We give algorithms to find shortest paths homotopic to given disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time O(k log n+n √n), and the randomized version in time O(k log n+n(log n)1(1+ε)), where k is the input plus output sizes of the paths.

We introduce a new type of diagram called the VV (c)-diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning natural-looking paths for a robot translating amidst polygonal obstacles in the plane. A natural-looking path is short, smooth, and keeps — where possible — an amount of clearance c from the obstacles. The VV (c)-diagram contains such paths. We also propose an algorithm that is capable of preprocessing a scene of configuration-space polygonal obstacles and constructs a data structure called the VV-complex. The VVcomplex can be used to efficiently plan motion paths for any start and goal configuration and any clearance value c, without having to explicitly construct the VV (c)-diagram for that c-value. The preprocessing time is O(n 2 log n), where n is the total number of obstacle vertices, and the data structure can be queried directly for any c-value by merely performing a Dijkstra search. We have implemented a Cgal-based software package for computing the VV (c)-diagram in an exact manner for a given clearance value, and used it to plan natural-looking paths in various applications.

Abstract. We unveil an alluring alternative to parametric search that applies to both the non-geodesic and geodesic Fréchet optimization problems. This randomized approach is based on a variant of red-blue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search. We present the first algorithm for the geodesic Fréchet distance between two polygonal curves A and B inside a simple bounding polygon P. The geodesic Fréchet decision problem is solved almost as fast as its non-geodesic sibling and requires O(N 2 log k) time and O(k + N) space after O(k) preprocessing, where N is the larger of the complexities of A and B and k is the complexity of P. The geodesic Fréchet optimization problem is solved by a randomized approach in O(k +N 2 log kN log N) expected time and O(k +N 2) space. This runtime is only a logarithmic factor larger than the standard non-geodesic Fréchet algorithm [4]. Results are also presented for the geodesic Fréchet distance in a polygonal domain with obstacles and the geodesic Hausdorff distance for sets of points or sets of line segments inside a simple polygon P. 1.

We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and information traveling through the network. We present an efficient linear-time algorithm which draws edges and vertices of varying 2-dimensional areas to represent the amount of information flowing through them. The algorithm avoids all occlusions of nodes and edges, while still drawing the graph on a compact integer grid.

INTRODUCTION We present an algorithm for growing fat graphs. Traditionally, graph drawing algorithms represent vertices as circles and edges as closed curves connecting the vertices. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional information. We show how to grow fat graphs with edges of variable thickness. For the purpose of the demonstration we focus on a restricted class of graphs that occur in VLSI wire routing. This class corresponds to planar, max-degree-1 graphs. The underlying algorithm also extends to general planar graphs as shown in [2]. In VLSI wire routing it is often desirable to maximize the distance between di#erent wires. Maximizing the distance between wires is equivalent to finding the drawing in which the edges are drawn as thick as possible, i.e., allowing the graph to grow as fat as possible. The continuous homotopic routing problem [1, 3, 5] is a classic VLSI problem. The input is an initial sk

We propose a layout algorithm for micro/macro graphs, i.e. relational structures with two levels of detail. While the micro-level graph is given, the macro-level graph is induced by a given partition of the micro-level vertices. A typical example is a social network of employees organized into different departments. We do not impose restrictions on the macro-level layout other than sufficient thickness of edges and vertices, so that the micro-level graph can be placed on top of the macrolevel graph. For the micro-level graph we define a combinatorial multicircular embedding and present corresponding layout algorithms based on edge crossing reduction strategies.

Abstract. We study methods for drawing trees with perfect angular resolution, i.e., with angles at each vertex, v, equal to 2π/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution. 1

Abstract. We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and information traveling through the network. We present an efficient linear-time algorithm which draws edges and vertices of varying 2-dimensional areas to represent the amount of information flowing through them. The algorithm avoids all occlusions of nodes and edges, while still drawing the graph on a compact integer grid. 1

The (Vietoris-)Rips complex of a discrete point-set P is an abstract simplicial complex in which a subset of P defines a simplex if and only if the diameter of that subset is at most 1. We describe an efficient algorithm to determine whether a given cycle in a planar Rips complex is contractible. Our algorithm requires O(m log n) time to preprocess a set of n points in the plane in which m pairs have distance at most 1; after preprocessing, deciding whether a cycle of k Rips edges is contractible requires O(k) time. We also describe an algorithm to compute the shortest non-contractible cycle in a planar Rips complex in O(n 2 log n + mn) time. ∗ See

Social network analysis investigates the structure of relations amongst social actors. A general approach to detect patterns of interaction and to filter out irregularities is to classify actors into groups and to analyze the relational structure between and within the various classes. The first part of this paper presents methods to define and compute structural network positions, i. e., classes of actors dependent on the network structure. In the second part we present techniques to visualize a network together with a given assignment of actors into groups, where specific emphasis is given to the simultaneous visualization of micro and macro structure.