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Register a new userColor Spanning Annulus: Square, Rectangle and Equilateral Triangle
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Added by
Ankush Acharyya
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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In this paper, we study different variations of minimum width color-spanning annulus problem among a set of points $P={p_1,p_2,ldots,p_n}$ in $I!!R^2$, where each point is assigned with a color in ${1, 2, ldots, k}$. We present algorithms for finding a minimum width color-spanning axis parallel square annulus $(CSSA)$, minimum width color spanning axis parallel rectangular annulus $(CSRA)$, and minimum width color-spanning equilateral triangular annulus of fixed orientation $(CSETA)$. The time complexities of computing (i) a $CSSA$ is $O(n^3+n^2klog k)$ which is an improvement by a factor $n$ over the existing result on this problem, (ii) that for a $CSRA$ is $O(n^4log n)$, and for (iii) a $CSETA$ is $O(n^3k)$. The space complexity of all the algorithms is $O(k)$.
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In this paper, we address the minimum-area rectangular and square annulus problem, which asks a rectangular or square annulus of minimum area, either in a fixed orientation or over all orientations, that encloses a set $P$ of $n$ input points in the plane. To our best knowledge, no nontrivial results on the problem have been discussed in the literature, while its minimum-width variants have been intensively studied. For a fixed orientation, we show reductions to well-studied problems: the minimum-width square annulus problem and the largest empty rectangle problem, yielding algorithms of time complexity $O(nlog^2 n)$ and $O(nlog n)$ for the rectangular and square cases, respectively. In arbitrary orientation, we present $O(n^3)$-time algorithms for the rectangular and square annulus problem by enumerating all maximal empty rectangles over all orientations. The same approach is shown to apply also to the minimum-width square annulus problem and the largest empty square problem over all orientations, resulting in $O(n^3)$-time algorithms for both problems. Consequently, we improve the previously best algorithm for the minimum-width square annulus problem by a factor of logarithm, and present the first algorithm for the largest empty square problem in arbitrary orientation. We also consider bicriteria optimization variants, computing a minimum-width minimum-area or minimum-area minimum-width annulus.
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Given a set $P$ of $n$ red and blue points in the plane, a emph{planar bichromatic spanning tree} of $P$ is a spanning tree of $P$, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree $T$, such that the length of the longest edge in $T$ is minimized. In this paper, we show that this problem is NP-hard for points in general position. Moreover, we present a polynomial-time $(8sqrt{2})$-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck $lambda$ can be converted to a planar bichromatic spanning tree of bottleneck at most $8sqrt{2}lambda$.
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