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Showing papers in "Discrete and Computational Geometry in 1990"


Journal ArticleDOI
TL;DR: Upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike are presented and it is proved that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m2/3n 2/3 +n), and that it isO(m3/2β(m) forn unit-circles.
Abstract: We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges boundingm cells in an arrangement ofn lines is ?(m2/3n2/3 +n), and that it isO(m2/3n2/3s(n) +n) forn unit-circles, wheres(n) (and laters(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up toO(m3/5n4/5s(n) +n). The same bounds (without thes(n)-terms) hold for the maximum sum of degrees ofm vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees ofm vertices in an arrangement ofn spheres in three dimensions isO(m4/7n9/7s(m, n) +n2), in general, andO(m3/4n3/4s(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances amongm points in three dimensions isO(m3/2s(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.

362 citations


Journal ArticleDOI
TL;DR: There is a constantc (≤((1+√5)/2) π≈5.08) independent ofS andN such that % MathType!MTEF!2!1!-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9g
Abstract: LetS be any set ofN points in the plane and let DT(S) be the graph of the Delaunay triangulation ofS. For all pointsa andb ofS, letd(a, b) be the Euclidean distance froma tob and let DT(a, b) be the length of the shortest path in DT(S) froma tob. We show that there is a constantc (≤((1+?5)/2) ??5.08) independent ofS andN such that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaie% aacaWFebGaa8hvaiaa-HcacaWGHbGaaiilaiaadkgacaWFPaaabaGa% amizaiaa-HcacaWGHbGaaiilaiaadkgacaWFPaaaaiabgYda8iaado% gacaGGUaaaaa!4248! $$\frac{{DT(a,b)}}{{d(a,b)}}< c.$$

233 citations


Journal ArticleDOI
TL;DR: It is shown that the adjacency graph (and poset of regions) of an arrangement determines the associated oriented matroid and hence in particular the lattice of intersections in the case of supersolvable arrangements.
Abstract: A hyperplane arrangement is a finite set of hyperplanes through the origin in a finite-dimensional real vector space. Such an arrangement divides the vector space into a finite set of regions. Every such region determines a partial order on the set of all regions in which these are ordered according to their combinatorial distance from the fixed base region. We show that the base region is simplicial whenever the poset of regions is a lattice and that conversely this condition is sufficient for the lattice property for three-dimensional arrangements, but not in higher dimensions. For simplicial arrangements, the poset of regions is always a lattice. In the case of supersolvable arrangements (arrangements for which the lattice of intersections of hyperplanes is supersolvable), the poset of regions is a lattice if the base region is suitably chosen. We describe the geometric structure of such arrangements and derive an expression for the rank-generating function similar to a known one for Coxeter arrangements. For arrangements with a lattice of regions we give a geometric interpretation of the lattice property in terms of a closure operator defined on the set of hyperplanes. The results generalize to oriented matroids. We show that the adjacency graph (and poset of regions) of an arrangement determines the associated oriented matroid and hence in particular the lattice of intersections.

151 citations


Journal ArticleDOI
TL;DR: An O(n4, log logn) algorithm to find a shortest watchman route in a simple polygon through a point,s, in its boundary is presented.
Abstract: In this paper we present an O(n4, log logn) algorithm to find a shortest watchman route in a simple polygon through a point,s, in its boundary. A watchman route is a route such that each point in the interior of the polygon is visible from at least one point along the route.

139 citations


Journal ArticleDOI
TL;DR: The proof takes an algorithmic approach, that is, an algorithm is described for the calculation of thesem faces and the upper bound for the total number of edges is derived from the analysis of the algorithm.
Abstract: We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m2/3??n2/3+2?+n) for any?>0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of thesem faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and its expected time complexity isO(m2/3??n2/3+2? logn+n logn logm). If instead of lines we have an arrangement ofn line segments, then the maximum number of edges ofm faces isO(m2/3??n2/3+2?+n? (n) logm) for any?>0, where?(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and takes expected timeO(m2/3??n2/3+2? log+n?(n) log2n logm).

108 citations


Journal ArticleDOI
TL;DR: The algorithm extends and combines the techniques of Leven and Sharir and of Sifrony andSharir used for the case in which B is a line segment and makes use of the results of Kedem and SharIR on the planning of translational motion of B amidst polygonal obstacles.
Abstract: We present an efficient algorithm for planning the motion of a convex polygonal bodyB in two-dimensional space bounded by a collection of polygonal obstacles. Our algorithm extends and combines the techniques of Leven and Sharir and of Sifrony and Sharir used for the case in whichB is a line segment (a "ladder"). It also makes use of the results of Kedem and Sharir on the planning of translational motion ofB amidst polygonal obstacles, and of a recent result of Leven and Sharir on the number of free critical contacts ofB with such polygonal obstacles. The algorithm runs in timeO(kn?6(kn) logkn), wherek is the number of sides ofB, n is the number of obstacle edges, and ?,(q) is an almost linear function ofq yielding the maximal number of connected portions ofq continuous functions which compose the graph of their lower envelope, where it is assumed that each pair of these functions intersect in at mosts points.

80 citations


Journal ArticleDOI
TL;DR: It is shown that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness, and this characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1-Tough.
Abstract: We show that nondegenerate Delaunay triangulations satisfy a combinatorial property called 1-toughness. A graphG is1-tough if for any setP of vertices,c(G?P)≤|G|, wherec(G?P) is the number of components of the graph obtained by removingP and all attached edges fromG, and |G| is the number of vertices inG. This property arises in the study of Hamiltonian graphs: all Hamiltonian graphs are 1-tough, but not conversely. We also show that all Delaunay triangulationsT satisfy the following closely related property: for any setP of vertices the number of interior components ofT?P is at most |P|?2, where an interior component ofT?P is a component that contains no boundary vertex ofT. These appear to be the first nontrivial properties of a purely combinatorial nature to be established for Delaunay triangulations. We give examples to show that these bounds are best possible and are independent of one another. We also characterize the conditions under which a degenerate Delaunay triangulation can fail to be 1-tough. This characterization leads to a proof that all graphs that can be realized as polytopes inscribed in a sphere are 1-tough. One consequence of the toughness results is that all Delaunay triangulations and all inscribable graphs have perfect matchings.

73 citations


Journal ArticleDOI
TL;DR: It is proved that S can realize at most 2n−2 geometric permutations, and the upper bound is tight.
Abstract: LetS be a collection ofn convex, closed, and pairwise nonintersecting sets in the Euclidean plane labeled from 1 ton. A pair of permutations $$\left\{ {(i_1 ,i_2 ,...,i_{n - 1} ,i_n ,),(i_n ,i_{n - 1} ,...,i_2 ,i_1 ,)} \right\}$$ is called ageometric permutation of S if there is a line that intersects all sets ofS in this order. We prove thatS can realize at most 2n?2 geometric permutations. This upper bound is tight.

71 citations


Journal ArticleDOI
TL;DR: The method is constructive as well, and it produces double-lattice packings consisting of translates ofK and translates of−K, an improvement of the bounds obtained previously in [5] and [6].
Abstract: Mahler [7] and Fejes Toth [2] proved that every centrally symmetric convex plane bodyK admits a packing in the plane by congruent copies ofK with density at least ?3/2. In this paper we extend this result to all, not necessarily symmetric, convex plane bodies. The methods of Mahler and Fejes Toth are constructive and produce lattice packings consisting of translates ofK. Our method is constructive as well, and it produces double-lattice packings consisting of translates ofK and translates of?K. The lower bound of ?3/2 for packing densities produced here is an improvement of the bounds obtained previously in [5] and [6].

69 citations


Journal ArticleDOI
TL;DR: This work considers the problem of packingn equal circles (i.e., pennies) in the plane so as to minimize the second momentU about their centroid, and appears to produce optimal packings for infinitely many values ofn.
Abstract: We consider the problem of packingn equal circles (i.e., pennies) in the plane so as to minimize the second momentU about their centroid. These packings are also minimal-energy two-dimensional codes. Adding one penny at a time according to the greedy algorithm produces a unique sequence of packings for the first 75 pennies, and appears to produce optimal packings for infinitely many values ofn. Several other conjectures are proposed, and a table is given of the best packings known forn≤500. For largen, U~?3n2/(4?).

58 citations



Journal ArticleDOI
Rephael Wenger1
TL;DR: It is shown that the directed lines inRd, d ≥ 3, can be partitioned into 12n such sets, which bounds the number of geometric permutations onA by 1/2φd ford≥3 and by 6n ford=2.
Abstract: LetA be a family ofn pairwise disjoint compact convex sets inRd. Let % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aaS% baaSqaaiaadsgaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0JaaGOm% aiabfo6atnaaDaaaleaacaWGPbGaeyypa0JaaGimaaqaaiaadsgacq% GHsislcaaIXaaaaOWaaeWaaeaadaqhaaWcbaGaiaiG0caaamyAaaqa% aiaad2gacqGHsislcaaIXaaaaaGccaGLOaGaayzkaaaaaa!4A12! $$\Phi _d (m) = 2\Sigma _{i = 0}^{d - 1} \left( {_i^{m - 1} } \right)$$ . We show that the directed lines inRd, d ? 3, can be partitioned into % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aaS% baaSqaaiaadsgaaeqaaOWaaeWaaeaadaqadaqaamaaDaaaleaacaaI% YaaabaGaamOBaaaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa!3CFF! $$\Phi _d \left( {\left( {_2^n } \right)} \right)$$ sets such that any two directed lines in the same set which intersect anyA?⊆A generate the same ordering onA?. The directed lines inR2 can be partitioned into 12n such sets. This bounds the number of geometric permutations onA by 1/2?d ford?3 and by 6n ford=2.



Journal ArticleDOI
TL;DR: A new duality between order-k Voronoi diagrams inEd and convex hulls inEd+1 is established, which implies a reasonably simple algorithm for computing the order- k diagram forn points in the plane in O(k2n logn) time and optimalO(k(n−k)) space.
Abstract: A new duality between order-k Voronoi diagrams inEd and convex hulls inEd+1 is established. It implies a reasonably simple algorithm for computing the order-k diagram forn points in the plane inO(k2n logn) time and optimalO(k(n?k)) space.


Journal ArticleDOI
TL;DR: An anO(n logn)-time algorithm is presented to determine whether a set of segments, constrained so that each segment has at least one endpoint on the boundary of the convex hull of the segments, admits a simple circuit.
Abstract: We address the problem of connecting line segments to form the boundary of a simple polygon--a simple circuit. However, not every set of segments can be so connected. We present anO(n logn)-time algorithm to determine whether a set of segments, constrained so that each segment has at least one endpoint on the boundary of the convex hull of the segments, admits a simple circuit. Furthermore, this technique can also be used to compute a simple circuit of minimum perimeter, or a simple circuit that bounds the minimum area, with no increase in computational complexity.

Journal ArticleDOI
TL;DR: This paper resolves a problem posed in [10] by proving that not every realizable simplicial chirotope admits a solvability sequence and shows that there is no easy combinatorial method for proving nonrealizability and thus justifies the final polynomial approach.
Abstract: This paper deals with a class of computational problems in real algebraic geometry. We introduce the concept of final polynomials as a systematic approach to prove nonrealizability for oriented matroids and combinatorial geometries. Hilbert's Nullstellensatz and its real analogue imply that an abstract geometric object is either realizable or it admits a final polynomial. This duality has first been applied by Bokowski in the study of convex polytopes [7] and [11], but in these papers the resulting final polynomials were given without their derivations. It is the objective of the present paper to fill that gap and to describe an algorithm for constructing final polynomials for a large class of nonrealizable chirotopes. We resolve a problem posed in [10] by proving that not every realizable simplicial chirotope admits a solvability sequence. This result shows that there is no easy combinatorial method for proving nonrealizability and thus justifies our final polynomial approach.


Journal ArticleDOI
TL;DR: A simple data structure is presented, a dynamic contour search tree, which contains all the points in the plane and maintains an embedded linked list of maximal points so thatm maximal points are accessible inO(m) time.
Abstract: A pointpi=(xi, yi) in thex---y plane ismaximal if there is no pointpj=(xj, yj) such thatxj>xi andyj>yi. We present a simple data structure, a dynamic contour search tree, which contains all the points in the plane and maintains an embedded linked list of maximal points so thatm maximal points are accessible inO(m) time. Our data structure dynamically maintains the set of points so that insertions takeO(logn) time, a speedup ofO(logn) over previous results, and deletions takeO((logn)2) time.

Journal ArticleDOI
Rephael Wenger1
TL;DR: Given an ordered family of compact convex sets in the plane, if every three sets can be intersected by some directed line “consistent” with the ordering, then there exists a common transversal of the family.
Abstract: Given an ordered family of compact convex sets in the plane, if every three sets can be intersected by some directed line "consistent" with the ordering, then there exists a common transversal of the family. This generalizes Hadwiger's Transversal Theorem to families of compact convex sets which are not necessarily pairwise disjoint. If every six sets can be intersected by some directed line "consistent" with the ordering, then there exists a common transversal which is "consistent" with the ordering. If the family is pairwise disjoint and every four sets can be intersected by some directed line "consistent" with the ordering, then there exists a common transversal which is "consistent" with the ordering.

Journal ArticleDOI
TL;DR: In this article, it was shown that the number of points in a combinatorial geometry of rank n representable over GF(3) and GF(q) is at mostn 2.
Abstract: Letq be a prime power not divisible by 3. We show that the number of points (or rank-1 flats) in a combinatorial geometry (or simple matroid) of rankn representable over GF(3) and GF(q) is at mostn2. Whenq is odd, this bound is sharp and is attained by the Dowling geometries over the cyclic group of order 2.


Journal Article
TL;DR: In this article, the authors consider arrangements of curves that intersect pairwise in at most k points and show that a curve can sweep any such arrangement and maintain the k-intersection property if and only if k equals 1 or 2.
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.


Journal ArticleDOI
TL;DR: It is shown that ifm X-ray pictures with directions in some plane are given, then the problem is well posed provided the number of the holes is less than or equal tom and the set of the directions satisfies a suitable condition.
Abstract: We give some uniqueness results for the problem of determining a finite set in the plane knowing its projections alongm directions. We apply the results to the problem of the reconstruction of a homogeneous convex body with a finite set of spherical disjoint holes. Ifm X-ray pictures with directions in some plane are given, then the problem is well posed provided the number of the holes is less than or equal tom and the set of the directions satisfies a suitable condition.

Journal ArticleDOI
TL;DR: The aim of this paper is to show that a convexd-polytope (d≥3) is a simplex if and only if its projection body and its difference set are polars.
Abstract: In a paper by the author and B. Weissbach it was proved that the projection body and the difference set of ad-simplex (d?2) are polars. Obviously, ford=2 a convex domain has this property if and only if its difference set is bounded by a so-called Radon curve. A natural question emerges about further classes of convex bodies inRd (d?3) inducing the mentioned polarity. The aim of this paper is to show that a convexd-polytope (d?3) is a simplex if and only if its projection body and its difference set are polars.

Journal ArticleDOI
TL;DR: In this paper, it is shown that cycling is impossible unless the two objective functions are related in a very special way to each other or to the constraints defining the feasible regionP. In particular, the avoidance of cycling does not require any restriction on the facial structure of P or on the algebraic relationships among the linear equalities and inequalities by means of which P is defined.
Abstract: In previous discussions of the Gass-Saaty algorithm, the possibility of cycling is avoided by making strong nondegeneracy assumptions or by incorporating a lexicographic decision rule. By analyzing the geometric ideas on which the algorithm is based, it is shown here that even without any "lexicography," cycling is impossible unless the two objective functions are related in a very special way to each other or to the constraints defining the feasible regionP. In particular, the avoidance of cycling does not require any restriction on the facial structure ofP or on the algebraic relationships among the linear equalities and inequalities by means of whichP is defined.