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, series-parallel digraphs, planar st-digraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straight-line, polyline, visibility), and update the drawing in a smooth way.

Shamos [1] recently showed that the diameter of a convex n-sided 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 minimum-area rectangle enclosing a polygon, (2) computing the maximum distance between two polygons, (3) performing the vector-sum 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.

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 non-directional 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 part--part 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 output-dependent ...

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 - ...

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 one-step translations, multi-step translations, and infinitesimal rigid motions. Several efficien...

Abstract. Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar st-graphs, 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. Key words. parallel algorithms, parallel computation, graph algorithms, planar st-graphs, transitive closure, reachability, planar point location, computational geometry, fractional cascading, graph drawing, visibility AMS(MOS) subject classi cations. 68E05, 68C05, 68C25 1. Introduction. Planar st-graphs

It has been conjectured that every configuration C of convex objects in 3-space 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).

Our work examines various complexity measures for two-handed 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 non-trivial. 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

Abstract. We consider the problem of removing a given disk from a collection of unit disks in the plane. At each step, we allow a disk to be removed by a collision-free translation to infinity, and the goal is to access a given disk using as few steps as possible. This Disks problem is a version of a common task in assembly sequencing, namely removing a given part from a fully assembled product. Recently there has been a focus on optimizing assembly sequences over various cost measures, however with very limited algorithmic success. We explain this lack of success, proving strong inapproximability results in this simple geometric setting. Namely, we show that approximating the number of steps required to within a factor of 2 log1−γ n for any γ>0 is quasi-NP-hard. This provides the first inapproximability results for assembly sequencing, realized in a geometric setting. As a stepping stone, we study the approximability of scheduling with and/or precedence constraints. The Disks problem can be formulated

We show some combinatorial and algorithmic results concerning sets of lines and polyhedral objects in 3-space. Our main results include: (1) An O(n 3 2 c p log n ) upper bound on the worst case complexity of the set of lines missing a star-shaped compact polyhedron with n edges, where c is a suitable constant. (2) An O(n 3 2 c p log n ) upper bound on the worst case complexity of the set of lines that can be moved to infinity without intersecting a set of n given lines, where c is a suitable constant. This bound is almost tight. (3) An O(n 1:5+ffl ) randomized expected time algorithm that tests whether a direction v exists along which a set of n red lines can be translated away from a set of n blue lines without collisions. (4) Computing the intersection of two polyhedral terrains in 3-space with n total edges in time O(n 4=3+ffl + k 1=3 n 1+ffl + k log 2 n), where k is the size of the output, and ffl ? 0 an arbitrary small but fixed constant. This algorithm ...