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25
A Framework for Dynamic Graph Drawing
 CONGRESSUS NUMERANTIUM
, 1992
"... Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows ..."
Abstract

Cited by 515 (40 self)
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Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows: ffl We devise a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing, and we show its applications. ffl We present several efficient dynamic drawing algorithms for trees, seriesparallel digraphs, planar stdigraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straightline, polyline, visibility), and update the drawing in a smooth way.
Solving geometric problems with the rotating calipers
, 1983
"... Shamos [1] recently showed that the diameter of a convex nsided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several se ..."
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Cited by 111 (14 self)
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Shamos [1] recently showed that the diameter of a convex nsided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be used on several convex polygons simultaneously. We then show that these generalizations allow us to obtain simple O(n) algorithms for solving a variety of problems defined on convex polygons. Such problems include (1) finding the minimumarea rectangle enclosing a polygon, (2) computing the maximum distance between two polygons, (3) performing the vectorsum of two polygons, (4) merging polygons in a convex hull finding algorithms, and (5) finding the critical support lines between two polygons. Finding the critical support lines, in turn, leads to obtaining solutions to several additional problems concerned with visibility, collision, avoidance, range fitting, linear separability, and computing the Grenander distance between sets. 1.
Efficient Binary Space Partitions for HiddenSurface Removal and Solid Modeling
, 1990
"... We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such a binary space partition, or BSP, is naturally considered as a binary tree where each internal node corresponds to a division. The goal is to choose the hyperplanes properly ..."
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Cited by 91 (0 self)
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We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such a binary space partition, or BSP, is naturally considered as a binary tree where each internal node corresponds to a division. The goal is to choose the hyperplanes properly so that the size of the BSP, i.e., the number of resulting fragments of the objects, is minimized. For the twodimensional case, we construct BSPs of size O(n log n) for n edges, while in three dimensions, we obtain BSPs of size O(n²) for n planar facets and prove a matching lower bound of Ω(n²). Two applications of efficient BSPs are given. The first is an O(n²)sized data structure for implementing a hiddensurface removal scheme of Fuchs et al. [6]. The second application is in solid modeling: given a polyhedron described by its n faces, we show how to generate an O(n²)sized CSG (constructivesolidgeometry) formula whose literals correspond to halfspaces supporting the faces of the polyhedron. The best previous results for both of these problems were O(n³).
On Geometric Assembly Planning
, 1992
"... This dissertation addresses the problem of generating feasible assembly sequences for a mechanical product from a geometric model of the product. An operation specifies a motion to bring two subassemblies together to make a larger subassembly. An assembly sequence is a sequence of operations that co ..."
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Cited by 71 (12 self)
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This dissertation addresses the problem of generating feasible assembly sequences for a mechanical product from a geometric model of the product. An operation specifies a motion to bring two subassemblies together to make a larger subassembly. An assembly sequence is a sequence of operations that construct the product from the individual parts. I introduce the nondirectional blocking graph, a succinct characterization of the blocking relationships between parts in an assembly. I describe efficient algorithms to identify removable subassemblies by constructing and analyzing the NDBG. For an assembly A of n parts and m partpart contacts equivalent to k contact points, a subassembly that can translate a small distance from the rest of A can be identified in O(mk 2 ) time. When rotations are allowed as well, the time bound is O(mk 5 ). Both algorithms are extended to find connected subassemblies in the same time bounds. All free subassemblies can be identified in outputdependent ...
Movable Separability of Sets
 Computational Geometry
, 1985
"... Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane to polyhedra in three dimensions, without allowing collisions betwee ..."
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Cited by 39 (4 self)
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Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane to polyhedra in three dimensions, without allowing collisions between the objects. One class of such problems considers the separability of sets of objects under different kinds of motions and various definitions of separation. This paper surveys this new area of research in a tutorial fashion, present new results, and provides a list of open problems and suggestions for further research. Key Words and Phrases: sofa problem, polygons, polyhedra, movable separability, visibility hulls, hidden lines, hidden surfaces, algorithms, complexity, computational geometry, spatial planning, collision avoidance, robotics, artificial intelligence. CR Categories: 3.36, 3.63, 5.25. 5.32. 5.5 * Research supported by NSERC Grant no. A9293 and FCAR Grant no.EQ1678.  2  ...
RAY SHOOTING AND OTHER APPLICATIONS OF SPANNING TREES WITH LOW STABBING NUMBER
, 1992
"... This paper considers the following problem: Given a set G of n (possibly intersecting) line segments in the plane, prcproccss it so that, given a query ray p emanating from a point p, one can quickly compute the intersection point &(G, p) of p with a segment of G that lies nearest to p. The paper pr ..."
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Cited by 31 (11 self)
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This paper considers the following problem: Given a set G of n (possibly intersecting) line segments in the plane, prcproccss it so that, given a query ray p emanating from a point p, one can quickly compute the intersection point &(G, p) of p with a segment of G that lies nearest to p. The paper presents an algorithm that preproccsses G, in time 0 ( 3/2 log n), into a data structure of size O(nc(n) log4 n), so that for a query ray p, /,(, p) can be computed in time O(v/nc(ni log2 n), where w is a constant < 4.33 and a(n) is a functional inverse of Ackermann’s function. If the given segments are nonintersecting, the storage goes down to O(n log3 n) and the query time becomes O(v/ log2 n). The main tool used is spanning trees (on the set of segment endpoints) with low stabbing number, i.e., with the property that no line intersects more than O(x/) edges of the tree. Such trees make it possible to obtain faster algorithms for several other problems, including implicit point location, polygon containment, and implicit hidden surface removal.
A General Framework for Assembly Planning: The Motion Space Approach
, 1998
"... Assembly planning is the problem of finding a sequence of motions to assemble a product from its parts. We present a general framework for finding assembly motions based on the concept of motion space. Assembly motions are parameterized such that each point in motion space represents a mating motion ..."
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Cited by 27 (5 self)
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Assembly planning is the problem of finding a sequence of motions to assemble a product from its parts. We present a general framework for finding assembly motions based on the concept of motion space. Assembly motions are parameterized such that each point in motion space represents a mating motion that is independent of the moving part set. For each motion we derive blocking relations that explicitly state which parts collide with other parts; each subassembly (rigid subset of parts) that does not collide with the rest of the assembly can easily be derived from the blocking relations. Motion space is partitioned into an arrangement of cells such that the blocking relations are fixed within each cell. In the first part of the paper we give background material, present the motion space approach and describe applications of the approach to assembly motions of several useful types, including onestep translations, multistep translations, and infinitesimal rigid motions. Several efficien...
Parallel transitive closure and point location in planar structures
 SIAM J. COMPUT
, 1991
"... Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of th ..."
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Cited by 23 (11 self)
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Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(log n) running time using n = log n processors in the EREW PRAM model, n being the number of vertices.
Objects That Cannot Be Taken Apart With Two Hands
 Proc. of the 9th ACM Symp. on Computational Geometry
, 1993
"... It has been conjectured that every configuration C of convex objects in 3space with disjoint interiors can be taken apart by translation with two hands: that is, some proper subset of C can be translated to infinity without disturbing its complement. We show that the conjecture holds for five or fe ..."
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Cited by 23 (2 self)
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It has been conjectured that every configuration C of convex objects in 3space with disjoint interiors can be taken apart by translation with two hands: that is, some proper subset of C can be translated to infinity without disturbing its complement. We show that the conjecture holds for five or fewer objects and give a counterexample with six objects. We extend the counterexample to a configuration that cannot be taken apart with two hands using arbitrary isometries (rigid motions).
Complexity Measures for Assembly Sequences
 In Proc. IEEE Int. Conf. on Robotics and Automation
, 1996
"... Our work examines various complexity measures for twohanded assembly sequences. For many products there exists an exponentially large set of valid sequences, and a natural goal is to use automated systems to select wisely from the choices. Since assembly sequencing is a preprocessing phase for a lo ..."
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Cited by 22 (3 self)
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Our work examines various complexity measures for twohanded assembly sequences. For many products there exists an exponentially large set of valid sequences, and a natural goal is to use automated systems to select wisely from the choices. Since assembly sequencing is a preprocessing phase for a long and expensive manufacturing process, any work towards ndinga\better" assembly sequence isofgreat value when it comes time to assemble the physical product in mass quantities. We take a step in this direction by introducing a formal framework for studying the optimization of several complexity measures. This framework focuses on the combinatorial aspect of the family of valid assembly sequences, while temporarily separating out the speci c geometric assumptions inherent to the problem. With an exponential number of possibilities, nding the true optimal cost solution is nontrivial. In fact in the most general case, our results show that even nding an approximate solution is hard. Furthermore, we can show several hardness results, even in simple geometric settings. Future work is directed towards using this model to study how the original geometric assumptions can be reintroduced toprove stronger approximation results. 1