Showing papers in "Discrete and Computational Geometry in 1992"

Classes of graphs which approximate the complete Euclidean graph

Abstract: LetS be a set ofN points in the Euclidean plane, and letd(p, q) be the Euclidean distance between pointsp andq inS. LetG(S) be a Euclidean graph based onS and letG(p, q) be the length of the shortest path inG(S) betweenp andq. We say a Euclidean graphG(S)t-approximates the complete Euclidean graph if, for everyp, q ?S, G(p, q)/d(p, q) ≤t. In this paper we present two classes of graphs which closely approximate the complete Euclidean graph. We first consider the graph of the Delaunay triangulation ofS, DT(S). We show that DT(S) (2?/(3 cos(?/6)) ? 2.42)-approximates the complete Euclidean graph. Secondly, we define?(S), the fixed-angle?-graph (a type of geometric neighbor graph) and show that?(S) ((1/cos?)(1/(1?tan?)))-approximates the complete Euclidean graph.

On the difficulty of triangulating three-dimensional nonconvex polyhedra.

Abstract: A number of different polyhedraldecomposition problems have previously been studied, most notably the problem of triangulating a simple polygon. We are concerned with thepolyhedron triangulation problem: decomposing a three-dimensional polyhedron into a set of nonoverlapping tetrahedra whose vertices must be vertices of the polyhedron. It has previously been shown that some polyhedra cannot be triangulated in this fashion. We show that the problem of deciding whether a given polyhedron can be triangulated 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 triangulate a polyhedron also turn out to be NP-hard.

Quantitative Steinitz's theorems with applications to multifingered grasping

Abstract: We prove the following quantitative form of a classical theorem of Steintiz: Letm be sufficiently large. If the convex hull of a subsetS of Euclideand-space contains a unit ball centered on the origin, then there is a subset ofS with at mostm points whose convex hull contains a solid ball also centered on the origin and havingresidual radius $$1 - 3d\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ The casem=2d was first considered by Baranyet al. [1]. We also show an upper bound on the achievable radius: the residual radius must be less than $$1 - \frac{1}{{17}}\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ These results have applications in the problem of computing the so-calledclosure grasps by anm-fingered robot hand. The above quantitative form of Steinitz's theorem gives a notion ofefficiency for closure grasps. The theorem also gives rise to some new problems in computational geometry. We present some efficient algorithms for these problems, especially in the two-dimensional case.

Area requirement and symmetry display of planar upward drawings

Abstract: In this paper we investigate the problem of constructing planar straight-line drawings of acyclic digraphs such that all the edges flow in the same direction, e.g., from bottom to top. Our contribution is twofold. First we show the existence of a family of planar acyclic digraphs that require exponential area for any such drawing. Second, motivated by the preceding lower bound, we relax the straight-line constraint and allow bends along the edges. We present a linear-time algorithm that produces drawings of planarst-graphs with a small number of bends, asymptotically optimal area, and such that symmetries and isomorphisms of the digraph are displayed. If the digraph has no transitive edges, then the drawing obtained has no bends. Also, a variation of the algorithm produces drawings with exact minimum area.

Random projections of regular simplices

Abstract: Precise asymptotic formulae are obtained for the expected number ofk-faces of the orthogonal projection of a regularn-simplex inn-space onto a randomly chosen isotropic subspace of fixed dimension or codimension, as the dimensionn tends to infinity.

Almost tight bounds for e-nets

Abstract: Given any natural numberd, 0

Inner and outer j -radii of convex bodies in finite-dimensional normed spaces

Abstract: This paper is concerned with the various inner and outer radii of a convex bodyC in ad-dimensional normed space. The innerj-radiusrj(C) is the radius of a largestj-ball contained inC, and the outerj-radiusRj(C) measures how wellC can be approximated, in a minimax sense, by a (d --j)-flat. In particular,rd(C) andRd(C) are the usual inradius and circumradius ofC, while 2r1(C) and 2R1(C) areC's diameter and width. Motivation for the computation of polytope radii has arisen from problems in computer science and mathematical programming. The radii of polytopes are studied in [GK1] and [GK2] from the viewpoint of the theory of computational complexity. This present paper establishes the basic geometric and algebraic properties of radii that are needed in that study.

Applications of random sampling to on-line algorithms in computational geometry

Abstract: This paper presents a general framework for the design and randomized analysis of geometric algorithms. These algorithms are on-line and the framework provides general bounds for their expected space and time complexities when averaging over all permutations of the input data. The method is general and can be applied to various geometric problems. The power of the technique is illustrated by new efficient on-line algorithms for constructing convex hulls and Voronoi diagrams in any dimension, Voronoi diagrams of line segments in the plane, arrangements of curves in the plane, and others.

Finding minimum areak-gons

Abstract: Given a setP ofn points in the plane and a numberk, we want to find a polygon[Figure not available: see fulltext.] with vertices inP of minimum area that satisfies one of the following properties: (1)[Figure not available: see fulltext.] is a convexk-gon, (2)[Figure not available: see fulltext.] is an empty convexk-gon, or (3)[Figure not available: see fulltext.] is the convex hull of exactlyk points ofP. We give algorithms for solving each of these three problems in timeO(kn3). The space complexity isO(n) fork=4 andO(kn2) fork?5. The algorithms are based on a dynamic programming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.

Almost tight bounds for $\epsilon$-nets

Abstract: Given any natural numberd, 0

An upper bound on the number of planar K -sets

Abstract: Given a setS ofn points, a subsetX of sizek is called ak-set if there is a hyperplane ? that separatesX fromS?X. We prove thatO(n?k/log*k) is an upper bound for the number ofk-sets in the plane, thus improving the previous bound of Erdos, Lovasz, Simmons, and Strauss by a factor of log*k.

The number of different distances determined by a set of points in the Euclidean plane

Abstract: In 1946 P. Erdos posed the problem of determining the minimum numberd(n) of different distances determined by a set ofn points in the Euclidean plane. Erdos provedd(n) ?cn1/2 and conjectured thatd(n)?cn/ ?logn. If true, this inequality is best possible as is shown by the lattice points in the plane. We showd(n)?n4/5/(logn)c.

Convex independent sets and 7-holes in restricted planar point sets

Abstract: For a finite setA of points in the plane, letq(A) denote the ratio of the maximum distance of any pair of points ofA to the minimum distance of any pair of points ofA. Fork>0 letc?(k) denote the largest integerc such that any setA ofk points in general position in the plane, satisfying % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeei0xd9Wq-Jb9% vqaqFfpe0db9as0-LqLs-Jirpepeei0-bs0Fb9pgea0db9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadghacaGGOa% GaamyqaiaacMcacqGH8aapiiaacqWFXoqydaGcaaqaaGqaciaa+Tga% aSqabaaaaa!3EAF! $$q(A)< \alpha \sqrt k $$ for fixed % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeei0xd9Wq-Jb9% vqaqFfpe0db9as0-LqLs-Jirpepeei0-bs0Fb9pgea0db9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGGaaiab-f7aHj% abgwMiZoaakaaabaGae8NmaiZaaOaaaeaacqWFZaWmaSqabaacbiGc% caGFVaGae8hWdahaleqaaOGaeSiuIiKaaGymaiaac6cacaaIWaGaaG% ynaaaa!4406! $$\alpha \geqslant \sqrt {2\sqrt 3 /\pi } \doteq 1.05$$ , contains at leastc convex independent points. We determine the exact asymptotic behavior ofc?(k), proving that there are two positive constantsβ=β(?),? such thatβk1/3≤c?(k)≤?k1/3. To establish the upper bound ofc?(k) we construct a set, which also solves (affirmatively) the problem of Alonet al. [1] about the existence of a setA ofk points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points fromA), satisfying % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeei0xd9Wq-Jb9% vqaqFfpe0db9as0-LqLs-Jirpepeei0-bs0Fb9pgea0db9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadghacaGGOa% GaamyqaiaacMcacqGH8aapiiaacqWFXoqydaGcaaqaaGqaciaa+Tga% aSqabaaaaa!3EAF! $$q(A)< \alpha \sqrt k $$ . The construction uses "Horton sets," which generalize sets without 7-holes constructed by Horton and which have some interesting properties.

Upper bounds for the diameter and height of graphs of convex polyhedra

Abstract: Let Δ(d, n) be the maximum diameter of the graph of ad-dimensional polyhedronP withn-facets. It was conjectured by Hirsch in 1957 that Δ(d, n) depends linearly onn andd. However, all known upper bounds for Δ(d, n) were exponential ind. We prove a quasi-polynomial bound Δ(d, n)≤n2 logd+3. LetP be ad-dimensional polyhedron withn facets, let ? be a linear objective function which is bounded onP and letv be a vertex ofP. We prove that in the graph ofP there exists a monotone path leading fromv to a vertex with maximal ?-value whose length is at most % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaW% baaSqabeaacaaIYaWaaOaaaeaacaWGUbaameqaaaaaaaa!3C85! $$n^{2\sqrt n } $$ .

AnO(n logn) algorithm for computing the link center of a simple polygon

Abstract: We present an algorithm that determines the link center of a simplen-vertex polygonP inO(n logn) time. The link center of a simple polygon is the set of pointsx insideP at which the maximal link-distance fromx to any other point inP is minimized. The link distance between two pointsx andy insideP is defined to be the smallest number of straight edges in a polygonal path insideP connectingx andy. Using our algorithm we also obtain anO(n logn)-time solution to the problem of determining the link radius ofP. The link radius ofP is the maximum link distance from a point in the link center to any vertex ofP. Both results are improvements over theO(n2) bounds previously established for these problems.

The different ways of stabbing disjoint convex sets

Abstract: We construct a family ofn disjoint convex set in ?d having (n/(d?1))d?1 geometric permutations. As well, we complete the enumeration problem for geometric permutations of families of disjoint translates of a convex set in the plane, settle the case for cubes in ?d, and construct a family ofd+1 translates in ?d admitting (d+1)!/2 geometric permutations.

Counting facets and incidences

Abstract: We show thatm distinct cells in an arrangement ofn planes in ?3 are bounded byO(m2/3n+n2) faces, which in turn yields a tight bound on the maximum number of facets boundingm cells in an arrangement ofn hyperplanes in ?d, for everyd?3. In addition, the method is extended to obtain tight bounds on the maximum number of faces on the boundary of all nonconvex cells in an arrangement of triangles in ?3. We also present a simpler proof of theO(m2/3nd/3+nd?1) bound on the number of incidences betweenn hyperplanes in ?d andm vertices of their arrangement.

Finding stabbing lines in 3-space

Abstract: A line intersecting all polyhedra in a set? is called a "stabber" for the set?. This paper addresses some combinatorial and algorithmic questions about the set?(?) of all lines stabbing?. We prove that the combinatorial complexity of?(?) has an % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGpbGaai% ikaiaad6gadaahaaWcbeqaaiaaiodaaaGccaaIYaWaaWbaaSqabeaa% caWGJbWaaOaaaeaaciGGSbGaai4BaiaacEgacaWGUbaameqaaaaaki% aacMcaaaa!4368! $$O(n^3 2^{c\sqrt {\log n} } )$$ upper bound, wheren is the total number of facets in?, andc is a suitable constant. This bound is almost tight. Within the same time bound it is possible to determine if a stabbing line exists and to find one.

The Euler characteristic is the unique locally determined numerical homotopy invariant of finite complexes

Abstract: If a numerical homotopy invariant of finite simplicial complexes has a local formula, then, up to multiplication by an obvious constant, the invariant is the Euler characteristic. Moreover, the Euler characteristic itself has a unique local formula.

Maintaining the minimal distance of a point set in polylogarithmic time

Abstract: A dynamic data structure is given that maintains the minimal distance in a set ofn points ink-dimensional space inO((logn)k log logn) amortized time per update. The size of the data structure is bounded byO(n(logn)k). Distances are measured in the MinkowskiLt-metric, where 1 ≤t ≤ ?. This is the first dynamic data structure that maintains the minimal distance in polylogarithmic time for fully on-line updates.

Self-packing of centrally symmetric convex bodies in R 2

Abstract: LetB be a compact convex body symmetric around0 in Â?2 which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radiusÂ?(m,B) is the smallestt such thattB can be packed withm translates of the interior ofB. Form≤6 we show that the self-packing radiusÂ?(m,B)=1+2/Â?(m,B) whereÂ?(m,B) is the Minkowski length of the side of the largest equilateralm-gon inscribed inB (measured in the Minkowski metric determined byB). We showÂ?(6,B)=Â?(7,B)=3 for allB, and determine most of the largest and smallest values ofÂ?(m,B) form≤7. For allm we have % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaam% aalaaabaGaamyBaaqaaiabes7aKjaacIcaieqacaWFcbGaaiykaaaa% aiaawIcacaGLPaaadaahaaWcbeqaamaalyaabaGaaGymaaqaaiaaik% daaaaaaOGaeyOeI0YaaSaaaeaacaaIZaaabaGaaGOmaaaacqGHKjYO% cqaHbpGCcaGGOaGaamyBaiaacYcacaWFcbGaaiykaiabgsMiJoaabm% aabaWaaSaaaeaacaWGTbaabaGaeqiTdqMaaiikaiaa-jeacaGGPaaa% aaGaayjkaiaawMcaamaaCaaaleqabaWaaSGbaeaacaaIXaaabaGaaG% OmaaaaaaGccqGHRaWkcaaIXaGaaiilaaaa!576F! $$\left( {\frac{m}{{\delta (B)}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern- ulldelimiterspace} 2}} - \frac{3}{2} \leqslant \rho (m,B) \leqslant \left( {\frac{m}{{\delta (B)}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern- ulldelimiterspace} 2}} + 1,$$ whereÂ?(B) is the packing density ofB in Â?2.

Alternative proof of sine's theorem on the size of a regular polygon in Rn with the ℓź-metric

Abstract: It is shown that a regular polygon inR n with the (2n) n -metric has at most (2n) n vertices.

Circle packings and polyhedral surfaces

Abstract: Given a triangulated surface, a euclidean or hyperbolic polyhedral surface can be constructed by assigning radii to the vertices of the triangulation. We develop necessary and sufficient conditions for the existence of such a polyhedral surface having specified characteristics.

The worm problem of Leo Moser

Abstract: One of Leo Moser's geometry problems is referred to as the Worm Problem [10]: "What is the (convex) region of smallest area which will accommodate (or cover) every planar arc of length 1?" For example, it is easy to show that the circular disk with diameter 1 will cover every planar arc of length 1. The area of the disk is approximately 0.78539. Here we show that a solution to the Worm Problem of Moser is a region with area less than 0.27524.

A cone of inhomogeneous second-order polynomials

Abstract: Let ?n be the cone of quadratic function % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWFgb% GaaGymaiaac6cacaWFGaGaamOzaiabg2da9iaadAgadaWgaaWcbaGa% aGimaaqabaGccqGHRaWkdaaeabqaaiaadAgadaWgaaWcbaGaamyAaa% qabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaaqabeqaniabggHiLdGc% cqGHRaWkdaaeabqaaiaadAgadaWgaaWcbaGaamyAaiaadQgaaeqaaO% GaamiEamaaBaaaleaacaWGPbaabeaaaeqabeqdcqGHris5aOGaamiE% amaaBaaaleaacaWGQbaabeaakiaacYcacaWGMbWaaSbaaSqaaiaadM% gacaWGQbaabeaakiabg2da9iaadAgadaWgaaWcbaGaamOAaiaadMga% aeqaaOGaaiilaaaa!59ED! $$F1. f = f_0 + \sum {f_i x_i } + \sum {f_{ij} x_i } x_j ,f_{ij} = f_{ji} ,$$ on ?n that satisfy the additional condition % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWFgb% Gaa8Nmaiaac6cacaWFGaGaamOzaiaacIcacaWG6bGaaiykaeXafv3y% SLgzGmvETj2BSbacfaGae4xzImRaaGimaiaacYcacaWG6bGaeyicI4% 8efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiyqacqqFKeIw% daahaaWcbeqaaiaad6gaaaGccaGGSaaaaa!570C! $$F2. f(z) \geqslant 0,z \in \mathbb{Z}^n ,$$ where ? denotes the integers. The coefficients and variables are assumed to be real and 1?i, j?n. The extent to which information on the convex structure of ?n can be used to determine the integer solutions of the equationf=0 forf ? ?n has been studied. Theroot figure off ? ?n, denotedRf, is the set ofn-vectorsz ? ?n satisfying the equationf(z)=0. The root figures relate to the convex structure of ?n in an obvious way: ifR is a root figure, then[InlineEquation not available: see fulltext.] is a relatively open face with closure {q??n|q(r)=0,r?R}. However, such formulas do not hold for all the relatively open and closed faces; this relates to some subtleties in the structure of ?n. Enumeration of the possible root figures is the central problem in the theory of ?n. The groupG(?n), of affine transformations on ?n leaving ?n invariant, is the full symmetry group of ?n. Classification of the root figures up toG(?n)-equivalence provides a complete solution to this problem, and this paper is concerned with some basic questions relating to such a classification. The ideas in this study closely relate to the theory ofL-polytopes in lattices as developed by Voronoi [V1], [V2], Delone [De1], [De2], and Ryshkov [RB];L-polytopes, along with their circumscribing empty spheres (often referred to as holes in lattices), play a central role in the study of optimal lattice coverings of space. In addition, the theory of ?n makes contact with: (1) the theory of finite metric spaces, in particular hypermetric spaces [DGL1], [DGL2], and (2) a significant problem in quantum mechanical many-body theory related to the theory of reduced density matrices [E2]---[E4].

X-rays of polygons

Abstract: Various results are given concerning X-rays of polygons in ?2. It is shown that no finite set of X-rays determines every star-shaped polygon, partially answering a question of S. Skiena. For anyn, there are simple polygons which cannot be verified by any set ofn X-rays. Convex polygons are uniquely determined by X-rays at any two points. Finally, it is proved that given a convex polygon, certain sets of three X-rays will distinguish it from other Lebesgue measurable sets.

Neighborly maps with few vertices

Abstract: A neighborly map is a simplicial 2-complex which decomposes a closed 2-manifold without boundary, such that any two vertices are joined by an edge (1-cell) in the complex. We find and describe all the neighborly maps with Euler characteristicX>?10 (i.e., genusg<6, if orientable) or, equivalently, all the neighborly maps withV<12 vertices.

Tiling polygons with parallelograms

Abstract: Under what conditions can a simple polygon be tiled by parallelograms? In this paper we give matching necessary and sufficient conditions on the polygon to be tilable and characterize the set of possible tilings We also provide an efficient algorithm for constructing a tiling

Separating convex sets in the plane

Abstract: Given a setA inR2 and a collectionS of plane sets, we say that a lineL separatesA fromS ifA is contained in one of the closed half-planes defined byL, while every set inS is contained in the complementary closed half-plane. We prove that, for any collectionF ofn disjoint disks inR2, there is a lineL that separates a disk inF from a subcollection ofF with at least ?(n?7)/4? disks. We produce configurationsHn andGn, withn and 2n disks, respectively, such that no pair of disks inHn can be simultaneously separated from any set with more than one disk ofHn, and no disk inGn can be separated from any subset ofGn with more thann disks. We also present a setJm with 3m line segments inR2, such that no segment inJm can be separated from a subset ofJm with more thanm+1 elements. This disproves a conjecture by N. Alonet al. Finally we show that ifF is a set ofn disjoint line segments in the plane such that they can be extended to be disjoint semilines, then there is a lineL that separates one of the segments from at least ?n/3?+1 elements ofF.

Polygon triangulation inO(n log logn) time with simple data structures

Abstract: We give a newO(n log logn)-time deterministic algorithm for triangulating simplen-vertex polygons, which avoids the use of complicated data structures. In addition, for polygons whose vertices have integer coordinates of polynomially bounded size, the algorithm can be modified to run inO(n log*n) time. The major new techniques employed are the efficient location of horizontal visibility edges that partition the interior of the polygon into regions of approximately equal size, and a linear-time algorithm for obtaining the horizontal visibility partition of a subchain of a polygonal chain, from the horizontal visibility partition of the entire chain. The latter technique has other interesting applications, including a linear-time algorithm to convert a Steiner triangulation of a polygon into a true triangulation.