Showing papers presented at "Symposium on Computational Geometry in 1989"

Epsilon geometry: building robust algorithms from imprecise computations

Abstract: This thesis introduces Epsilon Geometry, a framework for the development of robust geometric algorithms using inaccurate primitives. Epsilon Geometry is based on a very general model of imprecise computations, which includes floating-point and rounded-integer arithmetic as special cases. The algorithms in this framework produce exact solutions for perturbed versions of their input and return a bound on the size of these implicit perturbations. The thesis begins with a formal description of the Epsilon Geometry framework. It introduces the notions of an epsilon predicate, a geometric predicate that can be satisfied by a perturbed version of its arguments; and a critical distance, the size of the perturbation required to make an epsilon predicate true. It also introduces epsilon and delta boxes, the implementations of these mathematical concepts as computer programs. The thesis describes some general rules for turning mathematical lemmas about epsilon predicates and critical distances into implementations of epsilon and delta boxes. Next, it presents a basic set of two-dimensional geometric predicates that are used for all of the algorithms in the sequel. The second half of the thesis examines how Epsilon Geometry can be applied to various geometric objects and algorithms in the plane. It defines the notions of an $\varepsilon$-convex polygon, a polygon that can be made convex by perturbing each of its vertices by $\varepsilon$ or less; and a ($-\varepsilon)$-convex polygon, a polygon that remains convex even if its vertices are all displaced in arbitrary directions by a distance of $\varepsilon$ or less. The thesis develops robust algorithms for testing point inclusion in both kinds of polygons, and for testing a polygon's degree of convexity. Finally, the thesis investigates the existence of approximate hulls for sets of points. It proves that for every point set there exists a ($-\varepsilon$)-convex polygon H such that every point of the set is at most 4$\varepsilon$ away from H, and it describes robust algorithms for computing such hulls.

Representing geometric structures in d dimensions: topology and order

Abstract: We develop a representation for the topological structure of subdivided manifolds (with and without boundary) of dimension d ≥ 1 which allows straightforward access of the available order information. It is shown that there exists a large amount of ordering information in subdivided manifolds: given a (k-2)-cell in the boundary of a (k+1)-cell, 1 ≤ k ≤ d, all of the k- and (k-1)-cells 'between them' can be ordered 'around' the (k-2)-cell. This includes the usual orderings in 2- and 3-dimensional objects. We introduce the 'cell-tuple structure', a simple, uniform representation of the incidence and ordering information in a subdivided manifold. It includes the quad-edge data structure of Guibas and Stolfi [GS 85] and the facet-edge data structure of Dobkin and Laszlo [DL 87] as special cases in dimensions 2 and 3, respectively.

Quasi-optimal range searching in spaces of finite VC-dimension

Abstract: The range-searching problems that allow efficient partition trees are characterized as those defined by range spaces of finite Vapnik-Chervonenkis dimension. More generally, these problems are shown to be the only ones that admit linear-size solutions with sublinear query time in the arithmetic model. The proof rests on a characterization of spanning trees with a low stabbing number. We use probabilistic arguments to treat the general case, but we are able to use geometric techniques to handle the most common range-searching problems, such as simplex and spherical range search. We prove that any set ofn points inEd admits a spanning tree which cannot be cut by any hyperplane (or hypersphere) through more than roughlyn1Â?1/d edges. This result yields quasi-optimal solutions to simplex range searching in the arithmetic model of computation. We also look at polygon, disk, and tetrahedron range searching on a random access machine. Givenn points inE2, we derive a data structure of sizeO(n logn) for counting how many points fall inside a query convexk-gon (for arbitrary values ofk). The query time isO(Â?kn logn). Ifk is fixed once and for all (as in triangular range searching), then the storage requirement drops toO(n). We also describe anO(n logn)-size data structure for counting how many points fall inside a query circle inO(Â?n log2n) query time. Finally, we present anO(n logn)-size data structure for counting how many points fall inside a query tetrahedron in 3-space inO(n2/3 log2n) query time. All the algorithms are optimal within polylogarithmic factors. In all cases, the preprocessing can be done in polynomial time. Furthermore, the algorithms can also handle reporting within the same complexity (adding the size of the output as a linear term to the query time).

Subdivisions of n-dimensional spaces and n-dimensional generalized maps

Abstract: This paper deals with the modeling of n-dimensional objects, more precisely with the modeling of subdivisions of n-dimensional topological spaces. We here study the notions of: n-dimensional generalized map (or n-G-map), for the modeling of the topology of any subdivision of any n-dimensional topological space (orientable or not orientable, with or without boundaries);n-dimensional map (or n-map), for the modeling of the topology of any subdivision of any orientable n-dimensional topological space, without boundaries.These two notions extend the notion of topological map, which has been used for the modeling of the topology of any subdivision of any surface.We study in this paper some properties of the n-G-maps and the n-maps (orientability, duality, relationships between n-G-maps and n-maps …), and we define also operations for constructing any n-G-map.

Triangulating a non-convex polytype

Abstract: This paper is concerned with the problem of partitioning a three-dimensional polytope into a small number of elementary convex parts. The need for such decompositions arises in tool design, computer-aided manufacturing, finite-element methods, and robotics. Our main result is an algorithm for decomposing a polytope with n vertices and r reflex edges into O(n+r2) tetrahedra. This bound is asymptotically tight in the worst case. The algorithm is simple and practical. Its running time is O(nr + r2 log r).

Higher-dimensional Voronoi diagrams in linear expected time

Abstract: This work is the first to validate theoretically the suspicions of many researchers — that the “average” Voronoi diagram is combinatorially quite simple and can be constructed quickly. Specifically, assuming that dimension d is fixed, and that n input points are chosen independently from the uniform distribution on the unit d-ball, it is proved that the expected number of simplices of the dual of the Voronoi diagram is T(n) (exact constants are derived for the high-order term), anda relatively simple algorithm exists for constructing the Voronoi diagram in T(n) time.It is likely that the methods developed in the analysis will be applicable to other related quantities and other probability distributions.

An acyclicity theorem for cell complexes in d dimensions

Abstract: Let C be a cell complex in d-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope in d + 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This paper shows that the in_front/behind relation defined for the faces of C with respect to any fixed viewpoint x is acyclic. This result has applications to hidden line/surface removal and other problems in computational geometry.

On the number of halving planes

Abstract: Let S ⊂ R3 be an n-set in general position. A plane containing three of the points is called a halving plane if it dissects S into two parts of equal cardinality. It is proved that the number of halving planes is at most O(n2.998).As a main tool, for every set Y of n points in the plane a set N of size O(n4) is constructed such that the points of N are distributed almost evenly in the triangles determined by Y.

On monotone paths among obstacles with applications to planning assemblies

Abstract: We study the class of problems associated with the detection and computation of monotone paths among a set of disjoint obstacles. We give an O(nE) algorithm for finding a monotone path (if one exists) between two points in the plane in the presence of polygonal obstacles. (Here, E is the size of the visibility graph defined by the n vertices of the obstacles.) If all of the obstacles are convex, we prove that there always exists a monotone path between any two points s and t. We give an O(n log n) algorithm for finding such a path for any s and t, after an initial O(E + n log n) preprocesing. We introduce the notions of “monotone path map”, and “shortest monotone path map” and give algorithms to compute them. We apply our results to a class of separation and assembly problems, yielding polynomial-time algorithms for planning an assembly sequence (based on separations by single translations) of arbitrary polygonal parts in two dimensions.

Binary partitions with applications to hidden surface removal and solid modelling

Abstract: We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such a binary partition is naturally considered as a binary tree where each internal node corresponds to a division and the leaves correspond to the resulting fragments of objects. The goal is to choose the hyperplanes properly so that the size of the binary partition, i.e., the number of resulting fragments of the objects, is minimized. We construct binary partitions of size O(n log n) for n edges in the plane, and of size O(n) if the edges are orthogonal. In three dimensions, we obtain binary partitions of size O(n2) for n planar facets, and prove a lower bound of O(n3/2). Two applications of efficient binary partitions are given. The first is an O(n2)-sized data structure for implementing a hidden-surface removal scheme of Fuchs, Kedem and Naylor [5]. The second application is in solid modelling: given a polyhedron described by its n faces, we show how to generate an O(n2)-sized CSG (constructive-solid-geometry) formula whose literals correspond to half-spaces supporting the faces of the polyhedron (see Peterson [9] and Dobkin et al. [3]). The best previous results for both of these problems were O(n3).

Sweeping arrangements of curves

Abstract: We consider arrangements of curves that intersect pairwise in at most k points. We show that a curve can sweep any such arrangement and maintain the k-intersection property if and only if k equals 1 or 2. We apply this result to an eclectic set of problems: finding Boolean formulae for polygons with curved edges, counting triangles and digons in arrangements of pseudocircles, and finding extension curves for arrangements. We also discuss implementing the sweep.

Finding tailored partitions

Abstract: We consider the following problem: given a planar set of points S, a measure m acting on S, and a pair of values m1 and m2, does there exist a bipartition S = S1 U S2 satisfying m(Si) ≤ mi for i = 1,2? We present algorithms of complexity O(n log n) for several natural measures, including the diameter (set measure), the area, perimeter or diagonal of the smallest enclosing axes-parallel rectangle (rectangular measure), and the side length of the smallest enclosing axes-parallel square (square measure). The problem of partitioning S into k subsets, where k ≥ 3, is known to be NP-complete for many of these measures.

Computing partial sums in multidimensional arrays

Abstract: This problem comes in two distinct avors In query mode preprocessing is allowed and q is a query to be an swered on line In o line mode we are given the array A and a set of d rectangles q qm and we must com pute the m sums A qi Partial sum is a special case of the classical orthogonal range searching problem Given n weighted points in d space and a query d rectangle q compute the cumulative weight of the points in q see e g The dynamic version of partial sum in query mode was studied by Fredman who showed that a mixed sequence of n insertions deletions and queries may require n log n time which is optimal Willard and Lueker This re sult was partially extended to groups by Willard in For the case where only insertions and queries are al lowed a lower bound of n log n log log n was proven in the one dimensional case Yao and later extended to n log n log log n d for any xed dimension d Chazelle Regarding static one dimensional partial sum Yao proved that if m units of storage are used then any query can be answered in time O m n which is optimal in the arithmetic model The function m n is the func tional inverse of Ackermann s function de ned by Tarjan See also Alon and Schieber for related upper and lower bounds Our main results are a nonlinear lower bound for one dimensional partial sum in o line mode and a space time tradeo for partial sum in query mode in any xed di mension More precisely we prove that for any n and m there exist m partial sums whose evaluations require n m m n time This is a rare case where the func tion arises in an o line problem Noticeable instances are the complexity of union nd Tarjan and the length of Davenport Schinzel sequences Hart and Sharir Agar wal et al Interestingly the proof technique we use does not involve reductions from these problems Our result im plies that given a sequence of n numbers computing partial sums over a well chosen set of n intervals requires a nonlin ear number of additions This might come as a surprise in light of the fact that there is a trivial linear time algorithm as soon as we allow subtraction The lower bound can be regarded as a generalization of a result of Tarjan con cerning the o line evaluation of functions de ned over the paths of a tree As in our result also leads to an im proved lower bound on the minimum depth of a monotone circuit for computing conjunctions The other contribution of this paper is an algorithm which can answer any partial sum query in time O d m n where m is the amount of storage available This generalizes Yao s one dimensional upper bound to xed arbitrary dimension d Since our algorithm works on a RAM we can use it as the inner loop of standard multidimensional search ing structures For example consider the classical orthogo nal range searching problem on n weighted points in d space Lueker and Willard have described a data structure of size O n log n which can answer any range query in time O log n over a semigroup We improve the time bound to O n log n The remainder of this abstract is devoted to the proofs of the lower and upper bounds Except for a few technical lemmas whose proofs have been omitted our exposition is complete and self contained

On the difficulty of tetrahedralizing 3-dimensional non-convex polyhedra

Abstract: A number of different polyhedral decomposition problems have previously been studied, most notably the problem of triangulating a simple polygon. We are concerned with the tetrahedralization problem: decomposing a 3-dimensional polyhedron into a set of non-overlapping tetrahedra whose vertices are chosen from the vertices of the polyhedron. It has previously been shown that some polyhedra cannot be tetrahedralized in this fashion. We show that the problem of deciding whether a given polyhedron can be tetrahedralized is NP-complete, and hence likely to be computationally intractable. The problem remains NP-complete when restricted to the case of star-shaped polyhedra. Various versions of the question of how many Steiner points are needed to tetrahedralize a polyhedron also turn out to be NP-complete.

An efficient algorithm for link-distance problems

Abstract: The link distance between two points inside a simple polygon P is defined to be the minimum number of edges required to form a polygonal path inside P that connects the points. A link furthest neighbor of a point p E P is a point of P whose link distance is the maximum from p. The link center of P is the collection of points whose link distances to their link furthest neighbors are minimized. We present an O(n log n) time and O(n) space algorithm for computing the link center of a simple polygon P, where n is the number of vertices of P. This improves the previous O(n2) time and space algorithm. Our algorithm essentially sweeps a chord through the polygon and spends O(log n) time at each step. We demonstrate that the output of the algorithm, a sequence of sets of chords, is a powerful tool for solving several other link distance problems.

A fast planar partition algorithm, II

Abstract: Randomized, optimal algorithms to find a partition of the plane induced by a set of algebraic segments of a bounded degree, and a set of linear chains of a bounded degree, are given. This paper also provides a new technique for clipping, called virtual clipping, whose overhead per window W depends logarithmically on the number if intersections between the borders of W and the input segments. In contrast, the overhead of the conventional clipping technique depends linearly on this number of intersections. As an application of virtual clipping, a new simple and efficient algorithm for plannar point location is given.

An efficient algorithm for one-step planar complaint motion planning with uncertainty

Abstract: Uncertainty in the execution of robot motion plans must be accounted for in the geometric computations from which plans are obtained, especially in the case where position sensing is inaccurate. We give an O(n2 log n) algorithm to find a single commanded motion direction which will guarantee a successful motion in the plane from a specified start to a specified goal whenever such a one-step motion is possible. The plans account for uncertainty in the start position and in robot control, and anticipate that the robot may stick on or slide along obstacle surfaces with which it comes in contact. This bound improves on the best previous bound by a quadratic factor, and is achieved in part by a new analysis of the geometric complexity of the backprojection of the goal as a function of commanded motion direction.

Area requirement and symmetry display in drawing graphs

Abstract: A classical result shows that every planar graph admits a planar drawing with straight-line edges (struight-line druwing) [S, 25,26,34]. However, the existence of planar straight-line drawings with vertices placed at grid points (i.e., with integer coordinates) and polynomial area has been one of the most important and intriguing open problems in this field [23]. This question has been positively settled by de Fraysseix, Path and Pollack [9], and, independently, by Schnyder [24], who show that every n-vertex planar graph admits a planar straight-line drawing with vertices placed at grid points and 0 (n2) area.

Placing the largest similar copy of a convex polygon among polygonal obstacles

Abstract: Given a convex polygon P and an environment consisting of polygonal obstacles, we find the largest similar copy of P that does not intersect any of the obstacles. Allowing translation, rotation, and change-of-size, our method combines a new notion of Delaunay triangulation for points and edges with the well-known functions based on Davenport-Schinzel sequences producing an almost quadratic algorithm for the problem. Namely, if P is a convex k-gon and if Q has n corners and edges then we can find the placement of the largest similar copy of P in the environment Q in time O(k4n l4(kn) log n), where l4 is one of the almost-linear functions related to Davenport-Schinzel sequences. If the environment consists only of points then we can find the placement of the largest similar copy of P in time O(k2n l3(kn) log n).

A deterministic algorithm for partitioning arrangements of lines and its application

Abstract: In this paper we consider the following problem: Given a set l of n lines in the plane, partition the plane into O(r2) triangles so that no triangle intersects more than O(n/r) lines of l. We present a deterministic algorithm for this problem with O(nr log n logor) running time, where o is a constant

Bounded diameter minimum spanning trees and related problems

Abstract: We consider the problem of finding a minimum diameter spanning tree (MDST) of a set of n points in the Euclidean plane. The diameter of a spanning tree is the maximum distance between any two points in the tree. We give a characterization of an MDST and present a t(n3 time algorithm for solving the problem. We also show that for a weighted undirected graph, the problem of determining if a spanning tree with total weight and diameter upper bounded, respectively, by two given parameters C and D exists is N P-complete. The geometrical minimum diameter Steiner tree problem, in which new points are allowed to be part of the spanning tree, is shown to be solvable in O(n) time.

Calculating approximate curve arrangements using rounded arithmetic

Abstract: We present here an algorithm for the curve arrangement problem: determine how a set of planar curves subdivides the plane. This algorithm uses rounded arithmetic and generates an approximate result. It can be applied to a broad class of planar curves, and it is based on a new definition of approximate curve arrangements. This result is an important step towards the creation of practical computer programs for reasoning about algebraic curves of high degree.

Good splitters for counting points in triangles

Abstract: A set A of n points in the plane has to be stored in such a way that for any query triangle t the number of points of A inside t can be computed efficiently. For this problem a solution is presented with O(√n log n) query time, O (n log n) space and O(n3/2 log n) preprocessing time. The constants in the asymptotic bounds are small, and the method is easy to implement.

Compliant motion in a simple polygon

Abstract: We consider motion planning under the compliant motion model, in which a robot directed to walk into a wall may slide along it. We examine several variants of compliant motion planning for a point robot inside a simple polygon with n sides, where the goal is a fixed vertex or edge. For the case in which the robot moves with perfect control, we build a data structure that lets us in O(log n) time determine the range of directions in which the robot can move from a query point to the goal in a single step. This structure lets us solve a variety of other problems: we can find a similar query data structure for multi-step paths; we can solve single-step problems allowing uncertainty in control and position sensing; and we can explicitly compute the set of all points that can reach the goal in a single step, even allowing uncertainty in control. Our algorithms run in O(n log n) time and linear space; they use a novel method for maintaining convex hulls of simple paths that may be of independent interest.

Computing the geodesic diameter of a 3-polytope

Abstract: We present an O(n14 log n) algorithm for computing the geodesic diameter of a 3-polytope of n vertices. The geodesic diameter is the greatest separation between two points on the surface, where distance is determined by the shortest (geodesic) path between two points. We assume a model of computation that permits finding roots of a one-variable polynomial of fixed degree in constant time. The key geometric result underlying the algorithm is that, although it may be that neither endpoint of the diameter is a vertex of the polytope, when this occurs, there must be at least five distinct equal-length paths between the diameter endpoints.

Stabbing pairwise disjoint translates in linear time

Abstract: In general, finding a line stabber for a family of n objects in the plane takes o(n log n) time. However, we show how to find a line stabber for a family of n pairwise disjoint convex translates in the plane in linear time. Our algorithm still runs in optimal O (n log n) time when the translates are not pairwise disjoint.

On the parallel decomposability of geometric problems

Abstract: There is a large and growing body of literature concerning the solution of geometric problems on mesh-connected arrays of processors [5,9,14,17]. Most of these algorithms are optimal (i.e., run in time O(n1/d) on a d-dimensional n-processor array), and they all assume that the parallel machine is trying to solve a problem of size n on an n-processor array. What happens when we have parallel machine for efficiently solving a problem of size p, and we are interested in using it to solve a problem of size n