Showing papers in "Discrete and Computational Geometry in 1993"

On sparse spanners of weighted graphs

Abstract: Given a graphG, a subgraphG' is at-spanner ofG if, for everyu,v ?V, the distance fromu tov inG' is at mostt times longer than the distance inG. In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.

An optimal convex hull algorithm in any fixed dimension

Abstract: We present a deterministic algorithm for computing the convex hull ofn points inEd in optimalO(n logn+n?d/2?) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n?d/2?) time.

Cutting hyperplanes for divide-and-conquer

Abstract: Givenn hyperplanes inEd, a (1/r)-cutting is a collection of simplices with disjoint interiors, which together coverEd and such that the interior of each simplex intersects at mostn/r hyperplanes. We present a deterministic algorithm for computing a (1/r)-cutting ofO(rd) size inO(nrd?1) time. If we require the incidences between the hyperplanes and the simplices of the cutting to be provided, then the algorithm is optimal. Our method is based on a hierarchical construction of cuttings, which also provides a simple optimal data structure for locating a point in an arrangement of hyperplanes. We mention several other applications of our result, e.g., counting segment intersections, Hopcroft's line/point incidence problem, linear programming in fixed dimension.

Range searching with efficient hierarchical cuttings

Abstract: We present an improved space/query-time tradeoff for the general simplex range searching problem, matching known lower bounds up to small polylogarithmic factors. In particular, we construct a linear-space simplex range searching data structure withO(n1?1/d) query time, which is optimal ford=2 and probably also ford>2. Further, we show that multilevel range searching data structures can be built with only a polylogarithmic overhead in space and query time per level (the previous solutions require at least a small fixed power ofn). We show that Hopcroft's problem (detecting an incidence amongn lines andn points) can be solved in time % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaCa% aaleqabaGaaGinaiaac+cacaaIZaaaaOGaaGOmamaaCaaaleqabaGa% am4taiaacIcaciGGSbGaai4BaiaacEgadaahaaadbeqaaiaaisdaaa% WccaWGUbGaaiykaaaaaaa!40F5! $$n^{4/3} 2^{O(\log ^4 n)}$$ . In all these algorithms we apply Chazelle's results on computing optimal cuttings.

The upper envelope of voronoi surfaces and its applications

Abstract: Given a setS ofsources (points or segments) in ?211C;d, we consider the surface in ?211C;d+1 that is the graph of the functiond(x)=minp?S?(x, p) for some metric?. This surface is closely related to the Voronoi diagram, Vor(S), ofS under the metric?. The upper envelope of a set of theseVoronoi surfaces, each defined for a different set of sources, can be used to solve the problem of finding the minimum Hausdorff distance between two sets of points or line segments under translation. We derive bounds on the number of vertices on the upper envelope of a collection of Voronoi surfaces, and provide efficient algorithms to calculate these vertices. We then discuss applications of the methods to the problems of finding the minimum Hausdorff distance under translation, between sets of points and segments.

Representing geometric structures ind dimensions: Topology and order

Abstract: This work investigates data structures for representing and manipulatingd-dimensional geometric objects for arbitraryd ? 1. A class of geometric objects is defined, the "subdividedd-manifolds," which is large enough to encompass many applications. A new representation is given for such objects, the "cell-tuple structure," which provides direct access to topological structure, ordering information among cells, the topological dual, and boundaries. The cell-tuple structure gives a simple, uniform representation of subdivided manifolds which unifies the existing work in the field and provides intuitive clarity in all dimensions. The dual subdivision, and boundaries, are represented consistently. This work has direct applications in solid modeling, computer graphics, and computational geometry.

The reverse isoperimetric problem for Gaussian measure

Abstract: It is shown that ifg is the standard Gaussian density on ?n andC is a convex body in ?n $$\int_{\partial C} {g \leqslant 4n^{1/4} }$$ . The arguments presented raise several questions in integral geometry.

On ray shooting in convex polytopes

Abstract: LetP be a convex polytope withn facets in the Euclidean space of a (small) fixed dimensiond. We consider themembership problem forP (given a query point, decide whether it lies inP) and theray shooting problem inP (given a query ray originating insideP, determine the first facet ofP hit by it). It was shown in [AM2] that a data structure for the membership problem satisfying certain mild assumptions can also be used for the ray shooting problem, with a logarithmic overhead in query time. Here we show that some specific data structures for the membership problem can be used for ray shooting in a more direct way, reducing the overhead in the query time and eliminating the use of parametric search. We also describe an improved static solution for the membership problem, approaching the conjectured lower bounds more tightly.

Successive-minima-type inequalities

Abstract: We show analogues of Minkowski's theorem on successive minima, where the volume is replaced by the lattice point enumerator. We further give analogous results to some recent theorems by Kannan and Lovasz on covering minima.

Applications of a new space-partitioning technique

Abstract: We present several applications of a recent space-partitioning technique of Chazelle, Sharir, and Welzl (Proceedings of the 6th Annual ACM Symposium on Computational Geometry, 1990, pp. 23---33). Our results include efficient algorithms for output-sensitive hidden surface removal, for ray shooting in two and three dimensions, and for constructing spanning trees with low stabbing number.

Feasibility testing for systems of real quadratic equations

Abstract: We consider the problem of deciding whether a given system of quadratic homogeneous equations over the reals has nontrivial solution. We design an algorithm which, for a fixed number of equations, uses a number of arithmetic operations bounded by a polynomial in the number of variables only.

On convex body chasing

Abstract: A player moving in the plane is given a sequence of instructions of the following type: at stepi a planar convex setFi is specified, and the player has to move to a point inFi The player is charged for the distance traveled We provide a strategy for the player which is competitive, ie, for any sequenceFi the cost to the player is within a constant (multiplicative) factor of the "off-line" cost (ie, the least possible cost when allFi are known in advance) We conjecture that similar strategies can be developed for this game in any Euclidean space and perhaps even in all metric spaces The analogous statement where convex sets are replaced by more general families of sets in a metric space includes many on-line/off-line problems such as thek-server problem; we make some remarks on these more general problems

An upper bound for conforming Delaunay triangulations

Abstract: A plane geometric graphC in ?2conforms to another such graphG if each edge ofG is the union of some edges ofC. It is proved that, for everyG withn vertices andm edges, there is a completion of a Delaunay triangulation ofO(m2n) points that conforms toG. The algorithm that constructs the points is also described.

Diameter, width, closest line pair, and parametric searching

Abstract: We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improved solutions for them. We obtain, for any fixed ?>0, anO(n1+?) algorithm for computing the diameter of a point set in 3-space, anO(8/5+?) algorithm for computing the width of such a set, and onO(n8/5+?) algorithm for computing the closest pair in a set ofn lines in space. All these algorithms are deterministic.

Computing the volume, counting integral points, and exponential sums

Abstract: We design polynomial-time algorithms for some particular cases of the volume computation problem and the integral points counting problem for convex polytopes. The basic idea is a reduction to the computation of certain exponential sums and integrals. We give elementary proofs of some known identities between these sums and integrals and prove some new identities.

Edge insertion for optimal triangulations

Abstract: Edge insertion iteratively improves a triangulation of a finite point set in ?2 by adding a new edge, deleting old edges crossing the new edge, and retriangulating the polygonal regions on either side of the new edge. This paper presents an abstract view of the edge insertion paradigm, and then shows that it gives polynomial-time algorithms for several types of optimal triangulations, including minimizing the maximum slope of a piecewise-linear interpolating surface.

Extension spaces of oriented matroids

Abstract: We study the space of all extensions of a real hyperplane arrangement by a new pseudohyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the "Generalized Baues Problem" of Billera, Kapranov, and Sturmfels, via the Bohne-Dress theorem on zonotopal tilings. We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not known whether the extension space is connected for all realizable oriented matroids (hyperplane arrangements). We show that the subspace of realizable extensions is always connected but not necessarily spherical. Nonrealizable oriented matroids of rank 4 with disconnected extension spaces were recently constructed by Mnev and Richter-Gebert.

On the zone of a surface in a hyperplane arrangement

Abstract: LetH be a collection ofn hyperplanes in ?d, letA denote the arrangement ofH, and let ? be a (d?1)-dimensional algebraic surface of low degree, or the boundary of a convex set in ?d. Thezone of ? inA is the collection of cells ofA crossed by ?. We show that the total number of faces bounding the cells of the zone of ? isO(nd?1 logn). More generally, if ? has dimensionp, 0≤p

There exist 6n/13 ordinary points

Abstract: In 1958 L. M. Kelly and W. O. J. Moser showed that apart from a pencil, any configuration ofn lines in the real projective plane has at least 3n/7 ordinary or simple points of intersection, with equality in the Kelly-Moser example (a complete quadrilateral with its three diagonal lines). In 1981 S. Hansen claimed to have improved this ton/2 (apart from pencils, the Kelly-Moser example and the McKee example). In this paper we show that one of the main theorems used by Hansen is false, thus leavingn/2 open, and we improve the 3n/7 estimate to 6n/13 (apart from pencils and the Kelly-Moser example), with equality in the McKee example. Our result applies also to arrangements of pseudolines.

On-line steiner trees in the Euclidean plane

Abstract: Suppose we are given a sequence ofn points in the Euclidean plane, and our objective is to construct, on-line, a connected graph that connects all of them, trying to minimize the total sum of lengths of its edges. The points appear one at a time, and at each step the on-line algorithm must construct a connected graph that contains all current points by connecting the new point to the previously constructed graph. This can be done by joining the new point (not necessarily by a straight line) to any point of the previous graph (not necessarily one of the given points). The performance of our algorithm is measured by its competitive ratio: the supremum, over all sequences of points, of the ratio between the total length of the graph constructed by our algorithm and the total length of the best Steiner tree that connects all the points. There are known on-line algorithms whose competitive ratio isO(logn) even for all metric spaces, but the only lower bound known is of [IW] for some contrived discrete metric space. Moreover, for the plane, on-line algorithms could have been more powerful and achieve a better competitive ratio, and no nontrivial lower bounds for the best possible competitive ratio were known. Here we prove an almost tight lower bound of Ω(logn/log logn) for the competitive ratio of any on-line algorithm. The lower bound holds for deterministic algorithms as well as for randomized ones, and obviously holds in any Euclidean space of dimension greater than 2 as well.

The furthest-site geodesic voronoi diagram

Abstract: We present anO((n+k) log(n+k))-time,O(n+k)-space algorithm for computing the furthest-site Voronoi diagram ofk point sites with respect to the geodesic metric within a simplen-sided polygon.

Note on a conjecture of sierksma

Abstract: LetS(q, d) be the maximal numberv such that, for every general position linear maph: Δ(q?1)(d+1) ?Rd, there exist at leastv different collections {Δt1, ..., Δtq} of disjoint faces of Δ(q?1)(d+1) with the property thatf(Δt1) ? ... ?f(Δtq) ? O. Sierksma's conjecture is thatS(q, d)=((q?1)!)d. The following lower bound (Theorem 1) is proved assuming thatq is a prime number: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaacI% cacaWGXbGaaiilaiaadsgacaGGPaGaeyyzIm7aaSaaaeaacaaIXaaa% baGaaiikaiaadghacqGHsislcaaIXaGaaiykaiaacgcaaaWaaeWaae% aadaWcaaqaaiaadghaaeaacaaIYaaaaaGaayjkaiaawMcaamaaCaaa% leqabaWaaSGbaeaacaGGOaGaaiikaiaadghacqGHsislcaaIXaGaai% ykaiaacIcacaWGKbGaey4kaSIaaGymaiaacMcacaGGPaaabaGaaGOm% aaaaaaGccaGGUaaaaa!4F90! $$S(q,d) \geqslant \frac{1}{{(q - 1)!}}\left( {\frac{q}{2}} \right)^{{{((q - 1)(d + 1))} \mathord{\left/ {\vphantom {{((q - 1)(d + 1))} 2}} \right. \kern- ulldelimiterspace} 2}} .$$ Using the same technique we obtain (Theorem 2) a lower bound for the number of different splittings of a "generic" necklace.

How hard is half-space range searching?

Abstract: We investigate the complexity ofhalf-space range searching: givenn points ind-space, build a data structure that allows us to determine efficiently how many points lie in a query half-space. We establish a tradeoff between the storagem and the worst-case query timet in the Fredman/Yao arithmetic model of computation. We show thatt must be at least on the order of $$\frac{{(n/\log n)^{1 - (d - 1)/d(d + 1)} }}{{m^{1/d} }}$$ Although the bound is unlikely to be optimal, it falls reasonably close to the recent upper bound ofO(n/m1/d) established by Matousek. We also show that it is possible to devise a sequence ofn inserts and half-space range queries that require a total time ofn2-O(1/d). Our results imply the first nontrivial lower bounds for spherical range searching in any fixed dimension. For example, they show that, with linear storage, circular range queries in the plane require Ω(n1/3) time (modulo a logarithmic factor).

Minimum dissection of a rectilinear polygon with arbitrary holes into rectangles

Abstract: In this paper, the problem of dissecting a plane rectilinear polygon with arbitrary (possibly, degenerate) holes into a minimum number of rectangles is shown to be solvable inO(n3/2 logn) time. This fact disproves a famous assertion about the NP-hardness of the minimum rectangular dissection problem for rectilinear polygons with point holes.

Oriented matroids with few mutations

Abstract: This paper defines a "connected sum" operation on oriented matroids of the same rank. This construction is used for three different applications in rank 4. First it provides nonrealizable pseudoplane arrangements with a low number of simplicial regions. This contrasts the case of realizable hyperplane arrangements: by a classical theorem of Shannon every arrangement ofn projective planes in ?Pd-1 contains at leastn simplicial regions and every plane is adjacent to at leastd simplicial regions [17], [18]. We construct a class of uniform pseudoarrangements of 4n pseudoplanes in ?P3 with only 3n+1 simplicial regions. Furthermore, we construct an arrangement of 20 pseudoplanes where one plane is not adjacent to any simplicial region. Finally we disprove the "strong-map conjecture" of Las Vergnas [1]. We describe an arrangement of 12 pseudoplanes containing two points that cannot be simultaneously contained in an extending hyperplane.

A new bound on the local density of sphere packings

Abstract: It is shown that a packing of unit spheres in three-dimensional Euclidean space can have density at most 0.773055..., and that a Voronoi polyhedron defined by such a packing must have volume at least 5.41848... These bounds are superior to the best bounds previously published [5] (0.77836 and 5.382, respectively), but are inferior to the tight bounds of 0.7404... and 5.550... claimed by Hsiang [2]. Our bounds are proved by cutting a Voronoi polyhedron into cones, one for each of its faces. A lower bound is established on the volume of each cone as a function of its solid angle. Convexity arguments then show that the sum of all the cone volume bounds is minimized when there are 13 faces each of solid angle 4?/13.

Combinatorial models for the finite-dimensional Grassmannians

Abstract: Let ?n be a linear hyperplane arrangement in ?n. We define two corresponding posetsGk(?n andVk(?n) of oriented matroids, which approximate the GrassmannianGk(?n) and the Stiefel manifoldVk(?n). The basic conjectures are that the "OM-Grassmannian"Gk(?n) has the homotopy type ofGk(?n), and that the "OM-Stiefel bundle" Δ?: ΔVk(?n) ? ΔGk(?n) is a surjective map. These conjectures can be proved in some cases: we survey the known results and add some new ones. The conjectures fail if they are generalized to nonrealizable oriented matroids ?n.

A note on small linear-ordering polytopes

Abstract: In this paper we discuss the polyhedral structure of polytopes associated with the linear-ordering problem. We give explicit lists of facets of small linear-ordering polytopes for complete digraphs on up to seven nodes. For the latter we give a description that we believe to be complete.

Minimum steiner trees in normed planes

Abstract: A minimum Steiner tree for a given setX of points is a network interconnecting the points ofX having minimum possible total length. In this note we investigate various properties of minimum Steiner trees in normed planes, i.e., where the "unit disk" is an arbitrary compact convex centrally symmetric domainD having nonempty interior. We show that if the boundary ofD is strictly convex and differentiable, then each edge of a full minimum Steiner tree is in one of three fixed directions. We also investigate the Steiner ratio?(D) forD, and show that, for anyD, 0.623

A bound on local minima of arrangements that implies the upper bound theorem

Abstract: This paper shows that thei-level of an arrangement of hyperplanes inEd has at most $$\left( {\begin{array}{*{20}c} {i + d - 1} \\ {d - 1} \\ \end{array} } \right)$$ local minima. This bound follows from ideas previously used to prove bounds on (≤k)-sets. Using linear programming duality, the Upper Bound Theorem is obtained as a corollary, giving yet another proof of this celebrated bound on the number of vertices of a simple polytope inEd withn facets.