**Horizon theorems for lines and polygons**.

M. Bern, D. Eppstein, P. Plassman, and F. Yao.

*Discrete and Computational Geometry: Papers from the DIMACS Special Year*, J. Goodman, R. Pollack, and W. Steiger, eds.,*DIMACS Series in Discrete Mathematics and Theoretical Computer Science*6, Amer. Math. Soc., 1991, 45–66.The total complexity of the cells in a line arrangement that are cut by another line is at most 15

*n*/2. The complexity of cells cut by a convex*k*-gon is O(*n*α(*n,**k*)). The first bound is tight, but it remains open whether the second is, or whether only linear complexity is possible.**The expected extremes in a Delaunay triangulation**.

M. Bern, D. Eppstein, and F. Yao.

*18th Int. Coll. Automata, Languages and Programming,*Madrid, Spain, 1991.

Springer,*Lecture Notes in Comp. Sci.*510, 1991, 674–685.

*Int. J. Comp. Geom. & Appl.*1 (1): 79–92, 1991.Discusses the expected behavior of Delaunay triangulations for points chosen uniformly at random (without edge effects). The main result is that within a region containing

*n*points, the expected maximum degree is bounded to within a constant factor of log*n*/ log log*n.*(BibTeX – Citations – CiteSeer – ACM DL – ACM DL (2))

**Planar orientations with low out-degree and compaction of adjacency matrices**.

M. Chrobak and D. Eppstein.

*Theor. Comp. Sci.*86 (2): 243–266, 1991.Describes efficient sequential and parallel algorithms for orienting the edges of an undirected planar graph so that each vertex has few outgoing edges. From such an orientation one can test in constant time whether a given edge exists. One consequence is a parallel algorithm to list all subgraphs isomorphic to K3 or K4. More recently this paper has been cited for its applications to scheduling update operations in parallel finite element methods.

(BibTeX – Citations – CiteSeer – ACM DL)

**Equipartitions of graphs**.

D. Eppstein, J. Feigenbaum, and C.L. Li.

*Discrete Mathematics*91 (3): 239–248, 1991.Considers partitions of the vertices of a graph into equal subsets, with few pairs of subsets connected by edges. (Equivalently we view the graph as a subgraph of a product in which one factor is sparse.) A random graph construction shows that such a factorization does not always exist.

(BibTeX – Full paper – CiteSeer – ACM DL)

**Polynomial-size non-obtuse triangulation of polygons**.

M. Bern and D. Eppstein.

*7th ACM Symp. Comp. Geom.,*North Conway, New Hampshire, 1991, pp. 342–350.

*Int. J. Comp. Geom. & Appl.*2: 241–255, 1992 (special issue for 7th Symp. Comput. Geom.).Any simple polygon can be triangulated with quadratically many nonobtuse triangles. Mostly subsumed by recent results of Bern et al described in "Faster circle packing".

(BibTeX – Citations – CiteSeer)

**Efficient algorithms for sequence analysis**.

D. Eppstein, Z. Galil, R. Giancarlo, and G.F. Italiano.

*International Advanced Workshop on Sequences,*Positano, Italy, 1991.

*Sequences II: Methods in Communication, Security, and Computer Science,*R.M. Capocelli, A. De Santis, and U. Vaccaro, eds., Springer, 1993, pp. 225–244.

Surveys results on speeding up certain dynamic programs used for sequence comparison and RNA structure prediction.

(BibTeX – Citations – CiteSeer)

**Improved bounds for intersecting triangles and halving planes**.

D. Eppstein.

Tech. Rep. 91-60, ICS, UCI, 1991.

*J. Combinatorial Theory Ser. A*62: 176–182, 1993.Reduces the polylogarithmic term in an upper bound for the three-dimensional

*k*-set problem.A bug in the proof was corrected by Nivasch and Sharir.

(BibTeX – Citations – CiteSeer – ACM DL)

**Efficient sequential and parallel algorithms for computing recovery points in trees and paths**.

M. Chrobak, D. Eppstein, G.F. Italiano, and M. Yung.

*2nd ACM-SIAM Symp. Discrete Algorithms,*San Francisco, 1991, pp. 158–167.

ALCOM Report 91-74, University of Rome, 1991.Described slightly superlinear algorithms for partitioning a tree into a given number of subtrees, making them all as short as possible. Frederickson at the same conference further improved the sequential time to linear. There may still be something worth publishing in the parallel algorithms.

**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).

(BibTeX – Citations – CiteSeer – ACM DL)

**Persistence, offline algorithms, and space compaction**.

D. Eppstein.

Tech. Rep. 91-54, ICS, UCI, 1991.Considers persistence for a naive form of dynamic algorithm in which each update rebuilds a static solution. The space bounds for this can often be reduced by maintaining an offline solution over a sequence of updates constructed from an Euler tour of the persistent update history tree.

(BibTeX)

**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.

(BibTeX – Citations – CiteSeer)

**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.**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.(BibTeX – Citations – CiteSeer – ACM DL (JA))

**Subquadtratic nonobtuse triangulation of convex polygons**.

D. Eppstein.

Tech. Rep. 91-61, ICS, UCI, 1991.This was merged into "triangulating polygons without large angles". We find a grid-like structure in the input polygon, which is then thinned out using a complicated divide-and-conquer scheme. The results are largely subsumed by the method of Bern et al. described in "Faster circle packing".

(BibTeX)

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

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