**Trees in TeX**.

D. Eppstein.

*TUGboat*6 (1): 31, 1985.This note in the TeX user's group newsletter described a set of macros for drawing trees, using TeX's boxes-and-glue mechanisms to line up the nodes at each level of the tree.

(BibTeX – TeX source code – Citations – Nelson Beebe's TeX tree drawing bibliography)

**A heuristic approach to program inversion**.

D. Eppstein.

*9th Int. Joint Conf. Artificial Intelligence,*Los Angeles, 1985, pp. 219–221.Program transformation. Given a (lisp) program for an invertible function, how do we automatically find a program for the inverse function? Considers more general simultaneous inverses of multiple functions. The heuristic part involves type inference for finding conditionals to use in certain if statements.

(BibTeX – MacLISP source code – Kawabe's Common Lisp port – Citations – CiteSeer)

**On the NP-completeness of cryptarithms**.

D. Eppstein.

*SIGACT News*18 (3): 38–40, 1987.A cryptarithm (also known as an alphametic) is a puzzle in which the digits of a mathematical formula (typically addition of two large numbers) are replaced by letters; the goal is to determine which letter stands for which digits. If arithmetic is done in a polynomially large radix, the problem becomes NP-complete.

(BibTeX – Full paper – Citations)

**Efficient algorithms for sequence analysis with concave and convex gap costs**.

D. Eppstein.

Ph.D. thesis, Columbia University, 1989.Includes results from "Speeding up dynamic programming", "Sequence comparison with mixed convex and concave costs", and "Sparse dynamic programming".

(BibTeX)

**Sequence comparison with mixed convex and concave costs**.

D. Eppstein.

Tech. Rep. CUCS-382-88, Computer Science Dept., Columbia University, 1988.

*J. Algorithms*11 (1): 85–101, 1990.Gives an algorithm for finding the minimum number of mutations needed to transform one input string into another, in a general model in which substrings may be inserted or deleted at a cost depending nonlinearly on the substring length. The time bound depends on the number of times the second derivative of the cost function changes sign.

**Reset sequences for monotonic automata**.

D. Eppstein.

*15th Int. Coll. Automata, Languages and Programming,*Tampere, Finland, 1988.

Springer,*Lecture Notes in Comp. Sci.*317, 1988, pp. 230–238.

*SIAM J. Computing*19 (3): 500–510, 1990.Automata theory. A reset sequence for a DFA is an input such that, no matter which state the DFA starts in, it ends up after the input in a known state. These have been used by Natarajan and Goldberg for certain robot motion planning problems (in fact the conference version of this paper used the title "Reset sequences for finite automata with application to design of parts orienters"), and also in coding theory where they arise in the design of self-synchronizing codes. This paper considers DFAs in which the transition functions respect a given cyclic ordering of the states, and shows that their shortest reset sequences can be found quickly. It also considers parallel algorithms for the problem. There remains open a gap between

*n*and^{2}*n*in the maximum length of reset sequences for general automata.^{3}(BibTeX – Citations – CiteSeer – ACM DL (ICALP) – ACM DL (SJC))

**On reset sequence length**.

D. Eppstein.

Tech. Rep. CUCS-440-89, Computer Science Dept., Columbia University, 1989.(BibTeX)

**The farthest point Delaunay triangulation minimizes angles**.

D. Eppstein.

Tech. Rep. 90-45, ICS, UCI, 1990.

*Comp. Geom. Theory & Applications*1: 143–148, 1992.Given a collection of points in convex position, the sharpest angle determined by any triple can be found as a corner of a triangle in the farthest point Delaunay triangulation.

(BibTeX – Citations – CiteSeer – ACM DL)

**Parallel recognition of series parallel graphs**.

D. Eppstein.

*Information & Computation*98: 41–55, 1992.Characterizes two-terminal series graphs in terms of a tree-like structure in their ear decompositions. Uses this characterization to construct parallel algorithms that recognize these graphs and construct their series-parallel decompositions.

(BibTeX – Citations – MIT hypertext bibliography – CiteSeer – ACM DL)

**Finding the**.*k*smallest spanning trees

D. Eppstein.

*2nd Scand. Worksh. Algorithm Theory,*Bergen, Norway, 1990.

Springer,*Lecture Notes in Comp. Sci.*447, 1990, pp. 38–47.

*BIT*32: 237–248, 1992 (special issue for 2nd Scand. Worksh. Algorithm Theory).By removing edges not involved in some solution, and contracting edges involved in all solutions, we reduce the problem to one in a graph with O(

*k*) edges and vertices. This simplification step transforms any time bound involving*m*or*n*to one involving min(*m,**k*) or min(*n,**k*) respectively. This paper also introduces the geometric version of the*k*smallest spanning trees problem (the graph version was long known) which it solves using order (*k*+1) Voronoi diagrams.(BibTeX – Citations – ACM DL (SWAT) – ACM DL (BIT))

**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)

**Connectivity, graph minors, and subgraph multiplicity**.

D. Eppstein.

Tech. Rep. 92-06, ICS, UCI, 1992.

*J. Graph Th.*17: 409–416, 1993.It was known that planar graphs have O(

*n*) subgraphs isomorphic to K_{3}or K_{4}. That is, K_{3}and K_{4}have linear subgraph multiplicity. This paper shows that the graphs with linear subgraph multiplicity in the planar graphs are exactly the 3-connected planar graphs. Also, the graphs with linear subgraph multiplicity in the outerplanar graphs are exactly the 2-connected outerplanar graphs.More generally, let F be a minor-closed family, and let x be the smallest number such that some complete bipartite graph K

_{x,y}is a forbidden minor for F. Then the x-connected graphs have linear subgraph multiplicity for F, and there exists an (x − 1)-connected graph (namely K_{x − 1,x − 1}) that does not have linear subgraph multiplicity. When x ≤ 3 or when x = 4 and the minimal forbidden minors for F are triangle-free, then the graphs with linear subgraph multiplicity for F are exactly the x-connected graphs.Please refer only to the journal version, and not the earlier technical report: the technical report had a bug that was repaired in the journal version.

**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.**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.**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.

(BibTeX – Citations – CiteSeer – ACM DL)

**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))

**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.(BibTeX – Citations – Closest pair project page – CiteSeer)

**Ten algorithms for Egyptian fractions**.

D. Eppstein.

*Mathematica in Education and Research*4 (2): 5–15, 1995.Number theory. I survey and implement in

*Mathematica*several methods for representing rational numbers as sums of distinct unit fractions. One of the methods involves searching for paths in a certain graph using a*k*shortest paths heuristic.(BibTeX – Citations – Also available in HTML and Mathematica notebook formats)

**Asymptotic speed-ups in constructive solid geometry**.

D. Eppstein.

Tech. Rep. 92-87, ICS, UCI, 1992.

*Algorithmica*13: 462–471, 1995.Finds boundary representations of CSG objects. Uses techniques from dynamic graph algorithms, including a tree partitioning technique of Frederickson and a new data structure for maintaining the value of a Boolean expression with changing variables in time O(log

*n*/ log log*n*) per update.**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)

**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.

(BibTeX – SODA paper – Full paper – Citations – CiteSeer – ACM DL)

**Dynamic geometric optimization**.

D. Eppstein.

Invited talk,*3rd MSI Worksh. Computational Geometry*, 1993, p. 14.

**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.

**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.(BibTeX – Full paper – Citations – Graehl implementation – Jiménez-Marzal implementations – Shibuya implementation – Martins implementation – Cliff OpenStreetMap demo – CiteSeer: TR '94, SJC '98 – ACM DL)

**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.

(BibTeX – Citations – Citations from MIT hypertext bibliography – CiteSeer (SODA) – CiteSeer (JGAA))

**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.(BibTeX – Citations – CiteSeer – MIT hypertext bibliography)

**The diameter of nearest neighbor graphs**.

D. Eppstein.

Tech. Rep. 92-76, ICS, UCI, 1992.Any connected nearest neighbor forest with diameter

*D*has O(*D*^{6}) vertices. This was later further improved to O(*D*^{5}) and merged with results of Paterson and Yao into "On nearest neighbor graphs".(BibTeX – Citations – CiteSeer)

**Minimum range balanced cuts via dynamic subset sums**.

D. Eppstein.

Tech. Rep. 95-10, ICS, UCI, 1995.

*J. Algorithms*23: 375–385, 1997.Describes data structures for maintaining the solution of a dynamically changing subset sum problem, and uses them to find a cut in a graph minimizing the difference between the heaviest and lightest cut edge.

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

**Representing all minimum spanning trees with applications to counting and generation**.

D. Eppstein.

Tech. Rep. 95-50, ICS, UCI, 1995.Shows how to find for any edge weighted graph G an equivalent graph EG such that the minimum spanning trees of G correspond one-for-one with the spanning trees of EG. The equivalent graph can be constructed in time O(m+n log n) given a single minimum spanning tree of G. As a consequence one can find fast algorithms for counting, listing, and randomly generating MSTs. Also discusses similar equivalent graph constructions for shortest paths, minimum cost flows, and bipartite matching.

**Linear complexity hexahedral mesh generation**.

D. Eppstein.

Tech. Rep. 95-51, ICS, UCI, 1995.

*12th ACM Symp. Comp. Geom.,*Philadelphia, 1996, pp. 58–67.

arXiv:cs.CG/9809109.

*Comp. Geom. Theory & Applications*12: 3–16, 1999 (special issue for 12th SCG).Any simply connected polyhedron with an even number of quadrilateral sides can be partitioned into O(n) topological cubes, meeting face to face.

(BibTeX – Citations – SCG paper – SCG talk slides – CiteSeer – ACM DL (CGTA))

**Finding common ancestors and disjoint paths in DAGs**.

D. Eppstein.

Tech. Rep. 95-52, ICS, UCI, 1995.This paper describes algorithms for finding pairs of vertex-disjoint paths in a DAG, either connecting two given nodes to a common ancestor, or connecting two given pairs of terminals. The main results were merged into the journal version of "Finding the

*k*shortest paths".(BibTeX)

**Zonohedra and Zonotopes**.

D. Eppstein.

Tech. Rep. 95-53, ICS, UCI, 1995.

*Mathematica in Education and Research*5 (4): 15–21, 1996.I use Mathematica to construct zonotopes and display zonohedra. Many examples are shown, including one used for a lower bound in "The centroid of points with approximate weights". This paper is also available in HTML and

*Mathematica*notebook formats.(BibTeX – Citations – CiteSeer)

**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*log^{2}*n*).(BibTeX – SODA paper – Citations – DREI and SODA talk slides – CiteSeer)

**Dynamic connectivity in digital images**.

D. Eppstein.

Tech. Rep. 96-13, ICS, UCI, 1996.

*Inf. Proc. Lett.*62: 121–126, 1997.Any algorithm that maintains the connected components of a bitmap image must take Omega(log

*n*/ log log*n*) time per change to the image. The problem can be solved in O(log*n*) time per change using dynamic planar graph techniques. We discuss applications to computer Go and other games.(BibTeX – Citations – MIT hypertext bibliography – CiteSeer – ACM DL)

**On the parity of graph spanning tree numbers**.

D. Eppstein.

Tech. Rep. 96-14, ICS, UCI, 1996.Any bipartite Eulerian graph, any Eulerian graph with evenly many vertices, and any bipartite graph with evenly many vertices and edges, has an even number of spanning trees. More generally, a graph has evenly many spanning trees if and only if it has an Eulerian edge cut.

(BibTeX – Citations – CiteSeer)

**Beta-skeletons have unbounded dilation**.

D. Eppstein.

Tech. Rep. 96-15, ICS, UCI, 1996.

arXiv:cs.CG/9907031.

*Comp. Geom. Theory & Applications*23: 43–52, 2002.Beta-skeletons are geometric graphs used among other purposes for empirical network analysis and minimum weight triangulation. A fractal construction shows that, for any beta>0, the beta-skeleton of a point set can have arbitrarily large dilation: Omega(n

^{c}), where c is a constant depending on beta and going to zero in the limit as beta goes to zero. In particular this applies to the Gabriel graph. We also show upper bounds on dilation of the same form.**Spanning trees and spanners**.

D. Eppstein.

Tech. Rep. 96-16, ICS, UCI, 1996.

*Handbook of Computational Geometry,*J.-R. Sack and J. Urrutia, eds., Elsevier, 1999, pp. 425–461.Surveys results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs.

(BibTeX – Citations – CiteSeer)

**Computational geometry and parametric matroid optimization**.

D. Eppstein.

Invited talk, 5th Int. Symp. Parametric Optimization, Chiba, Japan, 1997.This talk surveys some connections from computational geometry to parametric matroids: the results of my paper "Geometric lower bounds", new upper bounds from a paper by Tamal Dey, and a problem from constructive solid geometry with the potential to lead to stronger lower bounds.

**Fast hierarchical clustering and other applications of dynamic closest pairs**.

D. Eppstein.

*9th ACM-SIAM Symp. Discrete Algorithms,*San Francisco, 1998, pp. 619–628.

arXiv:cs.DS/9912014.

*J. Experimental Algorithmics*5 (1): 1–23, 2000.This paper shows how to use my dynamic closest pair data structure from "Dynamic Euclidean minimum spanning trees" for some non-geometric problems including hierarchical clustering, greedy matching, and TSP heuristics. Experiments show variants of my data structures to be faster than previously used heuristics.

(Source code and experimental data – BibTeX – SODA paper – Citations – CiteSeer – ACM DL – JEA home page)

**Diameter and treewidth in minor-closed graph families**.

D. Eppstein.

arXiv:math.CO/9907126.

*Algorithmica*27: 275–291, 2000 (special issue on treewidth, graph minors, and algorithms).This paper introduces the

*diameter-treewidth property*(later known as*bounded local treewidth*): a functional relationship between the diameter of its graph and its treewidth. Previously known results imply that planar graphs have bounded local treewidth; we characterize the other minor-closed families with this property. Specifically, minor-closed family F has bounded local treewidth if and only if there exists an*apex graph*G that is not in F; an apex graph is a graph that can be made planar by removing a single vertex. The minor-free families that exclude an apex graph (and therefore have bounded local treewidth) include the bounded-genus graphs (for which, as with planar graphs, we show a linear bound for the treewidth as a function of the diameter) and K_{3,a}-free graphs. As a consequence, subgraph isomorphism for subgraphs of bounded size and approximations to several NP-hard optimization problems can be computed efficiently on these graphs, extending previous results for planar graphs.Some of these results were announced in the conference version of "subgraph isomorphism for planar graphs and related problems" but not included in the journal version. Since its publication, there have been many more works on local treewidth. The class of problems that could be solved quickly on graphs of bounded local treewidth was extended and classified by Frick and Grohe, "Deciding first-order properties of locally tree-decomposable structures",

*J. ACM*48: 1184–1206, 2001; the proof that bounded local treewidth is equivalent to having an excluded apex minor was simplified, and the dependence of the treewidth on diameter improved, by a subsequent paper of Demaine and Hajiaghayi, "Diameter and treewidth in minor-closed graph families, revisited",*Algorithmica*40: 211–215, 2004. The concept of local treewidth is the basis for the theory of*bidimensionality*, a general framework for fixed-parameter-tractable algorithms and approximation algorithms in minor-closed graph families; for a survey, see Demaine and Hajiaghayi, "The bidimensionality theory and its algorithmic applications",*The Computer J.*51: 292–302, 2008.**Incremental and decremental maintenance of planar width**.

D. Eppstein.

arXiv:cs.CG/9809038.

*10th ACM-SIAM Symp. Discrete Algorithms,*Baltimore, 1999, pp. S899-S900.

*J. Algorithms*37 (2): 570–577, 2000.We show how to maintain the width of a planar point set, subject to insertions or deletions (but not both) in time O(

*n*^{c}) per update for any*c*> 0. The idea is to apply our dynamic closest pair data structure to an appropriate measure of distance between pairs of convex hull features.(BibTeX – Citations – SODA talk slides – ACM DL)

**Setting parameters by example**.

D. Eppstein.

arXiv:cs.DS/9907001.

*40th IEEE Symp. Foundations of Comp. Sci.*, 1999, pp. 309–318.

*SIAM J. Computing*32 (3): 643–653, 2003.We introduce a class of "inverse parametric optimization" problems, in which one is given both a parametric optimization problem and a desired optimal solution; the task is to determine parameter values that lead to the given solution. We use low-dimensional linear programming and geometric sampling techniques to solve such problems for minimum spanning trees, shortest paths, and other optimal subgraph problems, and discuss applications in multicast routing, vehicle path planning, resource allocation, and board game programming.

(BibTeX – Citations – ACM DL (FOCS) – ACM DL (SJC))

**Tangent spheres and triangle centers**.

D. Eppstein.

arXiv:math.MG/9909152.

*Amer. Math. Monthly*108 (1): 63–66, 2001.Any four mutually tangent spheres determine three coincident lines through opposite pairs of tangencies. As a consequence, we define two new triangle centers. These centers arose as part of a compass-and-straightedge construction of Soddy circles; also available are some Mathematica calculations of trilinear coordinates for the new triangle centers.

(BibTeX – Citations – CiteSeer)

**Searching for spaceships**.

D. Eppstein.

arXiv:cs.AI/0004003.

MSRI Combinatorial Game Theory Research Worksh., July 2000.

*More Games of No Chance*, R. J. Nowakowski, ed., MSRI Publications 42, pp. 433–453.We describe software that searches for spaceships in Conway's Game of Life and related two-dimensional cellular automata. Our program searches through a state space related to the de Bruijn graph of the automaton, using a method that combines features of breadth first and iterative deepening search, and includes fast bit-parallel graph reachability and path enumeration algorithms for finding the successors of each state. Successful results include a new 2c/7 spaceship in Life, found by searching a space with 2^126 states.

(BibTeX – MSRI talk on streaming video and Slides – Citations – CiteSeer)

**Graphs for dynamic geometry**.

D. Eppstein.

Invited talk, Worksh. Dynamic Graph Algorithms, Victoria, Canada, 2000.This talk surveys work on computational geometry algorithms that use dynamic graph data structures, and the different kinds of geometric graph arising in this work.

**Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction**.

D. Eppstein.

arXiv:cs.DS/0009006.

*12th ACM-SIAM Symp. Discrete Algorithms,*Washington, 2001, pp. 329–337.Summarizes recent improvements to "3-Coloring in time O(1.3446

^{n}): a no-MIS algorithm". Merged with that paper for the journal version.(SODA talk slides – BibTeX – SODA paper – CiteSeer)

**Small maximal independent sets and faster exact graph coloring**.

D. Eppstein.

arXiv:cs.DS/0011009.

*7th Worksh. Algorithms and Data Structures,*Providence, Rhode Island, 2001.

Springer,*Lecture Notes in Comp. Sci.*2125, 2001, pp. 462–470.

*J. Graph Algorithms and Applications*(special issue for WADS'01) 7 (2): 131–140, 2003.We show that any graph can be colored in time O(2.415

^{n}), by a dynamic programming procedure in which we extend partial colorings on subsets of the vertices by adding one more color for a maximal independent set. The time bound follows from limiting our attention to maximal independent subsets that are small relative to the previously colored subset, and from bounding the number of small maximal independent subsets that can occur in any graph.(BibTeX – Citations – CiteSeer – WADS talk slides – ACM DL)

**Hinged kite mirror dissection**.

D. Eppstein.

arXiv:cs.CG/0106032.We show that any polygon can be cut into kites, connected into a chain by hinges at their vertices, and that this hinged assemblage can be unfolded and refolded to form the mirror image of the polygon.

**Topological issues in hexahedral meshing**.

D. Eppstein.

Invited talk at Conf. Algebraic Topology Methods in Computer Science, Stanford, 2001.We consider the problem of subdividing a polyhedral domain in R^3 into cuboids meeting face-to-face. For topological subdivisions (cell complexes in which each cell is combinatorially equivalent to a cube, but may not be embedded as a polyhedron) and simply-connected domains, a simple necessary and sufficient condition for the existence of a hexahedral mesh is known: a domain with quadrilateral faces can be meshed if and only if there is an even number of faces. However, conditions for the existence of polyhedral meshes remain open, as do the topological versions of the problem for more complicated domain topologies.

(Slides)

**Triangles and squares**.

D. Eppstein.

Invited talk at EuroConf. Combinatorics, Graph Theory, and Applications, Bellaterra, Spain, 2001, p. 114.Which unit-side-length convex polygons can be formed by packing together unit squares and unit equilateral triangles? For instance one can pack six triangles around a common vertex to form a regular hexagon. It turns out that there is a pretty set of 11 solutions. We describe connections from this puzzle to the combinatorics of 3- and 4-dimensional polyhedra, using illustrations from the works of M. C. Escher and others.

(Slides)

**Separating geometric thickness from book thickness**.

D. Eppstein.

arXiv:math.CO/0109195.We show that geometric thickness and book thickness are not asymptotically equivalent: for every

*t*, there exists a graph with geometric thickness two and book thickness__>__*t*.**Global optimization of mesh quality**.

D. Eppstein.

Tutorial at 10th Int. Meshing Roundtable, Newport Beach, 2001.Delaunay triangulation has been a staple of triangular mesh generation for a long time. Why? As well as being simple, fast, and visually pleasing, Delaunay triangulations can be shown to be optimal for various measures of mesh quality; for instance, they avoid sharp angles to the maximum extent possible. We review these and other results on construction of meshes that optimize given quality measures, including recent work on postprocessing tetrahedral meshes to eliminate slivers.

**Separating thickness from geometric thickness**.

D. Eppstein.

arXiv:math.CO/0204252.

10th Int. Symp. Graph Drawing, Irvine, 2002.

Springer,*Lecture Notes in Comp. Sci.*2528, 2002, pp. 150–161.

In*Towards a Theory of Geometric Graphs*, AMS, 2004, Contemporary Math 342, J. Pach, ed., pp. 75–86.We show that thickness and geometric thickness are not asymptotically equivalent: for every

*t*, there exists a graph with thickness three and geometric thickness__>__*t*. The proof uses Ramsey-theoretic arguments similar to those in "Separating book thickness from thickness".(BibTeX – GD'02 talk slides – Citations – ACM DL)

**Dynamic generators of topologically embedded graphs**.

D. Eppstein.

arXiv:cs.DS/0207082.

*14th ACM-SIAM Symp. Discrete Algorithms,*Baltimore, 2003, pp. 599–608.We describe a decomposition of graphs embedded on 2-dimensional manifolds into three subgraphs: a spanning tree, a dual spanning tree, and a set of leftover edges with cardinality determined by the genus of the manifold. This tree-cotree decomposition allows us to find efficient data structures for dynamic graphs (allowing updates that change the surface), better constants in bounded-genus graph separators, and efficient algorithms for tree-decomposition of bounded-genus bounded-diameter graphs.

(BibTeX – SODA talk slides – Citations)

**Möbius transformations in scientific computing**.

D. Eppstein.

Talk at SIAM Conf. on Computational Science and Engineering, San Diego, 2003.This talk, for the CSE session on combinatorial scientific computing, surveys my work with Marshall Bern on optimal Möbius transformation and Möbius-invariant natural neighbor transformation.

(Slides)

**The traveling salesman problem for cubic graphs**.

D. Eppstein.

arXiv:cs.DS/0302030.

*8th Worksh. Algorithms and Data Structures,*Ottawa, 2003.

Springer,*Lecture Notes in Comp. Sci.*2748, 2003, pp. 307–318.*J. Graph Algorithms and Applications*11 (1): 61–81, 2007.We find improved exponential-time algorithms for exact solution of the traveling salesman problem on graphs of maximum degree three and four. We also consider related problems including counting the number of Hamiltonian cycles in such graphs.

(BibTeX – WADS talk slides – Citations)

**Quasiconvex analysis of backtracking algorithms**.

D. Eppstein.

arXiv:cs.DS/0304018.

*15th ACM-SIAM Symp. Discrete Algorithms,*New Orleans, 2004, pp. 781–790.

*ACM Trans. Algorithms*2 (4): 492–509 (special issue for SODA 2004), 2006.We consider a class of multivariate recurrences frequently arising in the worst case analysis of Davis-Putnam-style exponential time backtracking algorithms for NP-hard problems. We describe a technique for proving asymptotic upper bounds on these recurrences, by using a suitable weight function to reduce the problem to that of solving univariate linear recurrences; show how to use quasiconvex programming to determine the weight function yielding the smallest upper bound; and prove that the resulting upper bounds are within a polynomial factor of the true asymptotics of the recurrence. We develop and implement a multiple-gradient descent algorithm for the resulting quasiconvex programs, using a real-number arithmetic package for guaranteed accuracy of the computed worst case time bounds.

The journal version uses the longer title "Quasiconvex analysis of multivariate recurrence equations for backtracking algorithms".

(BibTeX – SODA talk slides – Citations)

**Depth and arrangements**.

D. Eppstein.

Invited talk at DIMACS Worksh. on Data Depth, New Brunswick, NJ, 2003.

Invited talk at MSRI Introductory Worksh. Discrete & Computational Geometry, Berkeley, CA, 2003.Surveys projective duality and its uses in algorithms for robust regression. The MSRI talk used the alternative title "Computational geometry and robust statistics" but contained essentially the same content.

(DIMACS talk slides – video and slides of MSRI talk)

**Quasiconvex programming**.

D. Eppstein.

Invited talk at DIMACS Worksh. on Geometric Optimization, New Brunswick, NJ, 2003.

Plenary talk at ALGO 2004, Bergen, Norway, 2004.

arXiv:cs.CG/0412046.

*Combinatorial and Computational Geometry*, Goodman, Pach, and Welzl, eds., MSRI Publications 52, 2005, pp. 287–331.Defines

*quasiconvex programming*, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. Surveys algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.(DIMACS talk slides – ALGO talk slides)

**Guest editor's forward**.

D. Eppstein.

*Disc. Comp. Geom.*30: 1–2, 2003 (special issue for 17th Symp. Comp. Geom.)(BibTeX)

**Testing bipartiteness of geometric intersection graphs**.

D. Eppstein.

arXiv:cs.CG/0307023.

*15th ACM-SIAM Symp. Discrete Algorithms,*New Orleans, 2004, pp. 853–861.

*ACM Trans. Algorithms*5(2):15, 2009.We consider problems of partitioning sets of geometric objects into two subsets, such that no two objects within the same subset intersect each other. Typically, such problems can be solved in quadratic time by constructing the intersection graph and then applying a graph bipartiteness testing algorithm; we achieve subquadratic times for general objects, and O(n log n) times for balls in R^d or simple polygons in the plane, by using geometric data structures, separator based divide and conquer, and plane sweep techniques, respectively. We also contrast the complexity of bipartiteness testing with that of connectivity testing, and provide evidence that for some classes of object, connectivity is strictly harder due to a computational equivalence with Euclidean minimum spanning trees.

(BibTeX – Citations – SODA talk slides)

**Hyperbolic geometry, Möbius transformations, and geometric optimization**.

D. Eppstein.

Invited talk at MSRI Introductory Worksh. Discrete & Computational Geometry, Berkeley, CA, 2003.Describes extensions of computational geometry algorithms to hyperbolic geometry, including an output-sensitive 3d Delaunay triangulation algorithm of Boissonat et al. and my own research on optimal Möbius transformation.

**The lattice dimension of a graph.**

D. Eppstein.

arXiv:cs.DS/0402028.

*Eur. J. Combinatorics*26 (6): 585–592, 2005.Describes a polynomial time algorithm for isometrically embedding graphs into an integer lattice of the smallest possible dimension. The technique involves maximum matching in an auxiliary graph derived from a partial cube representation of the input.

**Algorithms for drawing media.**

D. Eppstein.

arXiv:cs.DS/0406020.

12th Int. Symp. Graph Drawing, New York, 2004.

Springer,*Lecture Notes in Comp. Sci.*3383, 2004, pp. 173–183.We describe two algorithms for finding planar layouts of partial cubes: one based on finding the minimum-dimension lattice embedding of the graph and then projecting the lattice onto the plane, and the other based on representing the graph as the planar dual to a weak pseudoline arrangement.

Due to editorial mishandling there will be no journal version of this paper: I submitted it to a journal in 2004, the reviews were supposedly sent back to me in 2005, but I didn't receive them and didn't respond to them, leading the editors to assume that I intended to withdraw the submission. Large portions of the paper have since been incorporated into my book

*Media Theory*, making journal publication moot.(GD04 talk slides – BibTeX – Citations – GDEA)

**All maximal independent sets and dynamic dominance for sparse graphs.**

D. Eppstein.

arXiv:cs.DS/0407036.

*16th ACM-SIAM Symp. Discrete Algorithms,*Vancouver, 2005, pp. 451–459.

*ACM Trans. Algorithms*5(4):A38, 2009.We show how to apply reverse search to list all maximal independent sets in bounded-degree graphs in constant time per set, in graphs from minor closed families in linear time per set, and in sparse graphs in subquadratic time per set. The latter two results rely on new data structures for maintaining a dynamic vertex set in a graph and quickly testing whether the set dominates all other vertices.

**Comment on "Location-Scale Depth".**

D. Eppstein.

*J. Amer. Stat. Assoc.*99 (468): 976–979, 2004.Discusses a paper by Mizera and Müller on depth-based methods for simultaneously fitting both a center and a radius to a set of sample points, by viewing the points as lying on the boundary of a model of a higher dimensional hyperbolic space. Reformulates their method in combinatorial terms more likely to be familiar to computational geometers, and discusses the algorithmic implications of their work.

**Nonrepetitive paths and cycles in graphs with application to Sudoku**.

D. Eppstein.

arXiv:cs.DS/0507053.We describe algorithms and hardness results for finding paths in edge-labeled graphs such that no two consecutive edges have the same label, and use our algorithms to implement heuristics for a program that automatically solves and generates Sudoku puzzles.

**Cubic partial cubes from simplicial arrangements.**

D. Eppstein.

arXiv:math.CO/0510263.

*Electronic J. Combinatorics*13(1) #R79, 2006.We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.

**Squarepants in a tree: sum of subtree clustering and hyperbolic pants decomposition.**

D. Eppstein.

arXiv:cs.CG/0604034.

*18th ACM-SIAM Symp. Discrete Algorithms,*New Orleans, 2007, pp. 29–38.

*ACM Trans. Algorithms*5(3): Article 29, 2009.We find efficient constant factor approximation algorithms for hierarchically clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition with approximately minimum total length.

(Slides)

**Upright-quad drawing of st-planar learning spaces.**

D. Eppstein.

arXiv:cs.CG/0607094.

14th Int. Symp. Graph Drawing, Karlsruhe, Germany, 2006.

Springer,*Lecture Notes in Comp. Sci.*4372, 2007, pp. 282–293.

*J. Graph Algorithms and Applications*12 (1): 51–72, 2008 (special issue for GD'06).We consider graph drawing algorithms for learning spaces, a type of st-oriented partial cube derived from antimatroids and used to model states of knowledge of students. We show how to draw any st-planar learning space so all internal faces are convex quadrilaterals with the bottom side horizontal and the left side vertical, with one minimal and one maximal vertex. Conversely, every such drawing represents an $st$-planar learning space. We also describe connections between these graphs and arrangements of translates of a quadrant.

**Happy endings for flip graphs.**

D. Eppstein.

arXiv:cs.CG/0610092.

*23rd ACM Symp. Comp. Geom.,*Gyeongju, South Korea, 2007, pp. 92–101.

*J. Computational Geometry*1 (1): 3–28, 2010.We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.

**Manhattan orbifolds.**

D. Eppstein.

arXiv:math.MG/0612109.

*Topology and its Applications*157 (2): 494–507, 2009.We investigate a class of metrics for 2-manifolds in which, except for a discrete set of singular points, the metric is locally isometric to an L

_{1}metric, and show that with certain additional conditions such metrics are injective. We use this construction to find the tight span of squaregraphs and related graphs, and we find an injective metric that approximates the distances in the hyperbolic plane analogously to the way the rectilinear metrics approximate the Euclidean distance.**Recognizing partial cubes in quadratic time**.

D. Eppstein.

arXiv:0705.1025.

*19th ACM-SIAM Symp. Discrete Algorithms,*San Francisco, 2008, pp. 1258–1266.

*J. Graph Algorithms and Applications*15 (2): 269–293, 2011.We show how to test whether a graph is a partial cube, and if so embed it isometrically into a hypercube, in time O(n

^{2}), improving previous O(nm)-time solutions; here n is the number of vertices and m is the number of edges. The ideas are to use bit-parallelism to speed up previous approaches to the embedding step, and to verify that the resulting embedding is isometric using an all-pairs shortest path algorithm from "algorithms for media".(slides)

**Geometry of partial cubes**.

D. Eppstein.

Invited talk at 6th Slovenian International Conference on Graph Theory, Bled, Slovenia, 2007.I survey some of my recent results on geometry of partial cubes, including lattice dimension, graph drawing, cubic partial cubes, and partial cube flip graphs of triangulations.

(Slides)

**The Complexity of Bendless Three-Dimensional Orthogonal Graph Drawing**.

D. Eppstein.

arXiv:0709.4087.

*Proc. 16th Int. Symp. Graph Drawing*, Heraklion, Crete, 2008.

Springer,*Lecture Notes in Comp. Sci.*5417, 2009, pp. 78–89.

*J. Graph Algorithms and Applications*17 (1): 35–55, 2013.Defines a class of orthogonal graph drawings formed by a point set in three dimensions for which axis-parallel line contains zero or two vertices, with edges connecting pairs of points on each nonempty axis-parallel line. Shows that the existence of such a drawing can be defined topologically, in terms of certain two-dimensional surface embeddings of the same graph. Based on this equivalence, describes algorithms, graph-theoretic properties, and hardness results for graphs of this type.

(Slides from talk at U. Arizona, February 2008 – Slides from GD08) )

**Learning sequences**.

D. Eppstein.

arXiv:0803.4030.

How to implement an antimatroid, with applications in computerized education.

**Principles of Graph Drawing**.

D. Eppstein.

Talk at Inst. for Mathematical Behavioral Sciences, May 2008.

Talk at Technical University of Eindhoven, September 2008.**Isometric diamond subgraphs**.

D. Eppstein.

arXiv:0807.2218.

*Proc. 16th Int. Symp. Graph Drawing*, Heraklion, Crete, 2008.

Springer,*Lecture Notes in Comp. Sci.*5417, 2009, pp. 384–389.

We describe polynomial time algorithms for determining whether an undirected graph may be embedded in a distance-preserving way into the hexagonal tiling of the plane, the diamond structure in three dimensions, or analogous structures in higher dimensions. The graphs that may be embedded in this way form an interesting subclass of the partial cubes.

(Slides)

**Finding large clique minors is hard**.

D. Eppstein.

arXiv:0807.0007.

*J. Graph Algorithms and Applications*13 (2): 197–204, 2009.Proves that it's NP-complete to compute the Hadwiger number of a graph.

**Paired approximation problems and incompatible inapproximabilities**.

D. Eppstein.

arXiv:0909.1870.

*21st ACM-SIAM Symp. Discrete Algorithms*, Austin, Texas, 2010, pp. 1076–1086.Considers situations in which two hard approximation problems are presented by means of a single input. In many cases it is possible to guarantee that one or the other problem can be approximated to within a better approximation ratio than is possible for approximating it as a single problem. For instance, it is possible to find either a (1+ε)-approximation to a 1-2 TSP defined from a graph or a n

^{ε}-approximation to the maximum independent set of the same graph, despite lower bounds showing nonexistence of approximation schemes for 1-2 TSP and nonexistence of approximations better than n^{1 − ε}for independent set. However, for some other pairs of problems, such as hitting set and set cover, we show that solving the paired problem approximately is no easier than solving either problem independently.(Slides)

**Optimally fast incremental Manhattan plane embedding and planar tight span construction**.

D. Eppstein.

arXiv:0909.1866.

*J. Computational Geometry*2 (1): 144–182, 2011.Shows that, when the tight span of a finite metric space is homeomorphic to a subset of the plane, it has the geometry of a Manhattan orbifold and can be constructed in time linear in the size of the input distance matrix. As a consequence, it can be tested in the same time whether a metric space is isometric to a subset of the L

_{1}plane.**Hyperconvexity and metric embedding**.

D. Eppstein.

Invited talk at Fifth William Rowan Hamilton Geometry and Topology Workshop, Dublin, Ireland, 2009.

Invited talk at IPAM Workshop on Combinatorial Geometry, UCLA, 2009.Surveys hyperconvex metric spaces, tight spans, and my work on Manhattan orbifolds, tight span construction, and optimal embedding into star metrics.

**Growth and decay in life-like cellular automata**.

D. Eppstein.

arXiv:0911.2890.

*Game of Life Cellular Automata*(Andrew Adamatzky, ed.), Springer, 2010, pp. 71–98.We classify semi-totalistic cellular automaton rules according to whether patterns can escape any finite bounding box and whether patterns can die out completely, with the aim of finding rules with interesting behavior similar to Conway's Game of Life. We survey a number of such rules.

**Regular labelings and geometric structures**.

D. Eppstein.

arXiv:1007.0221.

Invited to 22nd Canadian Conference on Computational Geometry (CCCG 2010).

Invited to*Proc. 21st International Symposium on Algorithms and Computation (ISAAC 2010)*, Jeju, Korea, 2010.

Springer,*Lecture Notes in Comp. Sci.*6506, 2010, p. 1.

We survey regular labelings for straight-line embedding of planar graphs on grids, rectangular partitions, and orthogonal polyhedra, and the many similarities between these different types of labeling.

**Densities of minor-closed graph families**.

D. Eppstein.

arXiv:1009.5633.

*Electronic J. Combinatorics*17(1), Paper R136, 2010.For every minor-closed graph family (such as the family of planar graphs), there is a constant c such that the maximum number of edges in an n-vertex graph in the family is

*c*(*n*+*o*(*n*); for instance, for planar graphs,*c*= 3. We call*c*the limiting density of the family, and we study the set of all limiting densities of all minor-closed graph families. We show that this set of limiting densities is closed and well-ordered, with order type at least ω^{ω}, and we find an exact description of the members of this set up to its first two cluster points 1 and 3/2.**Solving single-digit Sudoku subproblems**.

D. Eppstein.

arXiv:1202.5074.

*6th International Conference on Fun with Algorithms (FUN 2012)*, Venice, Italy, 2012.

Springer,*Lecture Notes in Comp. Sci.*7288, 2012, pp. 142–153.We find an algorithm for making all possible deductions based on the set of candidate locations for a single digit in a Sudoku puzzle; the problem is NP-hard, and our algorithm takes exponential time, but the mild form of the exponential allows it to be fast for practical problem sizes.

(Slides)

**Diamond-kite meshes: adaptive quadrilateral meshing and orthogonal circle packing**.

D. Eppstein.

arXiv:1207.5082.

*21st International Meshing Round Table*, San Jose, California, 2012, pp. 261–277.

*Engineering with Computers*30 (2): 223–235 (special issue for the 21st Int. Meshing Roundtable), 2014.We describe a recursive subdivision of the plane into quadrilaterals in the form of rhombi and kites with 60, 90, and 120 degree angles. The vertices of the resulting quadrilateral mesh form the centers of a set of circles that cross orthogonally for every two adjacent vertices, and it has many other properties that are important in finite element meshing.

(Slides)

**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.

**Planar Lombardi drawings for subcubic graphs**.

D. Eppstein.

arXiv:1206.6142.

*20th Int. Symp. Graph Drawing*, Redmond, Washington, 2012.

Springer,*Lecture Notes in Comp. Sci.*7704, 2013, pp. 126–137.

We show that every planar graph of maximum degree three has a planar Lombardi drawing and that some but not all 4-regular planar graphs have planar Lombardi drawings. The proof idea combines circle packings with a form of Möbius-invariant power diagram for circles, defined using three-dimensional hyperbolic geometry.

For the journal version, see "A Möbius-invariant power diagram and its applications to soap bubbles and planar lombardi drawing.".

(Slides)

**The graphs of planar soap bubbles**.

D. Eppstein.

arXiv:1207.3761.

*Proc. 29th ACM Symp. on Computational Geometry*, Rio de Janeiro, 2013, pp. 27–36.We characterize the graphs of two-dimensional soap bubble clusters as being exactly the bridgeless 3-regular planar graphs. The proof uses the Möbius invariance of the properties characterizing these clusters together with our previous circle packing method for constructing Lombardi drawings of graphs.

For the journal version, see "A Möbius-invariant power diagram and its applications to soap bubbles and planar lombardi drawing.".

(Slides)

**Antimatroids and balanced pairs**.

D. Eppstein.

arXiv:1302.5967.

*Order*31 (1): 81–99, 2014. Erratum (with V. Wiechert).We generalize the 1/3-2/3 conjecture, according to which every partial order should have a pair of items that are nearly equally likely to appear in either order in a random linear extension, to antimatroids, and we prove it for several specific types of antimatroid.

**Grid minors in damaged grids**.

D. Eppstein.

arXiv:1303.1136.

*Electronic J. Combinatorics*21(3), Paper P3.20, 2014.We give tight bounds on the size of the largest remaining grid minor in a grid graph from which a given number of vertices have been deleted, and study several related problems.

**Learning sequences: an efficient data structure for learning spaces**.

D. Eppstein.

In*Knowledge Spaces: Applications in Education*, J.-C. Falmagne, D. Albert, C. Doble, D. Eppstein, and X. Hu, eds.

Springer, 2013, pp. 287–304.We show how to represent a learning space by a small family of learning sequences, orderings of the items in a learning sequence that are consistent with their prerequisite relations. This representation allows for the rapid generation of the family of all consistent knowledge states and the efficient assessment of the state of knowledge of a human learner.

**Projection, decomposition, and adaption of learning spaces**.

D. Eppstein.

In*Knowledge Spaces: Applications in Education*, J.-C. Falmagne, D. Albert, C. Doble, D. Eppstein, and X. Hu, eds.

Springer, 2013, pp. 305–322.In another chapter of the same book we used learning sequences to represent learning spaces and perform efficient knowledge assessment of a human learning. In this chapter we show how to use the same data structure to build learning spaces on a sample of the items of a larger learning space (an important subroutine in knowledge assessment) and to modify a learning space to more accurately model students.

**A brief history of curves in graph drawing**.

D. Eppstein.

Invited survey talk at Workshop on Drawing Graphs and Maps with Curves, Dagstuhl, Germany, April 2013.

*Drawing Graphs and Maps with Curves (Dagstuhl Seminar 13151)*, S. G. Kobourov, M. Nöllenburg, and M. Teillaud, eds., Dagstuhl Reports 3(4): 40–46, 2013.(Slides)

**Drawing arrangement graphs in small grids, or how to play planarity**.

D. Eppstein.

arXiv:1308.0066.

*21st Int. Symp. Graph Drawing*, Bordeaux, France, 2013.

Springer,*Lecture Notes in Comp. Sci.*8242, 2013, pp. 436–447.

*J. Graph Algorithms & Applications*18 (2): 211–231, 2014 (special issue for GD'13).The planarity game involves rearranging a scrambled line arrangement graph to make it planar. We show that the resulting graphs have drawings in grids of area

*n*^{7/6}, much smaller than the quadratic size bound for grid drawings of planar graphs, and we provide a strategy for planarizing these graphs that is simple enough for human puzzle solving.**Structures in solution spaces: Three lessons from Jean-Claude**.

D. Eppstein.

Invited talk, Conference on Meaningfulness and Learning Spaces: A Tribute to the Work of Jean-Claude Falmagne

Irvine, California, 2014.This talk surveys my work on rectangular cartograms, the 1/3-2/3 conjecture for antimatroids, and flip distance for triangulations of point sets with no empty pentagon, and how this line of research stemmed from the work of Jean-Claude Falmagne on learning spaces.

.*k*-best enumeration

D. Eppstein.

*Encyclopedia of Algorithms*(Ming-Yang Kao, ed.), Springer, added 2014.

arXiv:1412.5075.

*Bull. EATCS*115, 2015.A brief survey of algorithms for finding the

*k*shortest paths and related*k*-best enumeration problems. The arXiv/EATCS version is significantly longer and with more references than the Springer version.**Simple recognition of Halin graphs and their generalizations**.

D. Eppstein.

arXiv:1502.05334.

*J. Graph Algorithms & Applications*20 (2): 323–346, 2016.We describe and implement a simple linear time algorithm for recognizing Halin graphs based on two simplifications of triples of degree-three vertices in such graphs. Removing some auxiliary data from the algorithm causes it to recognize a broader class of graphs that we call the D3-reducible graphs. We study the properties of these graphs, showing that they share many properties with the Halin graphs.

**The parametric closure problem**.

D. Eppstein.

arXiv:1504.04073.

14th Algorithms and Data Structures Symp. (WADS 2015), Victoria, BC.

Springer,*Lecture Notes in Comp. Sci.*9214 (2015), pp. 327–338.

*ACM Trans. Algorithms*14 (1): Article 2, 2018.We consider the minimum weight closure problem for a partially ordered set whose elements have weights that vary linearly as a function of a parameter. For several important classes of partial orders the number of changes to the optimal solution as the parameter varies is near-linear, and the sequence of optimal solutions can be found in near-linear time.

(Slides)

**Metric dimension parameterized by max leaf number**.

D. Eppstein.

arXiv:1506.01749.

*J. Graph Algorithms & Applications*19 (1): 313–323, 2015.We prove that when a graph of bounded size has its edges subdivided to form a larger graph, it is still possible to find its metric dimension by a fixed-parameter tractable algorithm, parameterized by the pre-subdivision size of the graph.

**Treetopes and their graphs**.

D. Eppstein.

arXiv:1510.03152.

*27th ACM-SIAM Symp. on Discrete Algorithms*, Arlington, 2016, pp. 969–984.

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.

(Slides)

**Cuckoo filter: simplification and analysis**.

D. Eppstein.

arXiv:1604.06067.

*Proc. 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)*, Reykjavik, Iceland.

Leibniz International Proceedings in Informatics (LIPIcs) 53, pp. 8.1–8.12.A cuckoo filter is an approximate set data structure that can be used in place of a Bloom filter, but with several practical advantages: it uses less space, has better locality of reference, and can handle element deletions. We provide the first theoretical analysis of a simplified variation of cuckoo filters, showing that these advantages can be guaranteed to hold theoretically and not just experimentally.

(Slides)

**Maximizing the sum of radii of disjoint balls or disks**.

D. Eppstein.

arXiv:1607.02184.

*Proc. 28th Canadian Conference on Computational Geometry*, Vancouver, BC, Canada, 2016, pp. 260–265.

*J. Computational Geometry*8 (1): 316–339, 2017.

We show how to find a system of disjoint balls with given centers, maximizing the sum of radius of the balls. Our algorithm takes cubic time in arbitrary metric spaces and can be sped up to subquadratic time in Euclidean spaces of any bounded dimension.

(Slides)

**Forbidden configurations in discrete geometry**.

D. Eppstein.

Paul Erdős Memorial Lecture, 29th Canadian Conference on Computational Geometry, Ottawa, Canada, 2017.

Invited talk, 20th Japan Conference on Discrete and Computational Geometry, Graphs, and Games, Tokyo, 2017.

Invited talk, 5th International Combinatorics Conference, Melbourne, Australia, 2017.

Cambridge University Press, to appear.We survey problems on finite sets of points in the Euclidean plane that are monotone under removal of points and depend only on the order-type of the points, and the subsets of points (forbidden configurations) that prevent a point set from having a given monotone property.

(CCCG talk slides – CCCG talk video)

**Triangle-free penny graphs: degeneracy, choosability, and edge count**.

D. Eppstein.

arXiv:1708.05152.

*Proc. 25th Int. Symp. Graph Drawing*, Boston, Massachusetts, 2017.

Springer,*Lecture Notes in Comp. Sci.*10692 (2018), pp. 506–513.

*J. Graph Algorithms & Applications*, to appear.A penny graph is the contact graph of unit disks: each disk represents a vertex of the graph, no two disks can overlap, and each tangency between two disks represents an edge in the graph. We prove that, when this graph is triangle free, its degeneracy is at most two. As a consequence, triangle-free penny graphs have list chromatic number at most three. We also show that the number of edges in any such graph is at most 2

*n*− Ω(√*n*).(Slides)

**The effect of planarization on width**.

D. Eppstein.

arXiv:1708.05155.

*Proc. 25th Int. Symp. Graph Drawing*, Boston, Massachusetts, 2017.

Springer,*Lecture Notes in Comp. Sci.*10692 (2018), pp. 560–572.

*J. Graph Algorithms & Applications*, to appear.We study what happens to nonplanar graphs of low width (for various width measures) when they are made planar by replacing crossings by vertices. For treewidth, pathwidth, branchwidth, clique-width, and tree-depth, this replacement can blow up the width from constant to linear. However, for bandwidth, cutwidth, and carving width, graphs of bounded width stay bounded when we planarize them.

(Slides)

**Faster evaluation of subtraction games**.

D. Eppstein.

arXiv:1804.06515.

*Proc. 9th Int. Conf. Fun with Algorithms (FUN 2018)*, La Maddalena, Italy, to appear.

We show how to evaluate the set of winning heap sizes in subtraction games like subtract-a-square in near-linear time, and how to compute the nim-values more quickly than naive dynamic programming. Additionally we perform computational experiments showing that the set of winning positions forms an unexpectedly dense square-difference-free set.

**Making change in 2048**.

D. Eppstein.

arXiv:1804.07396.

*Proc. 9th Int. Conf. Fun with Algorithms (FUN 2018)*, La Maddalena, Italy, to appear.

The 2048 puzzle, modified to use any sequence of integer tile values that has arbitrarily large gaps, always terminates. The proof relates the puzzle to the greedy algorithm for making change (suboptimally) using a given system of coins.

Co-authors – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine

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