Showing papers in "Discrete and Computational Geometry in 1986"
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TL;DR: A transfer map allows us to transfer properties of ϑ(P) to ℒ(P), and to transfer known inequalities involving linear extensions ofP to some new inequalities.
Abstract: Two convex polytopes, called theorder polytope ?(P) andchain polytope ?(P), are associated with a finite posetP. There is a close interplay between the combinatorial structure ofP and the geometric structure of ?(P). For instance, the order polynomial ?(P, m) ofP and Ehrhart polynomiali(?(P),m) of ?(P) are related by ?(P, m+1)=i(?(P),m). A "transfer map" then allows us to transfer properties of ?(P) to ?(P). In particular, we transfer known inequalities involving linear extensions ofP to some new inequalities.
470 citations
TL;DR: An upper bound for the number of points of local nonconvexity in the union ofm Minkowski sums of planar convex sets is obtained and can be applied to planning a collision-free translational motion of a convex polygonB amidst several polygonal obstacles.
Abstract: Let ?1,..., ?m bem simple Jordan curves in the plane, and letK1,...,Km be their respective interior regions. It is shown that if each pair of curves ?i, ?j,i ?j, intersect one another in at most two points, then the boundary ofK=?i=1mKi contains at most max(2,6m ? 12) intersection points of the curves?1, and this bound cannot be improved. As a corollary, we obtain a similar upper bound for the number of points of local nonconvexity in the union ofm Minkowski sums of planar convex sets. Following a basic approach suggested by Lozano Perez and Wesley, this can be applied to planning a collision-free translational motion of a convex polygonB amidst several (convex) polygonal obstaclesA1,...,Am. Assuming that the number of corners ofB is fixed, the algorithm presented here runs in timeO (n log2n), wheren is the total number of corners of theAi's.
422 citations
TL;DR: This work studies visibility representations of graphs, which are constructed by mapping vertices to horizontal segments, and edges to vertical segments that intersect only adjacent vertex-segments, and considers three types of visibility representations.
Abstract: We studyvisibility representations of graphs, which are constructed by mapping vertices to horizontal segments, and edges to vertical segments that intersect only adjacent vertex-segments Every graph that admits this representation must be planar We consider three types of visibility representations, and we give complete characterizations of the classes of graphs that admit them Furthermore, we present linear time algorithms for testing the existence of and constructing visibility representations of planar graphs Many applications of our results can be found in VLSI layout
415 citations
TL;DR: It turns out that the standard Euclidean Voronoi diagram of point sets inRd along with its order-k generalizations are intimately related to certain arrangements of hyperplanes, and this fact can be used to obtain new Vor onoi diagram algorithms.
Abstract: We propose a uniform and general framework for defining and dealing with Voronoi diagrams. In this framework a Voronoi diagram is a partition of a domainD induced by a finite number of real valued functions onD. Valuable insight can be gained when one considers how these real valued functions partitionD ×R. With this view it turns out that the standard Euclidean Voronoi diagram of point sets inRd along with its order-k generalizations are intimately related to certain arrangements of hyperplanes. This fact can be used to obtain new Voronoi diagram algorithms. We also discuss how the formalism of arrangements can be used to solve certain intersection and union problems.
346 citations
TL;DR: This work proposes a linear-time algorithm, a variant of one by Otten and van Wijk, that generally produces a more compact layout than theirs and allows the dual of the graph to be laid out in an interlocking way.
Abstract: We propose a linear-time algorithm for generating a planar layout of a planar graph. Each vertex is represented by a horizontal line segment and each edge by a vertical line segment. All endpoints of the segments have integer coordinates. The total space occupied by the layout is at mostn by at most 2n---4. Our algorithm, a variant of one by Otten and van Wijk, generally produces a more compact layout than theirs and allows the dual of the graph to be laid out in an interlocking way. The algorithm is based on the concept of abipolar orientation. We discuss relationships among the bipolar orientations of a planar graph.
335 citations
TL;DR: It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulated graph is maximum among all possible triangulations of the graph.
Abstract: We introduce the notion of generalized Delaunay triangulation of a planar straight-line graphG=(V, E) in the Euclidean plane and present some characterizations of the triangulation. It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulation is maximum among all possible triangulations of the graph. A general algorithm that runs inO(|V|2) time for computing the generalized Delaunay triangulation is presented. When the underlying graph is a simple polygon, a divide-and-conquer algorithm based on the polygon cutting theorem of Chazelle is given that runs inO(|V| log |V|) time.
282 citations
TL;DR: It follows as a corollary of the first result that there are no more thannd(d+1)n combinatorially distinct labeled simplicial polytopes inRd withn vertices, which improves the best previous upper bound ofncnd/2.
Abstract: We give a new upper bound onnd(d+1)n on the number of realizable order types of simple configurations ofn points inRd, and ofn2d2n on the number of realizable combinatorial types of simple configurations. It follows as a corollary of the first result that there are no more thannd(d+1)n combinatorially distinct labeled simplicial polytopes inRd withn vertices, which improves the best previous upper bound ofncnd/2.
145 citations
TL;DR: The volume of the convex hull of anym points of ann-dimensional ball with volumeV is at mostV·m/2n, which implies that no polynomial time algorithm can compute the volume of a convex set given by an oracle with less than exponential relative error.
Abstract: The volume of the convex hull of anym points of ann-dimensional ball with volumeV is at mostV·m/2n. This implies that no polynomial time algorithm can compute the volume of a convex set (given by an oracle) with less than exponential relative error. A lower bound on the complexity of computing width can also be deduced.
131 citations
TL;DR: In this article, it was shown that deciding whether two sets of planar points are circularly separable can be done in O(n) time using linear programming, where n is the number of points in the set.
Abstract: Two sets of planar pointsS1 andS2 are circularly separable if there is a circle that enclosesS1 but excludesS2. We show that deciding whether two sets are circularly separable can be accomplished inO(n) time using linear programming. We also show that a smallest separating circle can be found inO(n) time, and largest separating circles can be found inO(n logn) time. Finally we establish that all these results are optimal.
80 citations
TL;DR: The potato-peeling problem asks for the largest convex polygon contained inside a given simple polygon, and anO(n7) time algorithm is given to this problem, answering a question of Goodman.
Abstract: The potato-peeling problem asks for the largest convex polygon contained inside a given simple polygon. We give anO(n7) time algorithm to this problem, answering a question of Goodman. We also give anO(n6) time algorithm if the desired polygon is maximized with respect to perimeter.
77 citations
TL;DR: It is proved that for any centrally symmetric convex polygonal domainP and for any natural numberr, there exists a constantk=k(P, r) such that anyk-fold covering of the plane with translates ofP can be split intor simple coverings.
Abstract: It is proved that for any centrally symmetric convex polygonal domainP and for any natural numberr, there exists a constantk=k(P, r) such that anyk-fold covering of the plane with translates ofP can be split intor simple coverings
TL;DR: It is shown that the total number ofj-sets realized by a set ofn points inE3 isO(nk5); ak-set is any subset ofS of sizek which can be separated from the rest ofS by a plane.
Abstract: Given a fixed setS ofn points inE3 and a query plane?, the halfspace range search problem asks for the retrieval of all points ofS on a chosen side of?. We prove that withO(n(logn)8 (loglogn)4) storage it is possible to solve this problem inO(k+logn) time, wherek is the number of points to be reported. This result rests crucially on a new combinatorial derivation. We show that the total number ofj-sets (j=1, ...,k) realized by a set ofn points inE3 isO(nk5); ak-set is any subset ofS of sizek which can be separated from the rest ofS by a plane.
TL;DR: It is shown that there is a positive constantc such thatfK(S) is the number of subsets of S with cardinalityk εK which can be cut offS by a straight line.
Abstract: For a setS ofn points in the plane and forK ⊆ {1, 2, ..., [1/2n]}, letfK(S) denote the number of subsets ofS with cardinalityk ?K which can be cut offS by a straight line. We show that there is a positive constantc such thatfK(S)
TL;DR: A system of interpolating splines with first-order and approximate second-order geometric continuity and a specific family of parametric cubics is proposed to find aesthetically pleasing curves in a wide range of circumstances.
Abstract: We present a system of interpolating splines with first-order and approximate second-order geometric continuity. The curves are easily computed in linear time by solving a diagonally dominant, tridiagonal system of linear equations. Emphasis is placed on the need to find aesthetically pleasing curves in a wide range of circumstances; favorable results are obtained even when the knots are very unequally spaced or widely separated. The curves are invariant under translation, rotation, and scaling, and the effects of a local change fall off exponentially as one moves away from the disturbed knot.
Approximate second-order continuity is achieved by using a linear "mock curvature" function in place of the actual endpoint curvature for each spline segment and choosing tangent directions at knots so as to equalize these. This avoids extraneous solutions and other forms of undesirable behavior without seriously compromising the quality of the results.
The actual spline segments can come from any family of curves whose endpoint curvatures can be suitably approximated, but we propose a specific family of parametric cubics. There is freedom to allow tangent directions and "tension" parameters to be specified at knots, and special "curl" parameters may be given for additional control near the endpoints of open curves.
TL;DR: This paper describes a method to find a coordinatization for a large class of realizable cases and shows that all realizations found by this algorithm fulfill the isotopy property.
Abstract: Several important and hard realizability problems of combinatorial geometry can be reduced to the realizability problem of oriented matroids. In this paper we describe a method to find a coordinatization for a large class of realizable cases. This algorithm has been used successfully to decide several geometric realizability problems. It is shown that all realizations found by our algorithm fulfill the isotopy property.
TL;DR: It is proved that Prob (the origin belongs to the convA(2d+x→2d))=φ(x)+o(1) ifx is fixed andd → ∞.
Abstract: LetCd be the set of vertices of ad-dimensional cube,Cd={(x1, ...,xd):xi=±1}. Let us choose a randomn-element subsetA(n) ofCd. Here we prove that Prob (the origin belongs to the convA(2d+x?2d))=?(x)+o(1) ifx is fixed andd ? ?. That is, for an arbitrary?>0 the convex hull of more than (2+?)d vertices almost always contains 0 while the convex hull of less than (2-?)d points almost always avoids it.
TL;DR: To study how many essentially different common transversals a family of convex sets on the plane can have, this work considers the case where the family consists of pairwise disjoint translates of a single convex set.
Abstract: The object of this paper is to study how many essentially different common transversals a family of convex sets on the plane can have. In particular, we consider the case where the family consists of pairwise disjoint translates of a single convex set.
TL;DR: A combinatorial criterion for a toric variety to be projective is given which uses Gale-transforms and classes of nonprojective toric varieties are constructed.
Abstract: A combinatorial criterion for a toric variety to be projective is given which uses Gale-transforms. Furthermore, classes of nonprojective toric varieties are constructed.
TL;DR: It is shown that Dyck's map can be realized in E3 as a polyhedron of Kepler-Poinsot-type, i.e., with maximal symmetry and minimal self-intersections.
Abstract: Klein's and Dyck's regular maps on Riemann surfaces of genus 3 were one impetus for the theory of regular maps, automorphic functions, and algebraic curves. Recently a polyhedral realization inE3 of Klein's map was discovered [18], thereby underlining the strong analogy to the icosahedron. In this paper we show that Dyck's map can be realized inE3 as a polyhedron of Kepler-Poinsot-type, i.e., with maximal symmetry and minimal self-intersections. Furthermore some closely related polyhedra and a Kepler-Poinsot-type realization of Sherk's regular map of genus 5 are discussed.
TL;DR: It is stated that there is no polyhedron of genusg>0, such that its symmetry group acts transitively on its faces, and if this condition is slightly weakened, one obtains some interesting polyhedra with face-transitivity.
Abstract: In a remarkable paper [4] Grunbaum and Shephard stated that there is no polyhedron of genusg>0, such that its symmetry group acts transitively on its faces. If this condition is slightly weakened, one obtains some interesting polyhedra with face-transitivity (resp. vertex-transitivity).
TL;DR: Three theorems are proved that show that there exist always packings and coverings whose densities are equal to the corresponding packing and covering constants of a given collection of convex bodies.
Abstract: This article concerns packings and coverings that are formed by the application of rigid motions to the members of a given collectionK of convex bodies There are two possibilities to construct such packings and coverings: One may permit that the convex bodies fromK are used repeatedly, or one may require that these bodies should be used at most once In each case one can define the packing and covering constants ofK as, respectively, the least upper bound and the greatest lower bound of the densities of all such packings and coverings Three theorems are proved First it is shown that there exist always packings and coverings whose densities are equal to the corresponding packing and covering constants Then, a quantitative continuity theorem is proved which shows in particular that the packing and covering constants depend, in a certain sense, continuously onK Finally, a kind of a transference theorem is proved, which enables one to evaluate the packing and covering constants when no repetitions are allowed from the case when repetitions are permitted Furthermore, various consequences of these theorems are discussed
TL;DR: It is shown that if the unit square is covered byn rectangles, then at least one must have perimeter at least 4(2m+1)/(n+m( m+1)), wherem is the largest integer whose square is at mostn.
Abstract: We show that if the unit square is covered byn rectangles, then at least one must have perimeter at least 4(2m+1)/(n+m(m+1)), wherem is the largest integer whose square is at mostn This result is exact forn of the formm(m+1) (orm2)
TL;DR: A necessary and sufficient condition for a configuration of any type of infinite additive cellular automata to have periodic behavior in time is formulated.
Abstract: We formulate and study a necessary and sufficient condition for a configuration of any type of infinite additive cellular automata to have periodic behavior in time. The number of orbits with periodn is counted. Relations between spatial and temporal periods are discussed.
TL;DR: A subsetX of thed-dimensional Euclidean space ℝd can cover its shadows inRd, if every orthogonal projection ofX onto a (d−1)-dimensional linear subspace of �”d is contained in some congruent copy ofX.
Abstract: A subsetX of thed-dimensional Euclidean space ?d can cover its shadows inRd, if every orthogonal projection ofX onto a (d?1)-dimensional linear subspace of ?d is contained in some congruent copy ofX. Whereas every two-dimensional convex discC ?Rd has this property, no (d?1)-polytope does, provided thatd>-4.
TL;DR: It is proved that there are just four w.n.p. maps with Euler characteristic −1 and they are described.
Abstract: A weakly neighborly polyhedral map (w.n.p. map) is a two-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We prove that there are just four w.n.p. maps with Euler characteristic ?1 and we describe them.
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TL;DR: Extremal problems and the existence of designs is investigated in a new type of combinatorial structures, called squashed geometries, which are based on the model derived from Tournais' inequality.
Abstract: Extremal problems and the existence of designs is investigated in a new type of combinatorial structures, called squashed geometries.
TL;DR: Which disc of this class of all convex discs with areas and perimeters bounded by given constants has the least possible area deviation from ak-gon?
Abstract: We consider the class of all convex discs with areas and perimeters bounded by given constants Which disc of this class has the least possible area deviation from ak-gon? This and related questions are the subject of the present paper
TL;DR: Three equivalent formulations of a theorem of Seymour on nonnegative sums of circuits of a graph and a different (but not shorter) proof of Seymour's resut are discussed.
Abstract: We discuss three equivalent formulations of a theorem of Seymour on nonnegative sums of circuits of a graph, and present a different (but not shorter) proof of Seymour's resut.
TL;DR: It is proved that the density of a packing of translates ofu never exceeds thedensity of the densest lattice-packing.
Abstract: Letu be the union of two unit circles whose centers have a distance at most 2. Motivated by more general problems it is proved that the density of a packing of translates ofu never exceeds the density of the densest lattice-packing.