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Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of t ..."
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Cited by 149 (13 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
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 85 (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...
Selected Open Problems in Graph Drawing
"... In this manuscript, we present several challenging and interesting open problems in graph drawing. The goal of the listing in this paper is to stimulate future research in graph drawing. ..."
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Cited by 17 (2 self)
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In this manuscript, we present several challenging and interesting open problems in graph drawing. The goal of the listing in this paper is to stimulate future research in graph drawing.
A Sum of Squares Theorem for Visibility Complexes and Applications
, 2001
"... We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses sim ..."
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Cited by 11 (1 self)
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We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses simple data structures and only the leftturn or counterclockwise predicate; it relies, among other things, on a sum of squares like theorem for visibility complexes stated and proved in this paper. (The sum of squares theorem for a simple arrangement of lines states that the average value of the square of the number of vertices of a face of the arrangement is a O(1).)
Stretchability of Starlike PseudoVisibility Graphs
 In Proc. 15th Annu. ACM Sympos. Comput. Geom
, 1999
"... We present advances on the open problem of characterizing vertexedge visibility graphs (vegraphs), reduced by results of O'Rourke and Streinu to a stretchability question for pseudopolygons. We introduce starlike pseudopolygons as a special subclass containing all the known instances of nonst ..."
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Cited by 3 (1 self)
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We present advances on the open problem of characterizing vertexedge visibility graphs (vegraphs), reduced by results of O'Rourke and Streinu to a stretchability question for pseudopolygons. We introduce starlike pseudopolygons as a special subclass containing all the known instances of nonstretchable pseudopolygons. We give a complete combinatorial characterization and a lineartime decision procedure for starlike pseudopolygon stretchability and starlike vegraph recognition. To the best of our knowledge, this is the first problem in computational geometry for which a combinatorial characterization was found by first isolating the oriented matroid substructure and then separately solving the stretchability question. It is also the first class (as opposed to isolated examples) of oriented matroids for which an efficient stretchability decision procedure based on combinatorial criteria is given. The difficulty of the general stretchability problem implied by Mnev's Universal...
The Majority Rule and Combinatorial Geometry (via the Symmetric
 Group) Annales du LAMSADE n
"... Abstract. The Marquis du Condorcet recognized 200 years ago that majority rule can produce intransitive group preferences if the domain of possible (transitive) individual preference orders is unrestricted. We present results on the cardinality and structure of those maximal sets of permutations for ..."
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Cited by 1 (0 self)
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Abstract. The Marquis du Condorcet recognized 200 years ago that majority rule can produce intransitive group preferences if the domain of possible (transitive) individual preference orders is unrestricted. We present results on the cardinality and structure of those maximal sets of permutations for which majority rule produces transitive results (consistent sets). Consistent sets that contain a maximal chain in the Weak Bruhat Order inherit from it an upper semimodular sublattice structure. They are intrinsically related to a special class of hamiltonian graphs called persistent graphs. These graphs in turn have a clean geometric interpretation: they are precisely visibility graphs of staircase polygons. We highlight the main tools used to prove these connections and indicate possible social choice and computational research directions. 1.
NonStretchable PseudoVisibility Graphs
 PROC. 11TH CANADIAN CONF. COMP. GEOMETRY
, 1996
"... We exhibit a family of graphs which can be realized as pseudovisibility graphs of pseudopolygons, but not of straightline polygons. The construction is based on the characterization of vertexedge pseudovisibility graphs of O'Rourke and Streinu [ORS96] and extends recent results on nonstretchab ..."
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Cited by 1 (1 self)
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We exhibit a family of graphs which can be realized as pseudovisibility graphs of pseudopolygons, but not of straightline polygons. The construction is based on the characterization of vertexedge pseudovisibility graphs of O'Rourke and Streinu [ORS96] and extends recent results on nonstretchable vertexedge visibility graphs of Streinu [Str99]. We show that there is a pseudovisibility graphs for which there exists only one of vertexedge visibility graph compatible with it, which is then shown to be nonstretchable. The construction is then extended to an infinite family.