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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...
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...
Planar Upward Tree Drawings with Optimal Area
 Internat. J. Comput. Geom. Appl
, 1996
"... Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and pro ..."
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Cited by 19 (3 self)
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Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide lineartime algorithms for constructing optimal area drawings. Let T be a boundeddegree rooted tree with N nodes. Our results are summarized as follows: ffl We show that T admits a planar polyline upward grid drawing with area O(N ), and with width O(N ff ) for any prespecified constant ff such that 0 ! ff ! 1. ffl If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O(N log log N ). ffl We show that if T is ordered, it admits an O(N log N)area planar upward grid drawing that preserves the lefttoright ordering of the children of each node. ffl We show that all of the above area bounds are asymptotically optimal in the worst case. ffl ...
Optimal LinearTime Algorithm for the Shortest Illuminating Line Segment in a Polygon
 in a Polygon, Proc. 10th Annu. ACM Sympos. Comput. Geom
, 1994
"... Given a simple polygon, we present an optimal lineartime algorithm that computes the shortest illuminating line segment, if one exists; else it reports that none exists. This solves an intriguing open problem by improving the O(n log n)time algorithm [Ke87] for computing such a segment. 1 ..."
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Cited by 7 (2 self)
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Given a simple polygon, we present an optimal lineartime algorithm that computes the shortest illuminating line segment, if one exists; else it reports that none exists. This solves an intriguing open problem by improving the O(n log n)time algorithm [Ke87] for computing such a segment. 1
Symbol Spotting using Full Visibility Graph Representation
"... Abstract. In this paper, a method for matching symbols in linedrawings is presented. Facing both segmentation and recognition of symbols is a difficult challenge. Starting from the results of a vectorization procedure, a visibility graph is built to enhance the main geometric constraints which were ..."
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Cited by 4 (0 self)
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Abstract. In this paper, a method for matching symbols in linedrawings is presented. Facing both segmentation and recognition of symbols is a difficult challenge. Starting from the results of a vectorization procedure, a visibility graph is built to enhance the main geometric constraints which were specified during the construction of the initial document. The cliques detection, which correspond to a perceptual grouping of primitives, is used in the system to detect regions of particular interest. Both opened and perceptually closed curves are identified from aggregation of cliques. Finally, the recognition stage uses an attributed edit distance technique to match approximated curves within the host attributed relation graph and the ones from a collection of symbols.
On the Number of Directions in Visibility Representations of Graphs (Extended Abstract)
 Proc. Graph Drawing '94, Princeton NJ, 1994, Lecture Notes in Computer Science LNCS #894
, 1995
"... ) Evangelos Kranakis 1 (kranakis@scs.carleton.ca) Danny Krizanc 1 (krizanc@scs.carleton.ca) Jorge Urrutia 2 (jorge@csi.uottawa.ca) 1 Carleton University, School of Computer Science, Ottawa, ON, Canada 2 University of Ottawa, Department of Computer Science, Ottawa, ON, Canada Abstract. We c ..."
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Cited by 4 (0 self)
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) Evangelos Kranakis 1 (kranakis@scs.carleton.ca) Danny Krizanc 1 (krizanc@scs.carleton.ca) Jorge Urrutia 2 (jorge@csi.uottawa.ca) 1 Carleton University, School of Computer Science, Ottawa, ON, Canada 2 University of Ottawa, Department of Computer Science, Ottawa, ON, Canada Abstract. We consider visibility representations of graphs in which the vertices are represented by a collection O of nonoverlapping convex regions on the plane. Two points x and y are visible if the straightline segment xy is not obstructed by any object. Two objects A; B 2 O are called visible if there exist points x 2 A; y 2 B such that x is visible from y. We consider visibility only for a finite set of directions. In such a representation, the given graph is decomposed into a union of unidirectional visibility graphs, for the chosen set of directions. This raises the problem of studying the number of directions needed to represent a given graph. We study this number of directions as a graph paramet...
On the Minimum Size of Visibility Graphs
"... In this paper we give tight lower bounds on the size of the visibility graph, the contracted visibility graph, and the barvisibility graph of n disjoint line segments in the plane, according to their vertexconnectivity. ..."
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In this paper we give tight lower bounds on the size of the visibility graph, the contracted visibility graph, and the barvisibility graph of n disjoint line segments in the plane, according to their vertexconnectivity.