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13
Automatic Creation of BoundaryRepresentation Models from Single Line Drawings
, 2002
"... This thesis presents methods for the automatic creation of boundaryrepresentation models of polyhedral objects from single line drawings depicting the objects. This topic is important in that automated interpretation of freehand sketches would remove a bottleneck in current engineering design metho ..."
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Cited by 17 (11 self)
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This thesis presents methods for the automatic creation of boundaryrepresentation models of polyhedral objects from single line drawings depicting the objects. This topic is important in that automated interpretation of freehand sketches would remove a bottleneck in current engineering design methods. The thesis does not consider conversion of freehand sketches to line drawings or methods which require manual intervention or multiple drawings. Thge thesis contains a number of...
Graph and map isomorphism and all polyhedral embeddings in linear time
 IN PROCEEDINGS OF THE 40TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 2008
"... For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g), where g is the genus of S. Th ..."
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Cited by 16 (5 self)
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For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g), where g is the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S be a fixed surface. Given a graph G and an integer k≥3, we want to find an embedding of G in S of face width at least k, or conclude that such an embedding does not exist. It is known that this problem is NPhard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of facewidth at least k, up to Whitney equivalence. Here, the facewidth of an embedded graph G is the minimum number of points of G in which some noncontractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3connected graphs.
Planar graph isomorphism is in logspace
 In IEEE Conference on Computational Complexity
, 2009
"... Abstract. We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13]. 1 ..."
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Cited by 8 (1 self)
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Abstract. We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13]. 1
Algorithm and Experiments in Testing Planar Graphs for Isomorphism
, 2004
"... We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQRtrees. We provide a proof of the algorithm’s correctness and a complexity analysis. We determi ..."
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Cited by 6 (0 self)
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We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQRtrees. We provide a proof of the algorithm’s correctness and a complexity analysis. We determine the conditions in which the implemented algorithm outperforms other graph matchers, which do not impose topological restrictions on graphs. We report experiments with our planar graph matcher tested against McKay’s, Ullmann’s, and SUBDUE’s (a graphbased data mining system) graph matchers.
The Complexity of Planar Graph Isomorphism
"... The Graph Isomorphism problem restricted to planar graphs has been known to be solvable in polynomial time many years ago. In terms of complexity classes however, the exact complexity of the problem has been established only very recently. It was proved in [6] that planar graph isomorphism can be co ..."
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Cited by 2 (0 self)
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The Graph Isomorphism problem restricted to planar graphs has been known to be solvable in polynomial time many years ago. In terms of complexity classes however, the exact complexity of the problem has been established only very recently. It was proved in [6] that planar graph isomorphism can be computed within logarithmic space. Since there is a matching hardness result [12], this shows that the problem is complete for L. Although this could be considered as a result in algorithmics its proof relies on several important new developments in the area of logarithmic space complexity classes and reflects the close connections between algorithms and complexity theory. In this column we give an overview of this result mentioning the developments that led to it. 1
A logspace algorithm for canonization of planar graphs
, 2008
"... Planar graph canonization is known to be hard for L this directly follows from Lhardness of treecanonization [Lin92]. We give a logspace algorithm for planar graph canonization. This gives completeness for logspace under AC 0 manyone reductions and improves the previously known upper bound of A ..."
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Cited by 2 (1 self)
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Planar graph canonization is known to be hard for L this directly follows from Lhardness of treecanonization [Lin92]. We give a logspace algorithm for planar graph canonization. This gives completeness for logspace under AC 0 manyone reductions and improves the previously known upper bound of AC 1 [MR91]. A planar graph can be decomposed into biconnected components. We give a logspace procedure for the decomposition of a biconnected planar graph into a triconnected component tree. The canonization process is based on these decomposition steps. 1
Interpolating Polyhedral Models . . .
, 1997
"... Metamorphosis, or morphing, is the gradual transformation of one shape into another. It generally consists of two subproblems: the correspondence problem and the interpolation problem. This paper presents a solution to the interpolation problem of transforming one polyhedral model into another. It i ..."
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Metamorphosis, or morphing, is the gradual transformation of one shape into another. It generally consists of two subproblems: the correspondence problem and the interpolation problem. This paper presents a solution to the interpolation problem of transforming one polyhedral model into another. It is an extension of the intrinsic shape interpolation scheme (T. W. Sederberg, P. Gao, G. Wang and H. Mu, ‘2D shape blending: an intrinsic solution to the vertex path problem, SIGGRAPH ’93, pp. 15–18.) for 2D polygons. Rather than considering a polyhedron as a set of independent points or faces, our solution treats a polyhedron as a graph representing the interrelations between faces. Intrinsic shape parameters, such as dihedral angles and edge lengths that interrelate the vertices and faces in the two graphs, are used for interpolation. This approach produces more satisfactory results than the linear or cubic curve paths would, and is translation and rotation invariant.
Symmetries of Polyhedra: Detection and Applications
, 1994
"... This paper deals with the detection of symmetries of polyhedra and its applications. It consists of two parts. First, we review seven algorithms for symmetry detection of polyhedra. Since these algorithms supply symmetry information in quite different forms, we classify the three most common output ..."
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This paper deals with the detection of symmetries of polyhedra and its applications. It consists of two parts. First, we review seven algorithms for symmetry detection of polyhedra. Since these algorithms supply symmetry information in quite different forms, we classify the three most common output forms of the symmetry detection algorithms and discuss their relationships and the transformation from one form into another. For each algorithm, the following five aspects are considered: the output form, the computational complexity, the polyhedra class the algorithm can handle, the implementation, and the suitability for solving the related polyhedral congruity problem. Then, we compare the seven symmetry detection algorithms and give some recommendations as to which algorithm to choose for particular applications. In the second part of this paper we discuss some applications of symmetry information in robotics, geometric problemsolving, and computer vision. The conclusions of this revie...