David Eppstein – Publications

Journal and book chapter submissions

Better late than never: the complexity of arrangements of polyhedra.
B. Aronov, S. W. Bae, O. Cheong, D. Eppstein, C. Knauer, and R. Seidel.
41st European Workshop on Computational Geometry (EuroCG 2025), Liblice, Czech Republic, pp. 62:1–62:4.
arXiv:2506.03960.

My 1994 paper "On the number of minimal 1-Steiner trees" with Aronov and Bern, in some preliminary manuscript versions, included a bound of \(O(m^{\lceil d/2\rceil}n^{\lfloor d/2\rfloor})\) on the complexity of an arrangement of \(m\) convex polytopes given as intersections of a total of \(n\) halfspaces, interpolating between the upper bound theorem for polytopes and the complexity of hyperplane arrangements. However, the manuscript and the proof were lost. This paper re-proves the result, with better care for the degenerate cases.

(Blog post: Two recent preprints)

Decremental greedy polygons and polyhedra without sharp angles.
D. Eppstein.
arXiv:2507.04538.
Proc. 37th Canadian Conference on Computational Geometry, 2025, pp. 85–91.

For given elements and a quality function of an element in a subset, monotonically non-increasing with respect to removals of other elements, we can find the max-min-quality subset by a decremental greedy algorithm that repeatedly removes low-quality subsets. We apply this method to find the max-min-angle polygon for points in the plane, the max-min-weight cycle in a directed graph, and several generalizations of both problems.

(CCCG'25 decremental greedy slides – Blog post: Ready lists)

Entropy-bounded computational geometry made easier and sensitive to sortedness.
D. Eppstein, M. T. Goodrich, A. M. Illickan, and C. A. To.
arXiv:2508.20489.
Proc. 37th Canadian Conference on Computational Geometry, 2025, pp. 53–61.

We define a notion of structural entropy of point sets under which a set has low entropy when it can be covered by few disjoint triangles that are either entirely under the hull of the input or presorted, and show that we can find the hull in time sensitive to this entropy. Generalizations of the same technique apply to geometric maxima, lower envelopes, and visibility polygons. A preliminary title for this paper was "Instance-optimal computational geometry made easier and sensitive to sortedness".

(Blog post: Entropy in algorithm analysis)

Stabbing faces by a convex curve.
D. Eppstein.
arXiv:2508.17549.
33rd International Symposium on Graph Drawing and Network Visualization.
Leibniz International Proceedings in Informatics (LIPIcs) 357, 2025, pp. 29:1–29:8, doi:10.4230/LIPIcs.GD.2025.29.

For each smooth convex curve \(C\), and each planar graph \(G\), there exists a straight-line drawing of \(G\) all of whose faces are crossed by \(C\). This result serves as a counterexample to an argument that universal point sets for planar graph drawing cannot be covered by few convex polygons.

(Blog post: Five-arc fractal – GD'25 slides)