Abstract Let P be a set of n points in $$\mathbb {R}^3$$ R 3 in general position, and let RCH(P) be the rectilinear convex hull of P. In this paper we obtain an optimal $$O(n\log n)$$ O ( n log n ) time and O(n) space algorithm to compute RCH(P). We also obtain an efficient $$O(n\log ^2 n)$$ O ( n log 2 n ) time and $$O(n\log n)$$ O ( n log n ) space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of P as we rotate $${\mathbb {R}}^3$$ R 3 around the Z-axis. We study some combinatorial properties of the rectilinear convex hulls of point sets in $$\mathbb {R}^3$$ R 3 . Finally, as an application of the obtained results, we show an approximation algorithm to an optimization fitting problem in $$\mathbb {R}^3$$ R 3 .

Abstract We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let R and B be two disjoint sets of red and blue points in the plane, and $$\mathcal {O}$$ O be a set of $$k\ge 2$$ k ≥ 2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of $$\mathcal {O}$$ O for which the $$\mathcal {O}$$ O -convex hull of R contains no points of B. For $$k=2$$ k = 2 orthogonal lines we have the rectilinear convex hull. In optimal $$O(n\log n)$$ O ( n log n ) time and O(n) space, $$n = \vert R \vert + \vert B \vert $$ n = | R | + | B | , we compute the set of rotation angles such that, after simultaneously rotating the lines of $$\mathcal {O}$$ O around the origin in the same direction, the rectilinear convex hull of R contains no points of B. We generalize this result to the case where $$\mathcal {O}$$ O is formed by $$k \ge 2$$ k ≥ 2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of $$\mathcal {O}$$ O , let $$\alpha _i$$ α i be the angle required to clockwise rotate the ith line so it coincides with its successor. We solve the problem in this case in $$O({1}/{\Theta }\cdot N \log N)$$ O ( 1 / Θ · N log N ) time and $$O({1}/{\Theta }\cdot N)$$ O ( 1 / Θ · N ) space, where $$\Theta = \min \{ \alpha _1,\ldots ,\alpha _k \}$$ Θ = min { α 1 , … , α k } and $$N=\max \{k,\vert R \vert + \vert B \vert \}$$ N = max { k , | R | + | B | } . We finally consider the case in which $$\mathcal {O}$$ O is formed by $$k=2$$ k = 2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to $$\pi $$ π . We show that this last case can also be solved in optimal $$O(n\log n)$$ O ( n

Let $P$łabelpage1 be a set of $n$ points in the plane. We consider a variation of the classical Erdös--Szekeres problem, presenting efficient algorithms with $O(n^3)$ running time and $O(n^2)$ space complexity that compute (1) a subset $S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the maximum number of points from $P$, (2) a subset $S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the maximum number of points from $P$ and its interior contains no element of $P$, (3) a subset $S$ of $P$ such that the rectilinear convex hull of $S$ has maximum area and its interior contains no element of $P$, and (4) when each point of $P$ is assigned a weight, positive or negative, a subset $S$ of $P$ that maximizes the total weight of the points in the rectilinear convex hull of $S$. We also revisit the problems of computing a maximum area orthoconvex polygon and computing a maximum area staircase polygon, amidst a point set in a rectangular domain. We obtain new and simpler algorithms to solve both problems with the same complexity as in the state of the art.

For every $n\in \mathbb{N}$, we construct an $n$-vertex planar graph $G=(V,E)$ and $n$ distinct points $p(v)$, $v\in V$, in the plane such that in any crossing-free straight-line drawing of $G$, at most $O(n^{.4948})$ vertices $v\in V$ are embedded at points $p(v)$. This improves on an earlier bound of $O(\sqrt{n})$ by Goaoc et al. [Discrete Comput. Geom., 42 (2009), pp. 542--569].

Abstract We address the problem of discovering routes in strongly connected planar geometric networks with directed links. Motivated by the necessity for establishing communication in wireless ad hoc networks in which the only information available to a vertex is its immediate neighborhood, we are considering routing algorithms that use the neighborhood information of a vertex for routing with constant memory only. We solve the problem for three types of directed planar geometric networks: Eulerian (in which every vertex has the same number of incoming and outgoing edges), Outerplanar in which a single face contains all vertices of the network, and Strongly Face Connected, a new class of geometric networks that we define in the article, consisting of several faces, each face being a strongly connected outerplanar graph. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(1), 7–15 2006

Summary form only given. The problem of traversal of planar subdivisions or other graph-like structures without using mark bits is central to many real-world applications. The first such algorithms were able to traverse triangulated subdivisions. Later these algorithms were extended to traverse vertices of an arrangement or a convex polytope. The research progress culminated in an algorithm that can traverse any planar subdivision. We extend the notion of planar subdivision to quasiplanar subdivision in which we allow many edges to cross each other. We describe an algorithm to traverse any quasiplanar subdivision that satisfies a simple requirement. The worst case running time of our algorithm is O(|E| log |E|), which matches the running time of the traversal algorithm for planar subdivisions.

No abstract available.

We consider the problem of selecting the Kth smallest element of a set distributed among the sites of a communication network when the size of messages is bounded; that is, each message is a packet which contains at most c bits, where c/spl ges/1 is a constant. A general selection algorithm using packets is presented and its packet complexity is analyzed. Its complexity is shown to be a significant improvement for a large range of packet sizes over the existing bounds. The proposed technique is then instanciated for specific classes of network topologies; the resulting bounds either match or improve the ones of existing solutions for a large range of values of the packet size. Furthermore, it is bit optimal in star networks.

Article Flipping edges in triangulations Share on Authors: F. Hurtado Departamento de Matemática Aplicada II, Universitat Politécnica de Catalunya, Barcelona, Spain Departamento de Matemática Aplicada II, Universitat Politécnica de Catalunya, Barcelona, SpainView Profile , M. Noy Departamento de Matemática Aplicada II, Universitat Politécnica de Catalunya, Barcelona, Spain Departamento de Matemática Aplicada II, Universitat Politécnica de Catalunya, Barcelona, SpainView Profile , J. Urrutia Department of Computer Science, University of Ottawa, Ottawa, ON Canada Department of Computer Science, University of Ottawa, Ottawa, ON CanadaView Profile Authors Info & Claims SCG '96: Proceedings of the twelfth annual symposium on Computational geometryMay 1996 Pages 214–223https://doi.org/10.1145/237218.237367Online:01 May 1996Publication History 6citation419DownloadsMetricsTotal Citations6Total Downloads419Last 12 Months3Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Article Free Access Share on Voronoi diagrams and containment of families of convex sets on the plane Authors: M. Abellanas Universidad Politécnica de Madrid, Spain Universidad Politécnica de Madrid, SpainView Profile , G. Hernandez Universidad Politécnica de Madrid, Spain Universidad Politécnica de Madrid, SpainView Profile , R. Klein Fern Universität Hagen, Germany Fern Universität Hagen, GermanyView Profile , V. Neumann-Lara Universidad Nacional Autonoma de México, Mexico Universidad Nacional Autonoma de México, MexicoView Profile , J. Urrutia University of Ottawa, Canada University of Ottawa, CanadaView Profile Authors Info & Claims SCG '95: Proceedings of the eleventh annual symposium on Computational geometrySeptember 1995 Pages 71–78https://doi.org/10.1145/220279.220287Published:01 September 1995Publication History 3citation259DownloadsMetricsTotal Citations3Total Downloads259Last 12 Months12Last 6 weeks4 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Abstract In this paper we consider polygonizations that are robust when faced with changes in the vertices that are present or in their position. We analyze the dynamic maintenance of different types of polygonizations (monotone, star‐shaped…) and we introduce monotone half‐convex polygonizations that are specially interesting because they provide minimum cost per insertion or deletion. If we had to delete not only one point but several external layers of the set, then the onion polygonizations would be suited, because they can be updated in constant time. We also consider the case of points that can be moved to contiguous positions and we show how to polygonize the set for updating in linear time. We deal too with security problems for a polygon: What is the maximum distance the vertices of a polygon could be moved away of their position in such a way that the topology on the boundary of the polygon (or its convexity) remains the same?.

Given a family of objects in the plane, the line transversal problem is to compute a line that intersects every member of the family.In this paper we examine a variation of the line transversal problem that involves computing a shortest line segment that intersects every member of the family.In particular, we give O(n log n) time algorithms for computing a shortest transversal of a family of n lines and of a family of n line segments.We also present an O(n log2 n) time algorithm for computing a shortest transversal of a family of polygons with a total of n vertices.In general, finding a line transversal for a family of n objects takes fl(n log n) time.This time bound holds for a family of n line segments thus our shortest transversal algorithm for this family is optimal.

Ordered sets are used as a computational model for motion planning problems. Every ordered set has a two-directional point representation using subdivisions. These subdivision points correspond to direction changes along the path of motion.

Given two geometric sets A and B, it is said that A is containable in B provided A is isometric to a subset of B. Containability induces a partial order on any set of geometric figures, such as rectangles in the plane. A recent result states that for the set of rectangles in the plane, the containability partial order is of countably infinite dimension. In this paper the rectangle result is extended to other families of geometric figures and to a partial order obtained from quadratic polynomials.

The problem of assigningncarrier frequencies so as to avoid certain types (third and fifth order) of intermodulation interference is discussed. For the third-order case, close upper and lower bounds on the optimal solution are established; and close to optimal solutions are given forn < 100(previously, suboptimal solutions were known only forn \leq 23). For the fifth-order case, it is shown that some existing results can be applied to this problem, and suboptimal solutions obtained by this construction are given forn \leq 17(no solutions were known previously).

Abstract A new class of intersection graphs called circular permutation graphs is introduced and characterized. A circular permutation diagram for a permutation P (1),…, P ( n ) consists of two circles C 1 and C 2 ; the numbers 1′, 2′,…,n′ and P (1),…, P ( n ) on C 1 and C 2 , respectively; and a set of n chords 1, 2,…, n connecting i to i ′ such that two chords intersect each other at most once. A graph G represents a circular permutation diagram if there is a labeling of V ( G ) with {1,…, n} such that i is adjacement to j iff i and j intersect. Graphs which represent at least one permutation diagram are called circular permutation graphs. Circular permutation graphs generalize permutation graphs [2], [8] and are embedded in the set of comparability graphs [4]. The characterization leads to a recognition algorithm which requires O (δ|E|) steps where δ is the maximum degree of a vertex.

Abstract A circle diagram consists of a circle C and a set of n chords. This diagram defines a graph with n vertices where each vertex corresponds to a chord, and two vertices are adjacent if their corresponding chords intersect in C. A graph G is called a circle graph if it is defined by some circle diagram. An algorithm which requires O(n2) steps to generate one maximum clique is presented. The algorithm can also be used to generate all maximum cliques where the number of steps need to generate each additional maximum clique is linear in its size. This compares favourably with Gavril's algorithm [4] which works in O(n3) steps.