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Added by
Maria Saumell
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We study the cyclic color sequences induced at infinity by colored rays with apices being a given balanced finite bichromatic point set. We first study the case in which the rays are required to be pairwise disjoint. We derive a lower bound on the number of color sequences that can be realized from any such fixed point set and examine color sequences that can be realized regardless of the point set, exhibiting negative examples as well. We also provide a tight upper bound on the number of configurations that can be realized from a point set, and point sets for which there are asymptotically less configurations than that number. In addition, we provide algorithms to decide whether a color sequence is realizable from a given point set in a line or in general position. We address afterwards the variant of the problem where the rays are allowed to intersect. We prove that for some configurations and point sets, the number of ray crossings must be $Theta(n^2)$ and study then configurations that can be realized by rays that pairwise cross. We show that there are point sets for which the number of configurations that can be realized by pairwise-crossing rays is asymptotically smaller than the number of configurations realizable by pairwise-disjoint rays. We provide also point sets from which any configuration can be realized by pairwise-crossing rays and show that there is no configuration that can be realized by pairwise-crossing rays from every point set.
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We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A emph{red-blue-purple spanning graph} (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem can be solved in polynomial time using matroid techniques. In addition, we discuss more efficient algorithms for the case in which points are located on a line or a circle, and also describe a fast $(frac 12rho+1)$-approximation algorithm, where $rho$ is the Steiner ratio.
Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $operatorname{rb-index}(k)$ be the maximum of $operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $operatorname{rb-index}(k) eq k$, and we prove that for $kge 5$, [ frac{40lfloor (k-1)/2 rfloor -8}{19} %Birgit: leqoperatorname{rb-index}(k)leq 10 bigglfloorfrac{k}{7}biggrfloor + 11. ] Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 lfloorfrac{k}{7}rfloor + 11$ vertices can be computed in $O(nlog n)$ time.
Let $S$ be a finite set of geometric objects partitioned into classes or emph{colors}. A subset $Ssubseteq S$ is said to be emph{balanced} if $S$ contains the same amount of elements of $S$ from each of the colors. We study several problems on partitioning $3$-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are $2m$ lines of each color, there is a segment intercepting $m$ lines of each color. (b) Given $n$ red points, $n$ blue points and $n$ green points on any closed Jordan curve $gamma$, we show that for every integer $k$ with $0 leq k leq n$ there is a pair of disjoint intervals on $gamma$ whose union contains exactly $k$ points of each color. (c) Given a set $S$ of $n$ red points, $n$ blue points and $n$ green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of $S$.
K{a}rolyi, Pach, and T{o}th proved that every 2-edge-colored straight-line drawing of the complete graph contains a monochromatic plane spanning tree. It is open if this statement generalizes to other classes of drawings, specifically, to simple drawings of the complete graph. These are drawings where edges are represented by Jordan arcs, any two of which intersect at most once. We present two partial results towards such a generalization. First, we show that the statement holds for cylindrical simple drawings. (In a cylindrical drawing, all vertices are placed on two concentric circles and no edge crosses either circle.) Second, we introduce a relaxation of the problem in which the graph is $k$-edge-colored, and the target structure must be hypochromatic, that is, avoid (at least) one color class. In this setting, we show that every $lceil (n+5)/6rceil$-edge-colored monotone simple drawing of $K_n$ contains a hypochromatic plane spanning tree. (In a monotone drawing, every edge is represented as an $x$-monotone curve.)
This short contribution presents a method for generating $N$-point spherical configurations with low mesh ratios. The method extends Caspar-Klug icosahedral point-grids to non-icosahedral nets through the use of planar barycentric coordinates, which are subsequently interpreted as spherical area coordinates for spherical point sets. The proposed procedure may be applied iteratively and is parameterised by a sequence of integer pairs. For well-chosen input parameters, the proposed method is able to generate point sets with mesh ratios that are lower than previously reported for $N<10^6$.
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