Showing papers in "Discrete and Computational Geometry in 1987"

Abstract: We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ . These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.

Abstract: LetX be a given set ofn circular and straight line segments in the plane where two segments may interest only at their endpoints. We introduce a new technique that computes the Voronoi diagram ofX inO(n logn) time. This result improves on several previous algorithms for special cases of the problem. The new algorithm is relatively simple, an important factor for the numerous practical applications of the Voronoi diagram.

Abstract: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requiresO(sd+?) expected preprocessing time to build a search structure for an arrangement ofs hyperplanes ind dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query pointp, the cell of the arrangement containingp can be found inO(logs) worst-case time. (The bound holds for any fixed ?>0, with the constant factors dependent ond and ?.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expectedO(n[d/2]) time, wheren is the total number of vertices of the two polytopes. This matches previous results [10] for the cased = 3 and extends them. Another algorithm samples points in the plane to determine their orderk Voronoi diagram, and requires expectedO(s1+?k) time fors points. (It is assumed that no four of the points are cocircular.) This sharpens the boundO(sk2 logs) for Lee's algorithm [21], andO(s2 logs+k(s?k) log2s) for Chazelle and Edelsbrunner's algorithm [4]. Finally, random sampling is used to show that any set ofs points inE3 hasO(sk2 log8s/(log logs)6) distinctj-sets withj≤k. (ForS ?Ed, a setS? ?S with |S?| =j is aj-set ofS if there is a half-spaceh+ withS? =S ?h+.) This sharpens with respect tok the previous boundO(sk5) [5]. The proof of the bound given here is an instance of a "probabilistic method" [15].

Abstract: In this paper we show how certain geometric convolution operations can be computed efficiently. Here "efficiently" means that our algorithms have running time proportional to the input size plus the output size. Our convolution algorithms rely on new optimal solutions for certain reciprocal search problems, such as finding intersections between "blue" and "green" intervals, and overlaying convex planar subdivisions.

Abstract: For every polynomial time algorithm which gives an upper bound $$\overline {vol}$$ (K) and a lower boundvol(K) for the volume of a convex setK?Rd, the ratio $$\overline {vol}$$ (K)/vol(K) is at least (cd/logd)d for some convex setK?Rd.

Abstract: A criterion is given that decides, for a convex tilingC ofRd, whetherC is the projection of the faces in the boundary of some convex polyhedronP inRd+1. Its applicability is restricted neither by the properties nor by the dimension ofC. It turns out to be conceptually simpler than other criteria and allows the easy examination of various classes of cell complexes. In addition, the criterion is constructive, that is, it can be used to constructP provided it exists.

Abstract: We show that the number of critical positions of a convex polygonal objectB moving amidst polygonal barriers in two-dimensional space, at which it makes three simultaneous contacts with the obstacles but does not penetrate into any obstacle isO(kn?s(kn)) for somes≤6, wherek is the number of boundary segments ofB,n is the number of wall segments, and?s(q) is an almost linear function ofq yielding the maximal number of "breakpoints" along the lower envelope (i.e., pointwise minimum) of a set ofq continuous functions each pair of which intersect in at mosts points (here a breakpoint is a point at which two of the functions simultaneously attain the minimum). We also present an example where the number of such critical contacts is ?(k2n2), showing that in the worst case our upper bound is almost optimal.

Abstract: Fifty years ago Jarnik and Kossler showed that a Steiner minimal tree for the vertices of a regularn-gon contains Steiner points for 3 ≤n≤5 and contains no Steiner point forn=6 andn?13. We complete the story by showing that the case for 7≤n≤12 is the same asn?13. We also show that the set ofn equally spaced points yields the longest Steiner minimal tree among all sets ofn cocircular points on a given circle.

Abstract: A setP ofn points inRd is called simplicial if it has dimensiond and contains exactlyd + 1 extreme points. We show that whenP containsn? interior points, there is always one point, called a splitter, that partitionsP intod + 1 simplices, none of which contain more thandn?/(d + 1) points. A splitter can be found inO(d4 +nd2) time. Using this result, we give anO(nd4 log1+1/dn) algorithm for triangulating simplicial point sets that are in general position. InR3 we give anO(n logn +k) algorithm for triangulating arbitrary point sets, wherek is the number of simplices produced. We exhibit sets of 2n + 1 points inR3 for which the number of simplices produced may vary between (n ? 1)2 + 1 and 2n ? 2. We also exhibit point sets for which every triangulation contains a quadratic number of simplices.

Abstract: This paper investigates the existence of linear space data structures for range searching. We examine thehomothetic range search problem, where a setS ofn points in the plane is to be preprocessed so that for any triangleT with sides parallel to three fixed directions the points ofS that lie inT can be computed efficiently. We also look atdomination searching in three dimensions. In this problem,S is a set ofn points inE3 and the question is to retrieve all points ofS that are dominated by some query point. We describe linear space data structures for both problems. The query time is optimal in the first case and nearly optimal in the second.

Abstract: LetS = {A, B, C, D} consist of the four corner points of a convex quadrilateral where diagonals [A, C] and [B, D] intersect at the pointO. There are two possible full Steiner trees forS, theAB-CD tree hasA andB adjacent to one Steiner point, andC andD to another; theAD-BC tree hasA andD adjacent to one Steiner point, andB andC to another. Pollak proved that if both full Steiner trees exist, then theAB-CD (AD-BC) tree is the Steiner minimal tree if[Figure not available: see fulltext.]AOD>3 (<) 90°, and both are Steiner minimal trees if[Figure not available: see fulltext.]AOD=90°. While the theorem has been crucially used in obtaining results on Steiner minimal trees in general, its applicability is sometimes restricted because of the condition that both full Steiner trees must exist. In this paper we remove this obstacle by showing: (i) Necessary and sufficient conditions for the existence of either full Steiner tree forS. (ii) If[Figure not available: see fulltext.]AOD?90°, then theAB-CD tree is the SMT even if theAD-BC tree does not exist. (iii) If[Figure not available: see fulltext.]AOD<90° but theAD-BC tree does not exist, then theAB-CD tree cannot be ruled out as a Steiner minimal tree, though under certain broad conditions it can.

Abstract: A common structure arising in computational geometry is the subdivision of a plane defined by the faces of a straight-line planar graph. We consider a natural generalization of this structure on a polyhedral surface. The regions of the subdivision are bounded by geodesics on the surface of the polyhedron. A method is given for representing such a subdivision that is efficient both with respect to space and the time required to answer a number of different queries involving the subdivision. For example, given a pointx on the surface of the polyhedron, the region of the subdivision containingx can be determined in logarithmic time. Ifn denotes the number of edges in the polyhedron,m denotes the number of geodesics in the subdivision, andK denotes the number of intersections between edges and geodesics, then the space required by the data structure isO((n +m) log(n +m)), and the structure can be built inO(K + (n +m) log(n +m)) time. Combined with existing algorithms for computing Voronoi diagrams on the surface of polyhedra, this structure provides an efficient solution to the nearest-neighbor query problem on polyhedral surfaces.

Abstract: Consider a drawing in the plane ofKn, the complete graph onn vertices. If all edges are restricted to be straight line segments, the drawing is called rectilinear. Consider a Hamiltonian cycle in a drawing ofKn. If no pair of the edges of the cycle cross, it is called a crossing-free Hamiltonian cycle (cfhc). Let ?(n) represent the maximum number of cfhc's of any drawing ofKn, and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVC0xe9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpi0x% c9q8qqaqFj0df9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuOPdyKbae% baaaa!38CF! $$\bar \Phi$$ (n) the maximum number of cfhc's of any rectilinear drawing ofKn. The problem of determining ?(n) and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVC0xe9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpi0x% c9q8qqaqFj0df9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuOPdyKbae% baaaa!38CF! $$\bar \Phi$$ (n), and determining which drawings have this many cfhc's, is known as the optimal cfhc problem. We present a brief survey of recent work on this problem, and then, employing a recursive counting argument based on computer enumeration, we establish a substantially improved lower bound for ?(n) and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVC0xe9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpi0x% c9q8qqaqFj0df9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuOPdyKbae% baaaa!38CF! $$\bar \Phi$$ (n). In particular, it is shown that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVC0xe9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpi0x% c9q8qqaqFj0df9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuOPdyKbae% baaaa!38CF! $$\bar \Phi$$ (n) is at leastk × 3.2684n. We conjecture that both ?(n) and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVC0xe9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpi0x% c9q8qqaqFj0df9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuOPdyKbae% baaaa!38CF! $$\bar \Phi$$ (n) are at mostc × 4.5n.

Abstract: Bambah and Rogers proved that the area of a convex domain in the plane which can be covered byn translates of a given centrally symmetric convex domainC is at most (n?1)h(C)+a(C), whereh(C) denotes the area of the largest hexagon contained inC anda(C) stands for the area ofC. An improvement with a term of magnitude ?n is given here. Our estimate implies that ifC is not a parallelogram, then any covering of any convex domain by at least 26 translates ofC is less economic than the thinnest covering of the whole plane by translates ofC.

Abstract: If a graphG is embedded in a manifoldM such that all faces are cells bounded by simple closed curves we say that this is a closed 2-cell embedding ofG inM. We show how to generate the 2-cell embeddings in the projective plane from two minimal graphs and the 2-cell embeddings in the torus from six minimal graphs by vertex splitting and face splitting.

Abstract: The set of nonnegative solutions of a system of linear equations or inequalities is a convex polyhedron. If the coefficients of the system are chosen at random, the number of vertices of this polyhedron is a random variable. Its expected value, dependent on the probability distribution of the coefficients, which are assumed to be nonnegative throughout, is investigated, and a distribution-independent upper bound for this expected value is established.

Abstract: We provide simple arguments of a geometric nature to explain why the Mobius functions of certain lattices take only the values ?1, 0, 1.

Abstract: We study here the affine space generated by the extendedf-vectors of simplicial homology (d ? 1)-spheres which are balanced of a given type. This space is determined, and its dimension is computed, by deriving a balanced version of the Dehn-Sommerville equations and exhibiting a set of balanced polytopes whose extendedf-vectors span it. To this end, a unique minimal complex of a given type is defined, along with a balanced version of stellar subdivision, and such a subdivision of a face in a minimal complex is realized as the boundary complex of a balanced polytope. For these complexes, we obtain an explicit computation of their extendedh-vectors, and, implicitly,f-vectors.

Abstract: Let ? be a locally finite system of hyperplanes in ?d with the property that the cells of the induced cell complex decomposition ? of ?d have uniformly bounded diameters. If ? is simple and the density of the vertices in ? exists, then the density of thek-cells in ? exists and can be given explicitly (k = 1, ...,d). Also, the mean number ofj-faces of thek-cells in ? exists and can be calculated. For certain nonsimple systems ?, corresponding inequalities are obtained.

Abstract: Using a general construction method, sets of prototiles (with more than one element) are obtained which admit a countable infinity of distinct tilings. In contrast to the tilings described in a previous paper, in this case almost all the tilings are periodic. In particular, sets of two, three, four, and five prototiles are described.

Abstract: A cluster is the union of a finite number of cubes from the standard partition ofn-dimensional Euclidean space into unit cubes. If there is lattice tiling by translates of a cluster, then must there be a lattice tiling by translates of the cluster in which the translation vectors have only integer coordinates? In this article we prove that if the interior of the cluster is connected and the dimension is at most three, then the answer is affirmative.

Abstract: We show that the position of an input point in the Euclideand-dimensional space with respect to a given set of hyperplanes can be determined efficiently by linear decision trees. As an application, we prove that many concrete problems whose recognition versions are NP-complete, like the traveling salesman problem, many other shortest path problems, and integer programming, have polynomial-time upper bounds in the linear decision tree model of computation.

Abstract: Ifg andh are any nonzero functions on the class of convex polytopes then ?(Fi,Fj) =g(Fi)/h(Fj) is a valuation whose inverse is ?(Fi,Fj) = (?1)j?ih(Fi)/g(Fj). This is proved and a smaller class of valuations are characterized: those ?(Fi,Fj) which depend only oni andj and which have inverses of the same form.