We present a new algorithm to compute motorcycle graphs. It runs in O(n √ n log n) time when n is the number of motorcycles. We give a new characterization of the straight skeleton of a nondegenerate polygon. For a polygon with n vertices and h holes, we show that it yields a randomized algorithm that reduces the straight skeleton computation to a motorcycle graph computation in expected O(n √ h + 1 log 2 n) time. Combining these results, we can compute the straight skeleton of a nondegenerate polygon with h holes and with n vertices, among which r are reflex vertices, in O(n √ h + 1 log 2 n + r √ r log r ) expected time. In particular, we can compute the straight skeleton of a nondegenerate polygon with n vertices in O(n √ n log 2 n) expected time.

We give exact and approximation algorithms for computing the Gromov hyperbolicity of an n-point discrete metric space. We observe that computing the Gromov hyperbolicity from a fixed base-point reduces to a (max,min) matrix product. Hence, using the (max,min) matrix product algorithm by Duan and Pettie, the fixed base-point hyperbolicity can be determined in O(n 2.69 ) time. It follows that the Gromov hyperbolicity can be computed in O(n 3.69 ) time, and a 2-approximation can be found in O(n 2.69 ) time. We also give a (2 log 2 n)-approximation algorithm that runs in O(n 2 ) time, based on a tree-metric embedding by Gromov. We also show that hyperbolicity at a fixed base-point cannot be computed in O(n 2.05 ) time, unless there exists a faster algorithm for (max,min) matrix multiplication than currently known.

Our goal is to find an approximate shortest path for a point robot moving in a planar subdivision with n vertices. Let ρ 1 be a real number. Distances in each face of this subdivision are measured by a convex distance function whose unit disk is contained in a concentric unit Euclidean disk, and contains a concentric Euclidean disk with radius 1/ρ. Different convex distance functions may be used for different faces, and obstacles are allowed. These convex distance functions may be asymmetric. For any ε ∈ (0, 1) and for any two points v s and v d , we give an algorithm that finds a path from v s to v d whose cost is at most (1 + ε) times the optimal. Our algorithm runs in O ρ 2 log ρ ε 2 n 3 log ρn εtime. This bound does not depend on any other parameters; in particular it does not depend on the minimum angle in the subdivision. We give applications to two special cases that have been considered before: the weighted region problem and motion planning in the presence of uniform flows. For the weighted region problem with weights in [1, ρ] ∪ {∞}, the time bound of our algorithm improves to O ρ log ρ ε n 3 log ρn ε .

Given a set S of n points in R D , and an integer k such that 0 k < n, we show that a geometric graph with vertex set S, at most n − 1 + k edges, maximum degree five, and dilation O(n/(k + 1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n − 1 + k edges has dilation Ω(n/(k + 1)); a slightly weaker statement holds if the points of S are required to be in convex position. Preliminaries and introductionA geometric network is an undirected graph whose vertices are points in R D . Geometric networks, especially geometric networks of points in the plane, arise in many applications. Road networks, railway networks, computer networks-any collection of objects that have some connections between them can be modeled as a geometric network. A natural and widely studied type of geometric network is the Euclidean network, where the weight of an edge is simply the Euclidean distance between its two endpoints. Such networks for points in R D form the topic of study of our paper. When designing a network for a given set S of points, several criteria have to be taken into account. In particular, in many applications it is important to ensure a short connection between every two points in S. For this it would be ideal to have a direct connection between every two points; the network would then be a complete graph. In most applications, however, this is unacceptable due to the high costs. Thus the question arises: is it possible to construct a network that guarantees a reasonably short connection between every two points while not using too many edges? This leads to the well-studied concept of spanners, which we define next.The weight of an edge e = (u, v) in a Euclidean network G = (S, E) on a set S of n points is the Euclidean distance between u and v, which we denote by d (u, v). The graph distance d G (u, v) between two vertices u, v ∈ S is the length of a shortest path in G connecting u to v. The dilation (or stretch factor ) of G, denoted

Abstract. Voronoi diagrams of curved objects can show certain phenomena that are often considered artifacts: The Voronoi diagram is not connected; there are pairs of objects whose bisector is a closed curve or even a two-dimensional object; there are Voronoi edges between different parts of the same site (so-called self-Voronoi-edges); these self-Voronoiedges may end at seemingly arbitrary points not on a site, and, in the case of a circular site, even degenerate to a single isolated point. We give a systematic study of these phenomena, characterizing their differential-geometric and topological properties. We show how a given set of curves can be refined such that the resulting curves define a "well-behaved" Voronoi diagram. We also give a randomized incremental algorithm to compute this diagram. The expected running time of this algorithm is O(n log n).