Showing papers by "David Eppstein published in 2002"
Posted Content•
TL;DR: This work provides a data structure for maintaining an embedding of a graph on a surface and computing generators of the fundamental group of the surface, in amortized time O(log n + log g(log log g)3) per update on asurface of genus g, and applies similar ideas to improve the constant factor in a separator theorem for low-genus graphs, and to find in linear time a tree-decomposition of low-generation low-diameter graphs.
Abstract: We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(log n + log g(log log g)^3) per update on a surface of genus g; we can also test orientability of the surface in the same time, and maintain the minimum and maximum spanning tree of the graph in time O(log n + log^4 g) per update. Our data structure allows edge insertion and deletion as well as the dual operations; these operations may implicitly change the genus of the embedding surface. We apply similar ideas to improve the constant factor in a separator theorem for low-genus graphs, and to find in linear time a tree-decomposition of low-genus low-diameter graphs.
109 citations
Posted Content•
TL;DR: In this paper, the authors proposed a different web graph model with power law distribution that does not require incremental growth and provided a comparison of their model with several others in their ability to predict web graph clustering behavior.
Abstract: Power law distribution seems to be an important characteristic of web graphs Several existing web graph models generate power law graphs by adding new vertices and non-uniform edge connectivities to existing graphs Researchers have conjectured that preferential connectivity and incremental growth are both required for the power law distribution In this paper, we propose a different web graph model with power law distribution that does not require incremental growth We also provide a comparison of our model with several others in their ability to predict web graph clustering behavior
74 citations
26 Aug 2002
TL;DR: In this article, it was shown that graph-theoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and a geometric thickness t.
Abstract: We show that graph-theoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness ? t.
59 citations
TL;DR: Flip operations for quadrilateral and hexahedral meshes are defined and examined, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.
Abstract: We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.
57 citations
30 Mar 2002
TL;DR: A different web graph model with power law distribution that does not require incremental growth is proposed and a comparison of this model with several others in their ability to predict web graph clustering behavior is provided.
Abstract: Power law distribution seems to be an important characteristic of web graphs. Several existing web graph models generate power law graphs by adding new vertices and non-uniform edge connectivities to existing graphs. Researchers have conjectured that preferential connectivity and incremental growth are both required for the power law distribution. In this paper, we propose a different web graph model with power law distribution that does not require incremental growth. We also provide a comparison of our model with several others in their ability to predict web graph clustering behavior.
45 citations
Posted Content•
TL;DR: In this article, the fatness parameter of a 4-dimensional polytope P is defined as φ(P)=(f_1+f_2)/(f_0+f-3).
Abstract: We introduce the fatness parameter of a 4-dimensional polytope P, defined as \phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in 4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes bounded?
We describe and analyze a hyperbolic geometry construction that produces 4-polytopes with fatness \phi(P)>5.048, as well as the first infinite family of 2-simple, 2-simplicial 4-polytopes. Moreover, using a construction via finite covering spaces of surfaces, we show that fatness is not bounded for the more general class of strongly regular CW decompositions of the 3-sphere.
36 citations
TL;DR: A fractal construction shows that, for any β > 0, the β-skeleton of a point set can have arbitrarily large dilation, which applies to the Gabriel graph.
Abstract: A fractal construction shows that, for any β > 0, the β-skeleton of a point set can have arbitrarily large dilation. In particular this applies to the Gabriel graph.
35 citations
01 Jan 2002
TL;DR: Software that searches for spaceships in Conway's Game of Life and related two-dimensional cellular automata is described, 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.
Abstract: 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.
29 citations
05 Jun 2002
TL;DR: An algorithm is presented to unfold any triangulated 2-manifold into a non-overlap-linebreak ping, connected planar layout in linear time, and extended to establish a similar result for simplicial manifolds of arbitrary dimension.
Abstract: We present an algorithm to unfold any triangulated 2-manifold (in particular, any simplicial polyhedron) into a non-overlap-linebreak ping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles are connected at vertices, but not necessarily joined along edges. We extend our algorithm to establish a similar result for simplicial manifolds of arbitrary dimension.
24 citations
Posted Content•
TL;DR: An algorithm for finding the optimal gamut in time O(n3), where n denotes the number of projectors in the system, is developed.
Abstract: We consider the problem of finding a large color space that can be generated by all units in multi-projector tiled display systems. Viewing the problem geometrically as one of finding a large parallelepiped within the intersection of multiple parallelepipeds, and using colorimetric principles to define a volume-based objective function for comparing feasible solutions, we develop an algorithm for finding the optimal gamut in time O(n^3), where n denotes the number of projectors in the system. We also discuss more efficient quasiconvex programming algorithms for alternative objective functions based on maximizing the quality of the color space extrema.
24 citations
TL;DR: Simple linear time algorithms for coloring the squares of balanced and unbalanced quadtrees so that no two adjacent squares are given the same color are described.
Abstract: We describe simple linear time algorithms for coloring the squares of balanced and unbalanced quadtrees so that no two adjacent squares are given the same color. If squares sharing sides are defined as adjacent, we color balanced quadtrees with three colors, and unbalanced quadtrees with four colors; these results are both tight, as some quadtrees require this many colors. If squares sharing corners are defined as adjacent, we color balanced or unbalanced quadtrees with six colors; for some quadtrees, at least five colors are required.
01 Jan 2002
TL;DR: In this paper, it was shown that the set of configurations that can be reduced to a single peg forms a regular language, and that a linear-time algorithm exists for reducing any configuration to the minimum number of pegs.
Abstract: We solve the problem of one-dimensional Peg Solitaire. In particular, we show that the set of configurations that can be reduced to a single peg forms a regular language, and that a linear-time algorithm exists for reducing any configuration to the minimum number of pegs. We then look at the impartial two-player game, proposed by Ravikumar, where two players take turns making peg moves, and whichever player is left without a move loses. We calculate some simple nim-values and discuss when the game separates into a disjunctive sum of smaller games. In the version where a series of hops can be made in a single move, we show that neither the P-positions nor the N-positions (i.e. wins for the previous or next player) are described by a regular or context-free language.
01 Jan 2002
TL;DR: In this article, it was shown that the problem of determining whether a player has a move that immediately wins the game is NP-hard in the board game Phutball, whereas the problem is solvable in polynomial time in checkers.
Abstract: We show that, in John Conway’s board game Phutball (or Philosopher’s Football), it is NPcomplete to determine whether the current player has a move that immediately wins the game. In contrast, the similar problems of determining whether there is an immediately winning move in checkers, or a move that kings a man, are both solvable in polynomial time.
Posted Content•
TL;DR: An interpolation method invariant under Möbius transformations is proposed: interpolation followed by transformation gives the same result as transformation followed by interpolation.
Abstract: We propose an interpolation method that is invariant under Moebius transformations; that is, interpolation followed by transformation gives the same result as transformation followed by interpolation. The method uses natural (Delaunay) neighbors, but weights neighbors according to angles formed by Delaunay circles.
Journal Article•
TL;DR: In this paper, it was shown that graph-theoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric complexity > t.
Abstract: We show that graph-theoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness > t.
TL;DR: The min–min expectation selection problem is that of selecting k out of n given discrete probability distributions, to minimize the expectation of the minimum value resulting when independent random variables are drawn from the selected distributions and is polynomially solvable for constant d.
Abstract: We define the min-min expectation selection problem (resp. max-min expectation selection problem) to be that of selecting k out of n given discrete probability distributions, to minimize (resp. maximize) the expectation of the minimum value resulting when independent random variables are drawn from the selected distributions. We assume each distribution has finitely many atoms. Let d be the number of distinct values in the support of the distributions. We show that if d is a constant greater than 2, the min-min expectation problem is NP-complete but admits a fully polynomial time approximation scheme. For d an arbitrary integer, it is NP-hard to approximate the min-min expectation problem within any constant approximation factor. The max-min expectation problem is polynomially solvable for constant d; we leave open its complexity for variable d. We also show similar results for binary selection problems in which we must choose one distribution from each of n pairs of distributions.
31 Mar 2002
TL;DR: Eppstein et al. as mentioned in this paper introduced the fatness parameter of a 4-dimensional polytope P, defined as \phi(P)=(f_1+f_2)/(f_0+f-3).
Abstract: Author(s): Eppstein, David; Kuperberg, Greg; Ziegler, Gunter M. | Abstract: We introduce the fatness parameter of a 4-dimensional polytope P, defined as \phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in 4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes bounded? We describe and analyze a hyperbolic geometry construction that produces 4-polytopes with fatness \phi(P)g5.048, as well as the first infinite family of 2-simple, 2-simplicial 4-polytopes. Moreover, using a construction via finite covering spaces of surfaces, we show that fatness is not bounded for the more general class of strongly regular CW decompositions of the 3-sphere.
Posted Content•
TL;DR: This work introduces a new approach for drawing diagrams that allows groups of edges to be merged together and drawn as tracks (similar to train tracks), and offers two heuristic algorithms to test if a non-planar graph can be drawn efficiently in a confluent way.
Abstract: In this paper, we introduce a new approach for drawing diagrams that have applications in software visualization. Our approach is to use a technique we call confluent drawing for visualizing non-planar diagrams in a planar way. This approach allows us to draw, in a crossing-free manner, graphs--such as software interaction diagrams--that would normally have many crossings. The main idea of this approach is quite simple: we allow groups of edges to be merged together and drawn as "tracks" (similar to train tracks). Producing such confluent diagrams automatically from a graph with many crossings is quite challenging, however, so we offer two heuristic algorithms to test if a non-planar graph can be drawn efficiently in a confluent way. In addition, we identify several large classes of graphs that can be completely categorized as being either confluently drawable or confluently non-drawable.
Posted Content•
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TL;DR: The concept of a medium was introduced by Falmagne as discussed by the authors, a combinatorial object encompassing hyperplane arrangements, topological orderings, acyclic orientations, and many other familiar structures.
Abstract: Falmagne recently introduced the concept of a medium, a combinatorial object encompassing hyperplane arrangements, topological orderings, acyclic orientations, and many other familiar structures. We find efficient solutions for several algorithmic problems on media: finding short reset sequences, shortest paths, testing whether a medium has a closed orientation, and listing the states of a medium given a black-box description.
Posted Content•
TL;DR: It is shown that graph-theoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric Thickness t.
Abstract: We show that graph-theoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness >= t.