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15
Topologically Sweeping Visibility Complexes via Pseudotriangulations
, 1996
"... 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 run ..."
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Cited by 86 (9 self)
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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 redblack 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...
Efficient Algorithms for Approximating Polygonal Chains
"... 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 ..."
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Cited by 39 (2 self)
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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.
Computing the Visibility Graph via Pseudotriangulations
 In Proc. 11th Annu. ACM Sympos. Comput. Geom
, 1995
"... 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 pseudotriangulations, whose combinat ..."
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Cited by 31 (2 self)
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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 pseudotriangulations, 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 nontriv...
Repeated communication and Ramsey graphs
 IEEE Transactions on Information Theory
, 1995
"... We study the savings afforded by repeated use in two zeroerror communication problems. We show that for some random sources, communicating one instance requires arbitrarilymany bits, but communicating multiple instances requires roughly one bit per instance. We also exhibit sources where the numbe ..."
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Cited by 27 (14 self)
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We study the savings afforded by repeated use in two zeroerror communication problems. We show that for some random sources, communicating one instance requires arbitrarilymany 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
Data reduction, exact, and heuristic algorithms for clique cover
 In Proceedings 8th Workshop on Algorithm Engineering and Experiments ALENEX’06
, 2006
"... To cover the edges of a graph with a minimum number of cliques is an NPcomplete problem with many applications. The stateoftheart solving algorithm is a polynomialtime heuristic from the 1970’s. We present an improvement of this heuristic. Our main contribution, however, is the development of e ..."
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Cited by 18 (6 self)
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To cover the edges of a graph with a minimum number of cliques is an NPcomplete problem with many applications. The stateoftheart solving algorithm is a polynomialtime heuristic from the 1970’s. We present an improvement of this heuristic. Our main contribution, however, is the development of efficient and effective polynomialtime data reduction rules that, combined with a search tree algorithm, allow for exact problem solutions in competitive time. This is confirmed by experiments with realworld and synthetic data. Moreover, we prove the fixedparameter tractability of covering edges by cliques. 1
New Similarity Measures between Polylines with Applications to Morphing and Polygon Sweeping
, 2001
"... We introduce two new related metrics, the geodesic width and the link width, for measuring the \distance" between two nonintersecting 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 ) spac ..."
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Cited by 13 (2 self)
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We introduce two new related metrics, the geodesic width and the link width, for measuring the \distance" between two nonintersecting 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 closelyrelated combinatorial strutures: the shortestpath diagram and the link diagram of a simple polygon. The shortestpath (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 ACMSIAM Symposium on Discrete Algorithms [10] and the Proceedings of the 12th Annual ACMSIAM 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, GouldSimpson Building, The University of Arizona, PO Box 210077, Tucson AZ 857210077. Email: alon@cs.arizona.edu. WWW: http://www.cs.arizona.edu/people/alon/.
Dynamic Subgraph Connectivity with Geometric Applications
 Proc. 34th ACM Sympos. Theory Comput
, 2002
"... 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 ..."
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Cited by 12 (3 self)
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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.
Morphing between Polylines
 In Proc. 12th ACMSIAM Sympos. Discrete Algorithms
, 2000
"... We study the problem of continuously transforming or morphing two nonintersecting simple (not selfintersecting) 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 ..."
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Cited by 11 (2 self)
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We study the problem of continuously transforming or morphing two nonintersecting simple (not selfintersecting) 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 userdefined 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 277080129, 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...
Area Requirement of Visibility Representations of Trees
, 1996
"... We study the area requirement of barvisibility and rectanglevisibility representations of trees in the plane. We prove asymptotically tight lower and upper bounds on the area of such representations, and give lineartime algorithms that construct representations with asymptotically optimal area. ..."
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Cited by 11 (7 self)
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We study the area requirement of barvisibility and rectanglevisibility representations of trees in the plane. We prove asymptotically tight lower and upper bounds on the area of such representations, and give lineartime algorithms that construct representations with asymptotically optimal area.
Segment Endpoint Visibility Graphs are Hamiltonian
 COMPUT. GEOM
, 2002
"... We show that the segment endpoint visibility graph of any finite set of disjoint line segments in the plane admits a simple Hamiltonian polygon, if not all segments are collinear. This proves a conjecture of Mirzaian. ..."
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Cited by 9 (3 self)
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We show that the segment endpoint visibility graph of any finite set of disjoint line segments in the plane admits a simple Hamiltonian polygon, if not all segments are collinear. This proves a conjecture of Mirzaian.