arXiv subjects

David Eppstein

Publications and source records attributed to David Eppstein.

At least 217 records · Page 12Linked to original sources

arXiv2007

Edges and Switches, Tunnels and Bridges

Edge casing is a well-known method to improve the readability of drawings of non-planar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a "good" cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NP-hardness results.

cs.DS↗

arXiv2006

Choosing Colors for Geometric Graphs via Color Space Embeddings

Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. After the geometric embedding is specified, there is an additional step that is often overlooked or ignored: assigning display colors to the graph's vertices. We study the additional aesthetic criterion of assigning distinct colors to vertices of a geometric graph so that the colors assigned to adjacent vertices are as different from one another as possible. We formulate this as a problem involving perceptual metrics in color space and we develop algorithms for solving this problem by embedding the graph in color space. We also present an application of this work to a distributed load-balancing visualization problem.

cs.CG↗

arXiv2006

Trees with Convex Faces and Optimal Angles

We consider drawings of trees in which all edges incident to leaves can be extended to infinite rays without crossing, partitioning the plane into infinite convex polygons. Among all such drawings we seek the one maximizing the angular resolution of the drawing. We find linear time algorithms for solving this problem, both for plane trees and for trees without a fixed embedding. In any such drawing, the edge lengths may be set independently of the angles, without crossing; we describe multiple strategies for setting these lengths.

cs.CG↗

arXiv2006

Upright-Quad Drawing of st-Planar Learning Spaces

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.

cs.CG↗

arXiv2006

Drawings of Planar Graphs with Few Slopes and Segments

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on $n$ vertices has a plane drawing with at most ${5/2}n$ segments and at most $2n$ slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.

math.CO↗

arXiv2006

Guard Placement For Wireless Localization

Motivated by secure wireless networking, we consider the problem of placing fixed localizers that enable mobile communication devices to prove they belong to a secure region that is defined by the interior of a polygon. Each localizer views an infinite wedge of the plane, and a device can prove membership in the secure region if it is inside the wedges for a set of localizers whose common intersection contains no points outside the polygon. This model leads to a broad class of new art gallery type problems, for which we provide upper and lower bounds.

cs.CG↗

arXiv2006

Approximate Weighted Farthest Neighbors and Minimum Dilation Stars

We provide an efficient reduction from the problem of querying approximate multiplicatively weighted farthest neighbors in a metric space to the unweighted problem. Combining our techniques with core-sets for approximate unweighted farthest neighbors, we show how to find (1+epsilon)-approximate farthest neighbors in time O(log n) per query in D-dimensional Euclidean space for any constants D and epsilon. As an application, we find an O(n log n) expected time algorithm for choosing the center of a star topology network connecting a given set of points, so as to approximately minimize the maximum dilation between any pair of points.

cs.CG↗

arXiv2005

The Weighted Maximum-Mean Subtree and Other Bicriterion Subtree Problems

We consider problems in which we are given a rooted tree as input, and must find a subtree with the same root, optimizing some objective function of the nodes in the subtree. When this function is the sum of constant node weights, the problem is trivially solved in linear time. When the objective is the sum of weights that are linear functions of a parameter, we show how to list all optima for all possible parameter values in O(n log n) time; this parametric optimization problem can be used to solve many bicriterion optimizations problems, in which each node has two values xi and yi associated with it, and the objective function is a bivariate function f(SUM(xi),SUM(yi)) of the sums of these two values. A special case, when f is the ratio of the two sums, is the Weighted Maximum-Mean Subtree Problem, or equivalently the Fractional Prize-Collecting Steiner Tree Problem on Trees; for this special case, we provide a linear time algorithm for this problem when all weights are positive, improving a previous O(n log n) solution, and prove that the problem is NP-complete when negative weights are allowed.

cs.CG↗

arXiv2005

Cubic Partial Cubes from Simplicial Arrangements

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.

math.CO↗

arXiv2005

Delta-confluent Drawings

We generalize the tree-confluent graphs to a broader class of graphs called Delta-confluent graphs. This class of graphs and distance-hereditary graphs, a well-known class of graphs, coincide. Some results about the visualization of Delta-confluent graphs are also given.

cs.CG↗

arXiv2005

Nonrepetitive Paths and Cycles in Graphs with Application to Sudoku

We provide a simple linear time transformation from a directed or undirected graph with labeled edges to an unlabeled digraph, such that paths in the input graph in which no two consecutive edges have the same label correspond to paths in the transformed graph and vice versa. Using this transformation, we provide efficient algorithms for finding paths and cycles with no two consecutive equal labels. We also consider related problems where the paths and cycles are required to be simple; we find efficient algorithms for the undirected case of these problems but show the directed case to be NP-complete. We apply our path and cycle finding algorithms in a program for generating and solving Sudoku puzzles, and show experimentally that they lead to effective puzzle-solving rules that may also be of interest to human Sudoku puzzle solvers.

cs.DS↗

arXiv2005

The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data

We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R^2) or the skip octree (for point data in R^d, with constant d>2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined "box"-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location and approximate range queries.

cs.CG↗

arXiv2005

Skip-Webs: Efficient Distributed Data Structures for Multi-Dimensional Data Sets

We present a framework for designing efficient distributed data structures for multi-dimensional data. Our structures, which we call skip-webs, extend and improve previous randomized distributed data structures, including skipnets and skip graphs. Our framework applies to a general class of data querying scenarios, which include linear (one-dimensional) data, such as sorted sets, as well as multi-dimensional data, such as d-dimensional octrees and digital tries of character strings defined over a fixed alphabet. We show how to perform a query over such a set of n items spread among n hosts using O(log n / log log n) messages for one-dimensional data, or O(log n) messages for fixed-dimensional data, while using only O(log n) space per host. We also show how to make such structures dynamic so as to allow for insertions and deletions in O(log n) messages for quadtrees, octrees, and digital tries, and O(log n / log log n) messages for one-dimensional data. Finally, we show how to apply a blocking strategy to skip-webs to further improve message complexity for one-dimensional data when hosts can store more data.

cs.DC↗

arXiv2005

Confluent Layered Drawings

We combine the idea of confluent drawings with Sugiyama style drawings, in order to reduce the edge crossings in the resultant drawings. Furthermore, it is easier to understand the structures of graphs from the mixed style drawings. The basic idea is to cover a layered graph by complete bipartite subgraphs (bicliques), then replace bicliques with tree-like structures. The biclique cover problem is reduced to a special edge coloring problem and solved by heuristic coloring algorithms. Our method can be extended to obtain multi-depth confluent layered drawings.

cs.CG↗

arXiv2005

Improved Combinatorial Group Testing Algorithms for Real-World Problem Sizes

We study practically efficient methods for performing combinatorial group testing. We present efficient non-adaptive and two-stage combinatorial group testing algorithms, which identify the at most d items out of a given set of n items that are defective, using fewer tests for all practical set sizes. For example, our two-stage algorithm matches the information theoretic lower bound for the number of tests in a combinatorial group testing regimen.

cs.DS↗

arXiv2005

Minimum Dilation Stars

The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in R^d. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(n log n)-time algorithm for evaluating the dilation of a given star; a randomized O(n log n) expected-time algorithm for finding an optimal center in R^d; and for the case d=2, a randomized O(n 2^(alpha(n)) log^2 n) expected-time algorithm for finding an optimal center among the input points.

cs.CG↗

arXiv2004

Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey 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.

cs.CG↗

arXiv2004

All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs

We describe algorithms, based on Avis and Fukuda's reverse search paradigm, for listing all maximal independent sets in a sparse graph in polynomial time and delay per output. For bounded degree graphs, our algorithms take constant time per set generated; for minor-closed graph families, the time is O(n) per set, and for more general sparse graph families we achieve subquadratic time per set. We also describe new data structures for maintaining a dynamic vertex set S in a sparse or minor-closed graph family, and querying the number of vertices not dominated by S; for minor-closed graph families the time per update is constant, while it is sublinear for any sparse graph family. We can also maintain a dynamic vertex set in an arbitrary m-edge graph and test the independence of the maintained set in time O(sqrt m) per update. We use the domination data structures as part of our enumeration algorithms.

cs.DS↗