Showing papers in "Discrete and Computational Geometry in 1995"
TL;DR: A deterministic polynomial-time method for finding a set cover in a set system (X, ℛ) of dual VC-dimensiond such that the size of the authors' cover is at most a factor ofO(d log(dc)) from the optimal size,c.
Abstract: We give a deterministic polynomial-time method for finding a set cover in a set system (X, ?) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal size,c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives anO(logc) approximation factor. This improves the previous ?(log?X?) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in three-dimensional polytope approximation and two-dimensional disk covering, we can quickly findO(c)-sized covers.
541 citations
TL;DR: This lemma is a general “Localization Lemma” that reduces integral inequalities over then-dimensional space to integral inequalities in a single variable and is illustrated by showing how a number of well-known results can be proved using it.
Abstract: We study the smallest number ?(K) such that a given convex bodyK in ?n can be cut into two partsK1 andK2 by a surface with an (n?1)-dimensional measure ?(K) vol(K1)·vol(K2)/vol(K). LetM1(K) be the average distance of a point ofK from its center of gravity. We prove for the "isoperimetric coefficient" that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaqegWuDJLgzHbYqV52CVXwzaGGbciaa-H8acqGGOaak% cqWGlbWscqGGPaqkcqGHLjYSdaWcaaqaaiGbcYgaSjabc6gaUjabik% daYaqaaiabd2eannaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIaem4s% aSKaeiykaKcaaaaa!4EFC! $$\psi (K) \geqslant \frac{{\ln 2}}{{M_1 (K)}}$$ , and give other upper and lower bounds. We conjecture that our upper bound is the exact value up to a constant.
Our main tool is a general "Localization Lemma" that reduces integral inequalities over then-dimensional space to integral inequalities in a single variable. This lemma was first proved by two of the authors in an earlier paper, but here we give various extensions and variants that make its application smoother. We illustrate the usefulness of the lemma by showing how a number of well-known results can be proved using it.
489 citations
TL;DR: In this article, a simplicial complex dual to a decomposition of the union of balls using Voronoi cells is presented for computing topological, combinatorial, and metric properties of a union of finitely many spherical balls.
Abstract: Efficient algorithms are described for computing topological, combinatorial, and metric properties of the union of finitely many spherical balls in ?d. These algorithms are based on a simplicial complex dual to a decomposition of the union of balls using Voronoi cells, and on short inclusion-exclusion formulas derived from this complex. The algorithms are most relevant in ?3 where unions of finitely many balls are commonly used as models of molecules.
350 citations
TL;DR: It is proved that for a graphG withn vertices andk edges and for a dimensiond the image of the so-called rigidity map ℝdn→ℝk is a convex set in �”k provided by MathType!MTEF.
Abstract: A weighted graph is calledd-realizable if its vertices can be chosen ind-dimensional Euclidean space so that the Euclidean distance between every pair of adjacent vertices is equal to the prescribed weight. We prove that if a weighted graph withk edges isd-realizable for somed, then it isd-realizable for $$d = \left[ {\left( {\sqrt {8k + 1} - 1} \right)/2} \right]$$ (this bound is sharp in the worst case). We prove that for a graphG withn vertices andk edges and for a dimensiond the image of the so-called rigidity map ?dn??k is a convex set in ?k provided $$d \geqslant \left[ {\left( {\sqrt {8k + 1} - 1} \right)/2} \right]$$ . These results are obtained as corollaries of a general convexity theorem for quadratic maps which also extends the Toeplitz-Hausdorff theorem. The main ingredients of the proof are the duality for linear programming in the space of quadratic forms and the "corank formula" for the strata of singular quadratic forms.
270 citations
TL;DR: What is the tightest packing of N equal nonoverlapping spheres, in the sense of having minimal energy, i.e., smallest second moment about the centroid?
Abstract: What is the tightest packing ofN equal nonoverlapping spheres, in the sense of having minimal energy, i.e., smallest second moment about the centroid? The putatively optimal arrangements are described forN≤32. A number of new and interesting polyhedra arise.
130 citations
TL;DR: A general algorithm is given and its efficient implementations for specific geometric problems are discussed, for instance for the problem of computing the smallest circle enclosing all butk of the givenn points in the plane, anO(n logn+k3nε) algorithm.
Abstract: We investigate the problem of finding the best solution satisfying all butk of the given constraints, for an abstract class of optimization problems introduced by Sharir and Welzl--the so-calledLP-type problems. We give a general algorithm and discuss its efficient implementations for specific geometric problems. For instance for the problem of computing the smallest circle enclosing all butk of the givenn points in the plane, we obtain anO(n logn+k3n?) algorithm; this improves previous results fork small compared withn but moderately growing. We also establish some results concerning general properties ofLP-type problems.
127 citations
TL;DR: The technique of Gaussian-like measures on lattices is applied to obtain upper bounds for jh(U, V) in the case when U, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inlpn, 1≤p≤∞.
Abstract: LetL be a lattice and letU be ano-symmetric convex body inRn. The Minkowski functional Â? Â?U ofU, the polar bodyU0, the dual latticeL*, the covering radius μ(L, U), and the successive minima Â?i(L,U)i=1,...,n, are defined in the usual way. Let Â?n be the family of all lattices inRn. Given a pairU,V of convex bodies, we define[Figure not available: see fulltext.] and kh(U, V) is defined as the smallest positive numbers for which, given arbitraryLÂ?Â?n anduÂ?Rn/(L+U), somevÂ?L* with Â?vÂ?V≤sd(uv, Â?) can be found. Upper bounds for jh(U, U0), j=k, l, m, belong to the so-called transference theorems in the geometry of numbers. The technique of Gaussian-like measures on lattices, developed in an earlier paper [4] for euclidean balls, is applied to obtain upper bounds for jh(U, V) in the case whenU, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inlpn, 1≤p≤Â?. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most logn asnÂ?Â?. It is also proved that ifU is symmetric through each of the coordinate hyperplanes, then jh(U, U0) are less thanCn logn for some numerical constantC.
120 citations
TL;DR: A new proof that ifG is CP hyperbolic andD is any simply connected proper subdomain of the plane, then there is a disk packingP with contacts graphG such thatP is contained and locally finite inD.
Abstract: The contacts graph, or nerve, of a packing, is a combinatorial graph that describes the combinatorics of the packing. LetG be the 1-skeleton of a triangulation of an open disk.G is said to be CP parabolic (resp. CP hyperbolic) if there is a locally finite disk packingP in the plane (resp. the unit disk) with contacts graphG. Several criteria for deciding whetherG is CP parabolic or CP hyperbolic are given, including a necessary and sufficient combinatorial criterion. A criterion in terms of the random walk says that if the random walk onG is recurrent, theG is CP parabolic. Conversely, ifG has bounded valence and the random walk onG is transient, thenG is CP hyperbolic.
We also give a new proof thatG is either CP parabolic or CP hyperbolic, but not both. The new proof has the advantage of being applicable to packings of more general shapes. Another new result is that ifG is CP hyperbolic andD is any simply connected proper subdomain of the plane, then there is a disk packingP with contacts graphG such thatP is contained and locally finite inD.
118 citations
TL;DR: This work maintains the minimum spanning tree of a point set in the plane subject to point insertions and deletions, in amortized timeO(n1/2 log2n) per update operation, and uses a novel construction, theordered nearest neighbor path of a set of points.
Abstract: We maintain the minimum spanning tree of a point set in the plane subject to point insertions and deletions, in amortized timeO(n1/2 log2n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in timeO(ne) per update. Our algorithm uses a novel construction, theordered nearest neighbor path of a set of points. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining minima of binary functions, including the diameter of a point set and the bichromatic farthest pair.
100 citations
TL;DR: An anO(n logn)-time method for finding a bestk-link piecewise-linear function approximating ann-point planar point set using the well-known uniform metric to measure the error, ε≥0, of the approximation.
Abstract: We given anO(n logn)-time method for finding a bestk-link piecewise-linear function approximating ann-point planar point set using the well-known uniform metric to measure the error, ??0, of the approximation. Our methods is based upon new characterizations of such functions, which we exploit to design an efficient algorithm using a plane sweep in "? space" followed by several applications of the parametric-searching technique. The previous best running time for this problems wasO(n2).
87 citations
TL;DR: The result holds for arbitrary set systems as long as theimal shatter function isO(md) and matches known lower bounds, improving previous upper bounds by a factor.
Abstract: We show that the discrepancy of anyn-point setP in the Euclideand-space with respect to half-spaces is bounded byCdn1/2?1/2d, that is, a mapping ?:P?{?1,1} exists such that, for any half-space ?, ?, |Σp?P???(p)|≤Cdn1/2-1/2d. In fact, the result holds for arbitrary set systems as long as theprimal shatter function isO(md). This matches known lower bounds, improving previous upper bounds by a $$\sqrt {\log n} $$ factor.
TL;DR: It is shown that in three or more dimensions, reconfiguration is always possible, but that in dimension two this is not the case, and anO(n) algorithm is given for determining whether it is possible to move between two given configurations of a closed chain in the plane.
Abstract: Consider the problem of moving a closed chain ofn links in two or more dimensions from one given configuration to another. The links have fixed lengths and may rotate about their endpoints, possibly passing through one another. The notion of a "line-tracking motion" is defined, and it is shown that when reconfiguration is possible by any means, it can be achieved byO(n) line-tracking motions. These motions can be computed inO(n) time on real RAM. It is shown that in three or more dimensions, reconfiguration is always possible, but that in dimension two this is not the case. Reconfiguration is shown to be always possible in two dimensions if and only if the sum of the lengths of the second and third longest links add to at most the sum of the lengths of the remaining links. AnO(n) algorithm is given for determining whether it is possible to move between two given configurations of a closed chain in the plane and, if it is possible, for computing a sequence of line-tracking motions to carry out the reconfiguration.
TL;DR: A new technique for deriving optimal-sized polygonal schema for triangulated compact 2-manifolds without boundary inO(n) time, wheren is the size of the given triangulationT and Seifert-Van Kampen's theorem is used.
Abstract: We provide a new technique for deriving optimal-sized polygonal schema for triangulated compact 2-manifolds without boundary inO(n) time, wheren is the size of the given triangulationT. We first derive a polygonal schemaP embedded inT using Seifert-Van Kampen's theorem. A reduced polygonal schemaQ of optimal size is computed fromP, where a surjective map from the vertices ofP is retained to the vertices ofQ. This helps detecting null-homotopic (contractible to a point) cycles. Given a cycle of lengthk, we determine if it is null-homotopic inO(n+k logg) time and in ?(n+k) space, whereg is the genus of the given 2-manifold. The actual contraction for a null-homotopic cycle can be computed in ?(nk) time and space. This is a considerable improvement over the previous best-known algorithm for this problem.
TL;DR: It is proved that, under theLp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball, and an even tighter bound for M STs where the maximum degree is minimized is shown.
Abstract: Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that, under theLp norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where the maximum degree is minimized. We give the best-known bounds for the maximum MST degree for arbitraryLp metrics in all dimensions, with a focus on the rectilinear metric in two and three dimensions. We show that for any finite set of points in the rectilinear plane an MST exists with maximum degree of at most 4, and for three-dimensional rectilinear space the maximum possible degree of a minimum-degree MST is either 13 or 14.
TL;DR: A new type of randomized incremental algorithms suited for computing structures that have a “nonlocal” definition are introduced, and some results on random sampling are generalized to such situations.
Abstract: We introduce a new type of randomized incremental algorithms. Contrary to standard randomized incremental algorithms, theselazy randomized incremental algorithms are suited for computing structures that have a "nonlocal" definition. In order to analyze these algorithms we generalize some results on random sampling to such situations. We apply our techniques to obtain efficient algorithms for the computation of single cells in arrangements of segments in the plane, single cells in arrangements of triangles in space, and zones in arrangements of hyperplanes.
TL;DR: An algorithm is presented which places the [(n+h)/3] guards inO(n2) time to supervise an art gallery with holes.
Abstract: In this paper we consider the problem of placing guards to supervise an art gallery with holes. No gallery withn vertices andh holes requires more than [(n+h)/3] guards. For some galleries this number of guards is necessary. We present an algorithm which places the [(n+h)/3] guards inO(n2) time.
TL;DR: This work describes what may beall the best packings of nonoverlapping equal spheres in dimensionsn ≤10, where “best” means both having the highest density and not permitting any local improvement.
Abstract: We describe what may beall the best packings of nonoverlapping equal spheres in dimensionsn ≤10, where "best" means both having the highest density and not permitting any local improvement. For example, the best five-dimensional sphere packings are parametrized by the 4-colorings of the one-dimensional integer lattice. We also find what we believe to be the exact numbers of "uniform" packings among these, that is, those in which the automorphism group acts transitively. These assertions depend on certain plausible but as yet unproved postulates.
Our work may be regarded as a continuation of Laszlo Fejes Toth's work on solid packings.
TL;DR: An algorithm for triangulatingn-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than π/2, and the running time isO(n log2n).
Abstract: We give an algorithm for triangulatingn-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than ?/2. The number of triangles in the triangulation is onlyO(n), improving a previous bound ofO(n2), and the running time isO(n log2n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm.
TL;DR: An anO(|V(G)|)-time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graphG so that no two segments have an interior point in common, and two segments touch each other if and only if the corresponding vertices are adjacent.
Abstract: We give anO(|V(G)|)-time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graphG so that (i) no two segments have an interior point in common, and (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we obtain a strengthening of the following theorem of Ringel and Petrovia?. The edges of any maximal bipartite plane graphG with outer facebwb?w? can be colored by two colors such that the color classes form spanning trees ofG?b andG?b?, respectively. Furthermore, such a coloring can be found in linear time. Our method is based on a new linear-time algorithm for constructing bipolar orientations of 2-connected plane graphs.
TL;DR: It is shown that the cartesian product of two 5-cycles has crossing number 15, and the total number of points of intersection in is at least 2(m−1)n, where, and this bound is best possible.
Abstract: It[Figure not available: see fulltext.] and[Figure not available: see fulltext.] are two families of pairwise disjoint simple closed curves in the plane such that each curve in[Figure not available: see fulltext.] intersects each curve in[Figure not available: see fulltext.], then the total number of points of intersection in[Figure not available: see fulltext.] is at least 2(m?1)n, where[Figure not available: see fulltext.][Figure not available: see fulltext.], and this bound is best possible. We use this to show that the cartesian product of two 5-cycles has crossing number 15.
TL;DR: It is shown that for everyk and everyp≥q≥d+1 there is ac=c(k,p,q,d)<∞ such that the following holds.
Abstract: It is shown that for everyk and everyp?q?d+1 there is ac=c(k,p,q,d)
TL;DR: In the case whereS consists of the vertices of a regular polygon, an argument from hyperbolic geometry is used to exhibit an optimal net of sizeO(1/ε), which improves a previous bound of Capoyleas.
Abstract: LetS be a set ofn points in ?d. A setW is aweak ?-net for (convex ranges of)S if, for anyT⊆S containing ?n points, the convex hull ofT intersectsW. We show the existence of weak ?-nets of size % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGaem4ta8KaeiikaGIaeiikaGIaeGymaeJaei4la8Ia% eqyTdu2aaWbaaSqabeaacqWGKbazaaGccqGGPaqkcyGGSbaBcqGGVb% WBcqGGNbWzdaahaaWcbeqaaiabek7aInaaBaaameaacqWGKbazaeqa% aaaakiabcIcaOiabigdaXiabc+caViabew7aLjabcMcaPiabcMcaPa% aa!50FE! $$O((1/\varepsilon ^d )\log ^{\beta _d } (1/\varepsilon ))$$ , whereβ2=0,β3=1, andβd?0.149·2d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/?) log1.6(1/?)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/?), which improves a previous bound of Capoyleas.
TL;DR: This paper contains a variety of qualitative results that are related to the problems of finding a largest, a stable, or a rigidj-simplex in a givenn-dimensional convex body or convex polytope.
Abstract: Relative to a given convex bodyC, aj-simplexS inC islargest if it has maximum volume (j-measure) among allj-simplices contained inC, andS isstable (resp.rigid) if vol(S)?vol(S?) (resp. vol(S)>vol(S?)) for eachj-simplexS? that is obtained fromS by moving a single vertex ofS to a new position inC. This paper contains a variety of qualitative results that are related to the problems of finding a largest, a stable, or a rigidj-simplex in a givenn-dimensional convex body or convex polytope. In particular, the computational complexity of these problems is studied both for[Figure not available: see fulltext.]-polytopes (presented as the convex hull of a finite set of points) and for?-polytopes (presented as an intersection of finitely many half-spaces).
TL;DR: It is proved here that, asn→∞, almost all convex (1/n)ℤ2-lattice polygons lying in the square [−1, 1]2 are very close to a fixed convex set.
Abstract: It is proved here that, asn??, almost all convex (1/n)?2-lattice polygons lying in the square [?1, 1]2 are very close to a fixed convex set.
TL;DR: A criterion to decide if two tilings are in the same connected component, a simple formula for distances, and a method to construct geodesics in this graph, which is a CW-complex whose connected components are homotopically equivalent to points or circles.
Abstract: We consider the set of all tilings by dominoes (2×1 rectangles) of a surface, possibly with boundary, consisting of unit squares. Convert this set into a graph by joining two tilings by an edge if they differ by aflip, i.e., a 90° rotation of a pair of side-by-side dominoes. We give a criterion to decide if two tilings are in the same connected component, a simple formula for distances, and a method to construct geodesics in this graph. For simply connected surfaces, the graph is connected. By naturally adjoining to this graph higher-dimensional cells, we obtain a CW-complex whose connected components are homotopically equivalent to points or circles. As a consequence, for any region different from a torus or Klein bottle, all geodesics with common endpoints are equivalent in the following sense. Build a graph whose vertices are these geodesics, adjacent if they differ only by the order of two flips on disjoint squares: this graph is connected.
TL;DR: The problem of bounding the combinatorial complexity of a single cell in an arrangement ofn low-degree algebraic surface patches in 3-space is considered, and it is shown that this complexity isO(n2+ε), for any ε>0, where the constant of proportionality depends on ε and on the maximum degree of the given surfaces and of their boundaries.
Abstract: We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement ofn low-degree algebraic surface patches in 3-space. We show that this complexity isO(n2+?), for any ?>0, where the constant of proportionality depends on ? and on the maximum degree of the given surfaces and of their boundaries. This extends several previous results, almost settles a 9-year-old open problem, and has applications to motion planning of general robot systems with three degrees of freedom. As a corollary of the above result, we show that the overall complexity of all the three-dimensional cells of an arrangement ofn low-degree algebraic surface patches, intersected by an additional low-degree algebraic surface patch ? (the so-calledzone of ? in the arrangement) isO(n2+?), for any ?>0, where the constant of proportionality depends on ? and on the maximum degree of the given surfaces and of their boundaries.
TL;DR: An effective algorithm for a smooth (weak) stratification of a real semi-Pfaffian set is suggested, provided an oracle deciding consistency of a system of Pfaffian equations and inequalities is given.
Abstract: An effective algorithm for a smooth (weak) stratification of a real semi-Pfaffian set is suggested, provided an oracle deciding consistency of a system of Pfaffian equations and inequalities is given. An explicit estimate of the complexity of the algorithm and of the resulting stratification is given, in terms of the parameters of the Pfaffian functions defining the original semi-Pfaffian set. The algorithm is applied to sets defined by sparse polynomials and exponential polynomials.
TL;DR: A finite set of points in the plane is calledconvex if its points are vertices of a convex polygon.
Abstract: We show thatn random points chosen independently and uniformly from a parallelogram are in convex position with probability $$\left( {\frac{{\left( {\begin{array}{*{20}c} {2n - 2} \\ {n - 1} \\ \end{array} } \right)}}{{n!}}} \right)^2 $$ . A finite set of points in the plane is calledconvex if its points are vertices of a convex polygon. In this paper we show the following results:
TL;DR: The overall goal of both papers is to offer a characterization of visibility graphs, of convex fans, by describing a polynomial-time algorithm that recovers a representative maximal chain in the weak Bruhat order from a given persistent graph, thus characterizing the class of persistent graphs.
Abstract: The recognition problem for visibility graphs of simple polygons is not known to be in NP, nor is it known to be NP-hard. It is, however, known to be inPSPACE. Further, every such visibility graph can be dismantled as a sequence of visibility graphs of convex fans.
Any nondegenerated configuration ofn points can be associated with amaximal chain in the weak Bruhat order of the symmetric groupSn. The visibility graph ofany simple polygon defined on this configuration is completely determined by this maximal chain via a one-to-one correspondence between maximal chains andbalanced tableaux of a certain shape.
In the case of staircase polygons (special convex fans), we define a class of graphs calledpersistent graphs and show that the visibility graph of a staircase polygon is persistent. We then describe a polynomial-time algorithm that recovers a representative maximal chain in the weak Bruhat order from a given persistent graph, thus characterizing the class of persistent graphs.
The question of recovering a staircase polygon from a given persistent graph, via a maximal chain, is studied in the companion paper [4]. The overall goal of both papers is to offer a characterization of visibility graphs, of convex fans.
TL;DR: It is shown that in the worst case, Ω(nd) sidedness queries are required to determine whether a set ofn points in ℝd is affinely degenerate, i.e., whether it containsd+1 points on a common hyperplane.
Abstract: We show that in the worst case, Ω(nd) sidedness queries are required to determine whether a set ofn points in ?d is affinely degenerate, i.e., whether it containsd+1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing Ω(nd) "collapsible" simplices, any one of which can be made degenerate without changing the orientation of any other simplex. As an immediate corollary, we have an Ω(nd) lower bound on the number of sidedness queries required to determine the order type of a set ofn points in ?d. Using similar techniques, we also show that Ω(nd+1) in-sphere queries are required to decide the existence of spherical degeneracies in a set ofn points in ?d.