynomial multiple of the shortest edge length. However, if the child ordering is prescribed, exponential area might be necessary. We show that, if we relax the requirement of straight line edges and draw the edges as circular arcs (a style we call Lombardi drawing) then even with a prescribed child ordering we can achieve polynomial area.
Bounds on the complexity of halfspace intersections when the bounded faces have small dimension.
D. Eppstein and M. Löffler.
Proc. 27th ACM Symp. on Computational Geometry, Paris, 2011, pp. 361–368.
arXiv:1103.2575.
Discrete Comput. Geom. 50 (1): 1–21, 2013.Suppose that P is the intersection of n halfspaces in D dimensions, but that the bounded faces of P are at most d-dimensional, for some d that is much smaller than D. Then in this case we show that the number of vertices of P is O(nd), independent of D. We also investigate related bounds on the number of bounded faces of all dimensions of P, and algorithms for efficiently listing the vertices and bounded faces of P.
A Möbius-invariant power diagram and its applications to soap bubbles and planar Lombardi drawing.
D. Eppstein.
Invited talk at EuroGIGA Midterm Conference, Prague, Czech Republic, 2012.
Discrete Comput. Geom. 52 (3): 515–550, 2014 (Special issue for SoCG 2013).This talk and journal paper combines the results from "Planar Lombardi drawings for subcubic graphs" and "The graphs of planar soap bubbles". It uses three-dimensional hyperbolic geometry to define a partition of the plane into cells with circular-arc boundaries, given an input consisting of (possibly overlapping) circular disks and disk complements, which remains invariant under Möbius transformations of the input. We use this construction as a tool to construct planar Lombardi drawings of all 3-regular planar graphs; these are graph drawings in which the edges are represented by circular arcs meeting at equal angles at each vertex. We also use it to characterize the graphs of two-dimensional soap bubble clusters as being exactly the 2-vertex-connected 3-regular planar graphs.
Treetopes and their graphs.
D. Eppstein.
arXiv:1510.03152.
27th ACM-SIAM Symp. on Discrete Algorithms, Arlington, 2016, pp. 969–984.
Discrete Comput. Geom. 64 (2), 259–289, 2020 (special issue for Branko Grünbaum).We describe a class of polytopes of varying dimensions, whose restriction to three dimensions is the class of roofless polyhedra (Halin graphs). We call these polytopes treetopes. We show that the four-dimensional treetopes are closely related to clustered planar graph drawings, and we use this connection to recognize the graphs of four-dimensional treetopes in polynomial time.
Counting polygon triangulations is hard.
D. Eppstein.
arXiv:1903.04737.
Proc. 35th Int. Symp. on Computational Geometry, Portland, Oregon, June 2019.
Leibniz International Proceedings in Informatics (LIPIcs) 129, 2019, pp. 33:1–33:17.
Discrete Comput. Geom. 64 (4): 1210–1234, 2020 (special issue for SoCG 2019).Given a polygon with holes, it is #P-complete to determine how many triangulations it has.
Three-dimensional graph products with unbounded stack-number.
D. Eppstein, R. Robert Hickingbotham, L. Merker, S. Norin, M. T. Seweryn, and D. R. Wood.
arXiv:2202.05327.
Discrete Comput. Geom. 71: 1210–1237, 2024.The strong product of any three graphs of non-constant size has unbounded book thickness. In the case of strong products of three paths, and more generally of triangulations of \(n\times n\times n\) grid graphs obtained by adding a diagonal to each square of the grid, the book thickness is \(\Theta(n^{1/3})\). This is the first explicit example of a graph family with bounded maximum degree and unbounded book thickness.
Orthogonal dissection into few rectangles.
D. Eppstein.
arXiv:2206.10675.
34th Canadian Conference on Computational Geometry, 2022, pp. 143–150.
Discrete Comput. Geom. 73 (1): 129–148, 2025.The rank of the Dehn invariant of an orthogonal polygon equals the minimum number of rectangles into which it can be transformed by axis-parallel cuts, translation, and gluing. This allows the minimum number of rectangles to be calculated in polynomial time.