**Visibility with a moving point of view**.

M. Bern, D.P. Dobkin, D. Eppstein, and R. Grossman.

*1st ACM-SIAM Symp. Discrete Algorithms,*San Francisco, 1990, pp. 107–118.

*Algorithmica*11: 360–378, 1994.An investigation of 3d visibility problems in which the viewing position moves along a straight flight path, with various assumptions on the complexity of the viewed scene.

**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".

(Preliminary copy of journal version)

**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.**Tree-weighted neighbors and geometric**.*k*smallest spanning trees

D. Eppstein.

Tech. Rep. 92-77, ICS, UCI, 1992.

*Int. J. Comp. Geom. & Appl.*4: 229–238, 1994."Finding the

*k*smallest spanning trees" used higher order Voronoi diagrams to reduce the geometric*k*smallest spanning tree problem to the graph problem. Here I instead use nearest neighbors for a modified distance function where the bottleneck shortest path length is subtracted from the true distance between points. The result improves the planar time bounds and extends more easily to higher dimensions.**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*n*^{d}.**Arboricity and bipartite subgraph listing algorithms**.

D. Eppstein.

Tech. Rep. 94-11, ICS, UCI, 1994.

*Inf. Proc. Lett.*51: 207–211, 1994.For any sparse family of graphs, one can list in linear time all complete bipartite subgraphs of a graph in the family. There are O(n) complete bipartite subgraphs of a graph in the family, and the sum of the numbers of vertices in these subgraphs is also O(n).

Nowadays these results can also be interpreted as a form of formal concept analysis. If a set of objects and attributes is sparse (e.g., if it is generated by adding objects and attributes one at a time, where each newly-added object is given O(1) attributes and each newly-added attribute is held by O(1) objects) then the total size of all concepts in its concept lattice is linear, and this lattice may be generated in linear time.

**Offline algorithms for dynamic minimum spanning tree problems**.

D. Eppstein.

*2nd Worksh. Algorithms and Data Structures,*Ottawa, Canada, 1991.

Springer,*Lecture Notes in Comp. Sci.*519, 1991, pp. 392–399.

Tech. Rep. 92-04, ICS, UCI, 1992.

*J. Algorithms*17: 237–250, 1994.Given a sequence of edge insertions and deletions in a graph, finds the corresponding sequence of minimum spanning tree changes, in logarithmic time per update. Similarly solves the planar geometric version of the problem (using a novel "mixed MST" formulation in which part of the input is a graph and part is a point set) in time O(log

^{2}*n*) for the Euclidean metric and O(log*n*log log*n*) for the rectilinear metric.**Average case analysis of dynamic geometric optimization**.

D. Eppstein.

Tech. Rep. 93-18, ICS, UCI, 1993.

*5th ACM-SIAM Symp. Discrete Algorithms,*Arlington, 1994, pp. 77–86.

*Comp. Geom. Theory & Applications*6: 45–68, 1996.The Tech. Report used the more informative title "Updating widths and maximum spanning trees using the rotating caliper graph", which I also used for the journal submission, but the referees made me change it back. Dynamic geometry in a model of Mulmuley and Schwarzkopf in which insertions and deletions are chosen randomly among a worst-case pool of points. This paper introduces several fundamental techniques including the rotating caliper graph of a point set and a method for performing decomposible range queries in the average case setting. It has also since inspired the use of a similar model in dynamic graph algorithms.

(SODA paper – Full paper)

**Faster circle packing with application to nonobtuse triangulation**.

D. Eppstein.

Tech. Rep. 94-33, ICS, UCI 1994.

*Int. J. Comp. Geom. & Appl.*7 (5): 485–491, 1997.Speeds up a triangulation algorithm of Bern et al. ["Linear-Size Nonobtuse Triangulation of Polygons"] by finding a collection of disjoint circles which connect up the holes in a non-simple polygon. The method is to use a minimum spanning tree to find a collection of overlapping circles, then shrink them one by one to reduce the number of overlaps, using Sleator and Tarjan's dynamic tree data structure to keep track of the connectivity of the shrunken circles.

**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.

**Clustering for faster network simplex pivots**.

D. Eppstein.

Tech. Rep. 93-19, ICS, UCI, 1993.

*5th ACM-SIAM Symp. Discrete Algorithms,*Arlington, 1994, pp. 160–166.

*Networks*35 (3): 173–180, 2000.Speeds up the worst case time per pivot for various versions of the network simplex algorithm for minimum cost flow problems. Uses techniques from dynamic graph algorithms as well as some simple geometric data structures.

**Finding the**.*k*shortest paths

D. Eppstein.

*35th IEEE Symp. Foundations of Comp. Sci.,*Santa Fe, 1994, pp. 154–165.

Tech. Rep. 94-26, ICS, UCI, 1994.

*SIAM J. Computing*28 (2): 652–673, 1998.This paper presents an algorithm that finds multiple short paths connecting two terminals in a graph (allowing repeated vertices and edges in the paths) in constant time per path after a preprocessing stage dominated by a single-source shortest path computation. The paths it finds are the

*k*shortest in the graph, where*k*is a parameter given as input to the algorithm.The

*k*shortest paths problem has many important applications for finding alternative solutions to geographic path planning problems, network routing, hypothesis generation in computational linguistics, and sequence alignment and metabolic pathway finding in bioinformatics. Although there have been many papers on the*k*shortest paths problem before and after this one, it has become frequently cited in those application areas. Additionally, it marks a boundary in the theoretical study of the problem: prior theoretical work largely concerned how quickly the problem could be solved, a line of research that was closed off by the optimal time bounds of this paper. Subsequent work has focused instead on devising efficient algorithms for more complex alternative formulations of the problem that avoid the repeated vertices and other shortcomings of the alternative paths produced by this formulation.The journal version also includes material from a separate 1995 technical report, "Finding common ancestors and disjoint paths in DAGs".

(Full paper – Graehl implementation – Jiménez-Marzal implementations – Shibuya implementation – Martins implementation – Cliff OpenStreetMap demo)

**Dihedral bounds for mesh generation in high dimensions**.

M. Bern, L.P. Chew, D. Eppstein, and J. Ruppert.

*892nd Meeting Amer. Math. Soc.,*Brooklyn, 1994.

Abstract in*Abs. Amer. Math. Soc.*15, 1994, p. 366.

*6th ACM-SIAM Symp. Discrete Algorithms,*San Francisco, 1995, pp. 189–196.Any

*d*-dimensional point set can be triangulated with O(*n*^{ceil(d/2)}) simplices, none of which has an obtuse dihedral angle. No bound depending only on*n*is possible if we require the maximum dihedral angle to measure at most 90-epsilon degrees or the minimum dihedral to measure at least epsilon. Includes a classification of simplices in terms of their bad angles.**Subgraph isomorphism in planar graphs and related problems**.

D. Eppstein.

Tech. Rep. 94-25, ICS, UCI, 1994.

*6th ACM-SIAM Symp. Discrete Algorithms,*San Francisco, 1995, pp. 632–640.

arXiv:cs.DS/9911003.

*J. Graph Algorithms and Applications*3 (3): 1–27, 1999.Uses an idea of Baker to cover a planar graph with subgraphs of low treewidth. As a consequence, any fixed pattern can be found as a subgraph in linear time; the same methods can be used to solve other planar graph problems including vertex connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths. A companion paper, "Diameter and treewidth in minor-closed graph families", presents a result announced in the conference version of this paper, that exactly characterizes the minor-closed graph families for which the same techniques apply.

Years – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine

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