Solo papers
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.
(TeX source code – 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.
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.
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".
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 n2 and n3 in the maximum length of reset sequences for general automata.
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.
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.
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.
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.
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 \(\mathcal{F}\) 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 \(\mathcal{F}\). Then the \(x\)-connected graphs have linear subgraph multiplicity for \(\mathcal{F}\), and there exists an \((x-1)\)-connected graph (namely \(K_{x-1,x-1}\) that does not have linear subgraph multiplicity. When \(x\le 3\) or when \(x=4\) and the minimal forbidden minors for \(\mathcal{F}\) are triangle-free, then the graphs with linear subgraph multiplicity for \(\mathcal{F}\) 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).
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.
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.
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.
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.
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.
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.
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".
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)
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.
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 – Cliff OpenStreetMap demo)
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.
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.
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(D6) vertices. This was later further improved to O(D5) and merged with results of Paterson and Yao into "On nearest neighbor graphs".
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.
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.
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".
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.
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 log2 n).
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.
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.
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(nc), 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.
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.
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 K3,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(nc) 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.
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.
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.
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.
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(1.3446n): a no-MIS algorithm". Merged with that paper for the journal version.
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(2.415n), 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.
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.
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.
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".
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.
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.
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.
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".
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.
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.
Guest editor's forward.
D. Eppstein.
Disc. Comp. Geom. 30: 1–2, 2003 (special issue for 17th Symp. Comp. Geom.)
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.
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.
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), Paper 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.
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 L1 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(n2), 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".
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.
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.
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.
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.
Graph-theoretic solutions to computational geometry problems.
D. Eppstein.
Invited talk, the 35th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2009), Montpellier, France, 2009.
Springer, Lecture Notes in Comp. Sci. 5911, 2009, pp. 1–16.
arXiv:0908.3916.We survey problems in computational geometry that may be solved by constructing an auxiliary graph from the problem and solving a graph-theoretic problem on the auxiliary graph. The problems considered include the art gallery problem, partitioning into rectangles, minimum diameter clustering, bend minimization in cartogram construction, mesh stripification, optimal angular resolution, and metric embedding.
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 nε-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 n1 − 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.
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 L1 plane.
Hyperconvexity and metric embedding.
D. Eppstein.
Invited talk, 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 ωω, 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.
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.
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.".
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.".
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.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 n7/6, 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.
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.
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.
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.
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.
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.
Invited talk, Southern California Theory Day, Irvine, California, 2018.
Cambridge University Press, 2018.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 – SCTD talk slides – Updates, errata, and reviews)
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 22 (3): 483–499 (special issue for GD 2017), 2018.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 2n − Ω(√n). The journal version uses the alternative title "Edge Bounds and Degeneracy of Triangle-Free Penny Graphs and Squaregraphs".
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 22 (3): 461–481 (special issue for GD 2017), 2018.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.
Faster evaluation of subtraction games.
D. Eppstein.
arXiv:1804.06515.
Proc. 9th Int. Conf. Fun with Algorithms (FUN 2018), La Maddalena, Italy.
Leibniz International Proceedings in Informatics (LIPIcs) 100, pp. 20:1–20:12.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.
Leibniz International Proceedings in Informatics (LIPIcs) 100, pp. 21:1–21:13.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.
Which women mathematicians get written about on Wikipedia, and why.
D. Eppstein.
AWM Newsletter 48 (4): 12–14, July–August 2018.This short opinion piece explains Wikipedia's guidelines for inclusion of articles on academics, and outlines steps to help improve Wikipedia's coverage of women in mathematics.
Realization and connectivity of the graphs of origami flat foldings.
D. Eppstein.
arXiv:1808.06013.
Proc. 26th Int. Symp. Graph Drawing, Barcelona, 2018.
Springer, Lecture Notes in Comp. Sci. 11282 (2018), pp. 541–554.
J. Computational Geometry 10 (1): 257–280, 2019.If you fold a piece of paper flat and unfold it again, the resulting crease pattern forms a planar graph. We prove that a tree can be realized in this way (with its leaves as diverging rays that reach the boundary of the paper) if and only if all internal vertices have odd degree greater than two. On the other hand, for a folding pattern on an infinite sheet of paper with an added vertex at infinity as the endpoint of all its rays, the resulting graph must be 2-vertex-connected and 4-edge-connected.
Stable-matching Voronoi diagrams.
D. Eppstein.
Invited talk at 21st Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG3), Manila, Philippines, 2018.We survey the results from several of my earlier papers: Algorithms for stable matching and clustering in a grid, Defining equitable geographic districts in road networks via stable matching, Reactive proximity data structures for graphs, and Stable-matching Voronoi diagrams: combinatorial complexity and algorithms.
Cubic planar graphs that cannot be drawn on few lines.
D. Eppstein.
arXiv:1903.05256.
Proc. 35th Int. Symp. on Computational Geometry, Portland, Oregon, June 2019.
Leibniz International Proceedings in Informatics (LIPIcs) 129, 2019, pp. 32:1–32:15.
J. Computational Geometry 12 (1): 178–197, 2021.We construct planar graphs whose straight drawings require a large number of lines (at least the cube root of the number of vertices) to cover all vertices. We also find series-parallel graphs and apex-trees that require a non-constant number of lines.
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.
Bipartite and series-parallel graphs without planar Lombardi drawings.
D. Eppstein.
arXiv:1906.04401.
Proc. 31st Canadian Conference on Computational Geometry, Edmonton, Canada, 2019, pp. 17–22.
J. Graph Algorithms & Applications 25 (1): 549–562, 2021.Not every bipartite planar graph has a planar Lombardi drawing. Not every plane series parallel graph has a planar Lombardi drawing with the same embedding. The proof involves the properties of cycles of circular-arc-quadrilaterals all sharing the same two vertices, with equal and sharp vertex angles.
Graphs in nature.
D. Eppstein.
Invited talk at Symposium on Geometry Processing, Milan, Italy, July 2019.
Invited talk at Algorithms and Data Structures Symposium, Edmonton, Canada, August 2019.
Invited talk at International Colloquium on Graph Theory and Combinatorics, Montpellier, France, July 2022.I survey results on characterizing the graphs formed on planar surfaces according to several natural processes: motorcycle graphs, Gilbert tessellations, and the contact graphs of line segments from needle-like crystal growth and crack formation; the graphs of planar soap bubble foams; and graphs representing the fold patterns of crumpled paper.
Dynamic products of ranks.
D. Eppstein.
arXiv:2007.08123.
Proc. 32nd Canadian Conference on Computational Geometry, 2020, pp. 199–205.We provide data structures for the following problem: maintain a collection of points in the Euclidean plane, subject to insertions or deletions, and after each update find the point whose product of ranks according to the two sorted orders of the points by x- and y-coordinates is as small or as large as possible.
On polyhedral realization with isosceles triangles.
D. Eppstein.
arXiv:2009.00116.
Graphs and Combinatorics 37 (4): 1247–1269, 2021.We find convex polyhedra with all faces triangular that cannot be realized with all faces isosceles, and new families of convex polyhedra with congruent isosceles-triangle faces.
The graphs of stably matchable pairs.
D. Eppstein.
arXiv:2010.09230.
47th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2021).
Springer, Lecture Notes in Comp. Sci. 12911 (2021), pp. 349–360.If you form a bipartite graph from a stable matching instance, with an edge for each pair that can participate in a stable matching, what graphs can you get? They are matching-covered, but NP-hard to recognize, and their structure is related to the structure of the lattice of stable matchings of the same instance.
A stronger lower bound on parametric minimum spanning trees.
D. Eppstein.
arXiv:2105.05371.
17th Algorithms and Data Structures Symp. (WADS 2021).
Springer, Lecture Notes in Comp. Sci. 12808 (2021), pp. 343–356.
Algorithmica 85: 1738–1753, 2023 (special issue for WADS 2021).There exist graphs with edges labeled by linear real functions, such that the number of different minimum spanning trees obtained for different choices of the function argument is \(\Omega(m\log n)\).
Limitations on realistic hyperbolic graph drawing.
D. Eppstein.
arXiv:2108.07441.
Proc. 29th Int. Symp. Graph Drawing, Tübingen, Germany, 2021.
Springer, Lecture Notes in Comp. Sci. 12868 (2021), pp. 343–357.Graphs drawn in the hyperbolic plane can be forced to have features much closer together than unit distance, in the absolute distance scale of the plane. In particular this is true for planar drawings of maximal planar graphs or grid graphs, and for nonplanar drawings of arbitrary graphs.
Egyptian fractions with denominators from sequences closed under doubling.
D. Eppstein.
J. Integer Sequences 24: 21.8.8, 2021.
arXiv:2109.12217.We prove that if a set of positive integers includes multiples of all integers, and is closed under multiplication by two, then its reciprocals can form Egyptian fractions for every rational number up to the sum of reciprocals of the set.
Finding relevant points for nearest-neighbor classification.
D. Eppstein.
arXiv:2110.06163.
Proc. SIAM Symp. Simplicity in Algorithms, 2022, pp. 68–78; best paper award winner.The nearest-neighbor classification problem considered here takes as input a training set of points in a Euclidean space, each with a classification from some finite set of classes or colors, and then uses that input to predict the classification of new points as being equal to that of the nearest neighbor in the input training set. A training point is irrelevant when removing it from the training set would produce the same predicted classification for all possible new points that might be queried. We describe how to find all of the relevant points, in polynomial time, using a simple algorithm whose only components are construction of a minimum spanning tree of the training set and the computation of extreme points (convex hull vertices) of geometrically transformed subsets of points. For any constant dimension, with \(k\) relevant points resulting from a training set of \(n\) points, this method can be made to take time \(O(n^2+k^2n)\), using only simple algorithms for the minimum spanning tree and extreme point subroutines. For small dimensions, somewhat better but more complicated bounds are possible.
The complexity of iterated reversible computation.
D. Eppstein.
arXiv:2112.11607.
Invited talk at 15th Latin American Theoretical Informatics Symposium, Guanajuato, Mexico, November 2022.
TheoretiCS 2: A10:1–A10:41, 2023.We study a class of problems involving computing a value \(f^{(n)}(x)\), the \(n\)th iterate of a function \(f\), for polynomial time bijections. We prove that these problems are complete for \(\mathsf{P}^{\mathsf{PSPACE}}\), and include hard problems in circuit complexity, graph algorithms, the simulation of one- and two-dimensional automata, and dynamical systems. We also provide a polynomial time algorithm for the special case where the bijection is an interval exchange transformation.
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.
Reflections in an octagonal mirror maze.
D. Eppstein.
arXiv:2206.11413.
34th Canadian Conference on Computational Geometry, 2022, pp. 129–134.For two-dimensional scenes consisting of mirrors and opaque walls that are either axis-parallel or with slope ±1, with integer endpoints, it is possible to determine the eventual destination of rational light rays in time polynomial in the description complexity of the scene, even though the number of reflections before a ray reaches this destination may be exponential. The solution involves transforming the problem into one involving interval exchange transformations and from there into another problem involving normal curves on topological surfaces.
Locked and unlocked smooth embeddings of surfaces.
D. Eppstein.
arXiv:2206.12989.
34th Canadian Conference on Computational Geometry, 2022, pp. 135–142.
Computing in Geometry and Topology 2 (2): 5.1–5.20, 2023.If a subset of the plane has a continuous shrinking motion of itself, then every smooth isometric embedding of that subset into 3d can be smoothly flattened. However, there exist subsets of the plane with holes, for which some smooth embeddings that are topologically equivalent to a flat embedding cannot be smoothly flattened.
On the biplanarity of blowups.
D. Eppstein.
arXiv:2301.09246.
Proc. 31st Int. Symp. Graph Drawing and Network Visualization, Palermo, Italy, 2023 (best paper award).
Springer, Lecture Notes in Computer Science 14465 (part I), pp. 3–17.
J. Graph Algorithms & Applications 28 (2): 83–99, 2024.Expanding every vertex of a planar graph to two independent copies can produce a graph of thickness greater than two, disproving a conjecture of Ellen Gethner. We also investigate special cases of this construction for which the thickness is two.
Non-crossing Hamiltonian paths and cycles in output-polynomial time.
D. Eppstein.
arXiv:2303.00147.
Proc. 39th International Symposium on Computational Geometry (SoCG 2023).
Leibniz International Proceedings in Informatics (LIPIcs) 258, 2023, pp. 29:1–29:16.
Algorithmica 86: 3027–3053, 2024.For any point set, the numbers of non-crossing paths, non-crossing Hamiltonian paths, non-crossing surrounding polygons, and non-crossing Hamiltonian cycles can be bounded above and below by functions of two simple parameters: the minimum number of points whose deletion leaves a collinear subset, and the number of points interior to the convex hull. Because their bounds have the same form, the numbers of the two types of paths are bounded by polynomials of each other, as are the numbers of the two types of polygons. We use these relations to list non-crossing Hamiltonian paths and polygonalizations in time polynomial in the number of outputs.
Lower bounds for non-adaptive shortest path relaxation.
D. Eppstein.
arXiv:2305.09230.
Proc. 18th Algorithms and Data Structures Symposium (WADS 2023).
Springer, Lecture Notes in Computer Science 13079 (2023), pp. 416–429.The Bellman–Ford algorithm for single-source shortest paths operates by relaxation steps, in which it checks for a given edge whether the best path it knows to the start of the edge, plus the edge itself, is better than the path it already knows to the end of the edge. We prove that, up to constant factors, Bellman–Ford is optimal among algorithms that use relaxation in an edge ordering that does not depend on the results of earlier relaxation steps.
Widths of geometric graphs.
D. Eppstein.
Invited talk, Workshop on Parameterized Algorithms for Geometric Problems, CG Week 2023.Many computational geometry problems can be solved by transforming them into a graph problem on a geometric graph. When the graph has low width, for some form of graph width, this can often lead to efficient classical or parameterized algorithms. We survey the geometric graphs that always have low width, the geometric graphs that can have high width, and the opportunities for parameterization obtained by studying low-width instances of graphs that sometimes have high width.
A parameterized algorithm for flat folding.
D. Eppstein.
arXiv:2306.11939.
Proc. 35th Canadian Conference on Computational Geometry, 2023, pp. 35–42.Testing whether an origami folding pattern can be folded flat is fixed-parameter tractable when parameterized by two parameters: the ply (maximum number of layers) of the folding, and the treewidth of a planar graph describing the arrangement of polygons in the flat-folded result. The dependence on treewidth is optimal under the exponential-time hypothesis.
Merged into "Computational complexities of folding" for journal submission.
The widths of strict outerconfluent graphs.
D. Eppstein.
arXiv:2308.03967.
Proc. 31st Int. Symp. Graph Drawing and Network Visualization, Palermo, Italy, 2023 (poster session).
Springer, Lecture Notes in Computer Science 14466 (part II), pp. 244–247.
Discrete Mathematics & Theoretical Computer Science 26 (3) 20:1–20:17, 2024.We construct a family of strict outerconfluent graphs with unbounded clique-width. In contrast, we use a counting argument to prove that strict outerconfluent graphs have bounded twin-width.
Uniqueness in puzzles and puzzle solving.
D. Eppstein.
Minisymposium on Mathematical Puzzles and Games in Theoretical Computer Science, ICIAM, Waseda Univ., Tokyo, Japan, 2023.In puzzles such as Sudoku, one can often make use of an assumption that the solution is unique, in deduction rules that eliminate alternatives that would lead to a non-unique solution. However, these rules can lack the monotonicity properties of other deduction rules, under which a conclusion that can be deduced from some partially-solved state of the puzzle remains available to be deduced until it actually is deduced. I discuss a method of obtaining monotonic deductions in a planar precoloring extension puzzle, and ask whether there is some more general method for obtaining monotonicity in this way.
Games on game graphs.
D. Eppstein. AMS Special Session on Serious Recreational Mathematics, Joint Mathematics Meetings, San Francisco, 2024.This talk surveys graph parameters defined from pursuit-evasion games on graphs, including cop-number, treewidth, and flip-width, and the values of these parameters on graphs derived from games and puzzles.
Non-Euclidean Erdős–Anning theorems.
D. Eppstein.
arXiv:2401.06328.
41st International Symposium on Computational Geometry (SoCG 2025), Kanazawa, Japan.
Leibniz International Proceedings in Informatics (LIPIcs) 332, 2025, pp. 46:1–46:15.Non-collinear point sets with integer distances must be finite, for strictly convex distance functions on the plane, for two-dimensional complete Riemannian manifolds of bounded genus, and for geodesic distance on convex surfaces in 3d.
What is ... treewidth?.
D. Eppstein.
Notices of the American Mathematical Society 72 (2): 172–175, 2025.We survey graph treewidth at a high level. The focus is on applications of treewidth to various areas of mathematics.
Computational complexities of folding.
D. Eppstein.
Invited talk, 8th International Meeting on Origami in Science, Mathematics and Education, Melbourne, Australia, 2024.
Invited talk, 26th Japan Conference on Discrete and Computational Geometry, Graphs, and Games, Tokyo, 2024.
arXiv:2410.07666.
Journal of Information Processing 33: 954–973, 2025.We survey known hardness results on folding origami and prove several new ones: finding a flat-folded state is \(\mathsf{NP}\)-hard, but fixed-parameter tractable in a combination of ply and the treewidth of an associated graph. Finding a 3d-folded state cannot be expressed in closed form and cannot be computed by bounded-degree algebraic-root primitives. Reconfiguring certain systems of overlapping origami squares, hinged to a table at one edge, is \(\mathsf{PSPACE}\)-complete, and counting the number of configurations is \(\#\mathsf{P}\)-complete. Testing flat-foldability of infinite fractal folding patterns (even on normal square origami paper) is undecidable.
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.
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.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.
Hamiltonian cycles in subdivided doubles.
D. Eppstein.
arXiv:2510.18359.
Ars Mathematica Contemporanea, to appear.The subdivided double construction turns a 4-regular graph into a bigger 4-regular graph, by subdividing each edge and doubling each vertex. We prove that the resulting graphs, which include the Folkman graph, have the following curious property: every Hamiltonian cycle (of which there are exponentially many) is complementary to another Hamiltonian cycle.