arXiv subjects
Publications and source records attributed to David Eppstein.
We describe algorithms for drawing media, systems of states, tokens and actions that have state transition graphs in the form of partial cubes. Our algorithms are based on two principles: embedding the state transition graph in a low-dimensional integer lattice and projecting the lattice onto the plane, or drawing the medium as a planar graph with centrally symmetric faces.
Triangle strips have been widely used for efficient rendering. It is NP-complete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for creating a single triangle loop or strip from a triangulated model. Our method applies a dual graph matching algorithm to partition the mesh into cycles, and then merges pairs of cycles by splitting adjacent triangles when necessary. New vertices are introduced at midpoints of edges and the new triangles thus formed are coplanar with their parent triangles, hence the visual fidelity of the geometry is not changed. We prove that the increase in the number of triangles due to this splitting is 50% in the worst case, however for all models we tested the increase was less than 2%. We also prove tight bounds on the number of triangles needed for a single-strip representation of a model with holes on its boundary. Our strips can be used not only for efficient rendering, but also for other applications including the generation of space filling curves on a manifold of any arbitrary topology.
We show how to find a Hamiltonian cycle in a graph of degree at most three with n vertices, in time O(2^{n/3}) ~= 1.260^n and linear space. Our algorithm can find the minimum weight Hamiltonian cycle (traveling salesman problem), in the same time bound. We can also count or list all Hamiltonian cycles in a degree three graph in time O(2^{3n/8}) ~= 1.297^n. We also solve the traveling salesman problem in graphs of degree at most four, by randomized and deterministic algorithms with runtime O((27/4)^{n/3}) ~= 1.890^n and O((27/4+epsilon)^{n/3}) respectively. Our algorithms allow the input to specify a set of forced edges which must be part of any generated cycle. Our cycle listing algorithm shows that every degree three graph has O(2^{3n/8}) Hamiltonian cycles; we also exhibit a family of graphs with 2^{n/3} Hamiltonian cycles per graph.
In this paper we study the problem of how resilient networks are to node faults. Specifically, we investigate the question of how many faults a network can sustain so that it still contains a large (i.e. linear-sized) connected component that still has approximately the same expansion as the original fault-free network. For this we apply a pruning technique which culls away parts of the faulty network which have poor expansion. This technique can be applied to both adversarial faults and to random faults. For adversarial faults we prove that for every network with expansion alpha, a large connected component with basically the same expansion as the original network exists for up to a constant times alpha n faults. This result is tight in the sense that every graph G of size n and uniform expansion alpha(.), i.e. G has an expansion of alpha(n) and every subgraph G' of size m of G has an expansion of O(alpha(m)), can be broken into sublinear components with omega(alpha(n) n) faults. For random faults we observe that the situation is significantly different, because in this case the expansion of a graph only gives a very weak bound on its resilience to random faults. More specifically, there are networks of uniform expansion O(sqrt{n}) that are resilient against a constant fault probability but there are also networks of uniform expansion Omega(1/log n) that are not resilient against a O(1/log n) fault probability. Thus, a different parameter is needed. For this we introduce the span of a graph which allows us to determine the maximum fault probability in a much better way than the expansion can. We use the span to show the first known results for the effect of random faults on the expansion of d-dimensional meshes.
We describe a polynomial time algorithm for, given an undirected graph G, finding the minimum dimension d such that G may be isometrically embedded into the d-dimensional integer lattice Z^d.
We prove that the geometric thickness of graphs whose maximum degree is no more than four is two. All of our algorithms run in O(n) time, where n is the number of vertices in the graph. In our proofs, we present an embedding algorithm for graphs with maximum degree three that uses an n x n grid and a more complex algorithm for embedding a graph with maximum degree four. We also show a variation using orthogonal edges for maximum degree four graphs that also uses an n x n grid. The results have implications in graph theory, graph drawing, and VLSI design.
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.
We present memory-efficient deterministic algorithms for constructing epsilon-nets and epsilon-approximations of streams of geometric data. Unlike probabilistic approaches, these deterministic samples provide guaranteed bounds on their approximation factors. We show how our deterministic samples can be used to answer approximate online iceberg geometric queries on data streams. We use these techniques to approximate several robust statistics of geometric data streams, including Tukey depth, simplicial depth, regression depth, the Thiel-Sen estimator, and the least median of squares. Our algorithms use only a polylogarithmic amount of memory, provided the desired approximation factors are inverse-polylogarithmic. We also include a lower bound for non-iceberg geometric queries.
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.
A hinged dissection of a set of polygons S is a collection of polygonal pieces hinged together at vertices that can be folded into any member of S. We present a hinged dissection of all edge-to-edge gluings of n congruent copies of a polygon P that join corresponding edges of P. This construction uses kn pieces, where k is the number of vertices of P. When P is a regular polygon, we show how to reduce the number of pieces to ceiling(k/2)*(n-1). In particular, we consider polyominoes (made up of unit squares), polyiamonds (made up of equilateral triangles), and polyhexes (made up of regular hexagons). We also give a hinged dissection of all polyabolos (made up of right isosceles triangles), which do not fall under the general result mentioned above. Finally, we show that if P can be hinged into Q, then any edge-to-edge gluing of n congruent copies of P can be hinged into any edge-to-edge gluing of n congruent copies of Q.
We show it is possible to tile three-dimensional space using only tetrahedra with acute dihedral angles. We present several constructions to achieve this, including one in which all dihedral angles are less than $77.08^\circ$, and another which tiles a slab in space.
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.
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.
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.
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.
We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.
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.
We introduce the fatness parameter of a 4-dimensional polytope P, defined as ϕ(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 ϕ(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.