This paper describes a new algorithm for constructing the set of free bitangents of a collection of n disjoint convex obstacles of constant complexity. The algorithm runs in time O(n log n + k), where k is the output size, and uses O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using red--black trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of th...

We show that the k free bitangents of a collection of n pairwise disjoint convex plane sets can be computed in time O(k+n log n) and O(n) working space. The algorithm uses only one advanced data structure, namely a splittable queue. We introduce (weakly) greedy pseudo--triangulations, whose combinatorial properties are crucial for our method. 1 Introduction Consider a collection O of pairwise disjoint convex objects in the plane. We are interested in problems in which these objects arise as obstacles, either in connection with visibility problems where they can block the view from an other geometric object, or in motion planning, where these objects may prevent a moving object from moving along a straight line path. The visibility graph is a central object in such contexts. For polygonal obstacles the vertices of these polygons are the nodes of the visibility graph, and two nodes are connected by an arc if the corresponding vertices can see each other. [9] describes the first non-triv...

We consider the problem of approximating a polygonal chain C by another polygonal chain C ′ whose vertices are constrained to be a subset of the set of vertices of C. The goal is to minimize the number of vertices needed in the approximation C ′. Based on a framework introduced by Imai and Iri [25], we define an error criterion for measuring the quality of an approximation. We consider two problems. (1) Given a polygonal chain C and a parameter ε ≥ 0, compute an approximation of C, among all approximations whose error is at most ε, that has the smallest number of vertices. We present an O(n 4/3+δ)-time algorithm to solve this problem, for any δ>0; the constant of proportionality in the running time depends on δ. (2) Given a polygonal chain C and an integer k, compute an approximation of C with at most k vertices whose error is the smallest among all approximations with at most k vertices. We present a simple randomized algorithm, with expected running time O(n 4/3+δ), to solve this problem.

We study the savings afforded by repeated use in two zero-error communication problems. We show that for some random sources, communicating one instance requires arbitrarily-many bits, but communicating multiple instances requires roughly one bit per instance. We also exhibit sources where the number of bits required for a single instance is comparable to the source’s size, but two instances require only a logarithmic number of additional bits. We relate this problem to that of communicating information over a channel. Known results imply that some channels can communicate exponentially more bits in two uses than they can in one use. 1

To cover the edges of a graph with a minimum number of cliques is an NP-complete problem with many applications. The state-of-the-art solving algorithm is a polynomial-time heuristic from the 1970’s. We present an improvement of this heuristic. Our main contribution, however, is the development of efficient and effective polynomial-time data reduction rules that, combined with a search tree algorithm, allow for exact problem solutions in competitive time. This is confirmed by experiments with real-world and synthetic data. Moreover, we prove the fixed-parameter tractability of covering edges by cliques. 1

Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity : design a data structure for an undirected graph G = (V, E) and a subset of vertices S # V , to support insertions and deletions in S and connectivity queries (are two vertices connected?) in the subgraph induced by S. We develop the first sublinear, fully dynamic method for this problem for general sparse graphs, using an elegant combination of several simple ideas. Our method requires linear space, # O(|E| 4#/(3#+3) ) = O(|E| 0.94 ) amortized update time, and # O(|E| 1/3 ) query time, where # is the matrix multiplication exponent and # O hides polylogarithmic factors.

We study the problem of continuously transforming or morphing two non-intersecting simple (not self-intersecting) polylines in the plane. Our morphing strategies have the property that every intermediate polyline is also simple. We also guarantee that no portion of the polylines to be morphed is stretched or compressed by more than a user-defined parameter during the entire morphing. Our algorithms are driven by a new metric for measuring the similarity between two polylines, which may have other applications. We compute morphing schemes that minimize this metric and also approximate the minimum value efficiently. Department of Computer Science, D340 Levine Science Research Center, Duke University, Box 90129, Durham, NC 27708-0129, USA, sariel@cs.duke.edu http://www.cs.duke.edu/~sariel/ y Compaq Computer Corporation, Cambridge Research Lab, One Cambridge Center, Cambridge MA 02142, murali@crl.dec.com. 1 1 Introduction In the last few years, the problem of continuously morph...

We study the area requirement of bar-visibility and rectangle-visibility representations of trees in the plane. We prove asymptotically tight lower and upper bounds on the area of such representations, and give linear-time algorithms that construct representations with asymptotically optimal area.

We introduce two new related metrics, the geodesic width and the link width, for measuring the \distance" between two non-intersecting polylines in the plane. If the two polylines have n vertices in total, we present algorithms to compute the geodesic width of the two polylines in O(n ) space and the link width ) working space. Our computation of these metrics relies on two closely-related combinatorial strutures: the shortest-path diagram and the link diagram of a simple polygon. The shortest-path (resp., link) diagram encodes the Euclidean (resp., link) shortest path distance between all pairs of points on the boundary of the polygon. We use these algorithms to solve two problems: Compute a continuous transformation that \morphs" one polyline into another polyline. Our morphing strategies ensure that each point on a polyline moves Preliminary versions of this paper appeared in the Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms [10] and the Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms [11] The author did part of this research when he was aliated with Stanford University. Address: Room 747, Department of Computer Science, Gould-Simpson Building, The University of Arizona, PO Box 210077, Tucson AZ 85721-0077. Email: alon@cs.arizona.edu. WWW: http://www.cs.arizona.edu/people/alon/.

We study the problem of computing the visibility graph defined by a set P of n points inside a polygon Q: two points p, q ∈ P are joined by an edge if the segment pq ⊂ Q. Efficient output-sensitive algorithms are known for the case in which P is the set of all vertices of Q. We examine the general case in which P is an arbitrary set of points, interior or on the boundary of Q and study a variety of algorithmic questions. We give an output-sensitive algorithm, which is nearly optimal, when Q is a simple polygon. We introduce a notion of “fat ” or “robust ” visibility, and give a nearly optimal algorithm for computing visibility graphs according to it, in polygons Q that may have holes. Other results include an algorithm to detect if there are any visible pairs among P, and algorithms for output-sensitive computation of visibility graphs with distance restrictions, invisibility graphs, and rectangle visibility graphs.