Optimally Decomposing Coverings with Translates of a Convex Polygon

Gibson, Matt; Varadarajan, Kasturi

doi:10.1007/s00454-011-9353-9

Cited by 19 publications

(24 citation statements)

References 11 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It was shown recently [7] that for every convex polygon P , there exists a constant c, such that any point set in R 2 can be colored with k colors in such a way that any translation of P containing at least p(k) = ck points contains at least one point of each color. This improves on several previous intermediate results [15,17,2].…”

Section: Introductionmentioning

confidence: 99%

Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

Asinowski

Cardinal

Cohen

et al. 2013

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color.We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k) = 3k − 2.This can be interpreted as coloring point sets in R 2 with k colors such that any bottomless rectangle containing at least 3k − 2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set.For this problem, we also prove a lower bound p(k) > ck, where c > 1.67. Hence for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function p(k) exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Pálvölgyi on cover-decomposability of octants (2011, 2012).

show abstract

Section: Introductionmentioning

confidence: 99%

Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

Asinowski

Cardinal

Cohen

et al. 2013

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…To our knowledge, our result is the first polynomial bound for cover-decomposability of homothetic copies of a polygon. Linear upper bounds have been obtained for halfplanes [4,34], and translates of a convex polygon in the plane [35,29,3,16]. A restricted version of this problem involving unit balls is shown to be solvable using the probabilistic method in the well-known book from Alon and Spencer [2].…”

Section: Previous Resultsmentioning

confidence: 99%

Making Octants Colorful and Related Covering Decomposition Problems

Cardinal¹,

Knauer²,

Micek³

et al. 2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

Abstract. We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R 3 can be colored with k colors so that every translate of the negative octant containing at least k 6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition problems involving intervals appearing on a line. We prove that no algorithm can dynamically maintain a decomposition of a multiple covering by intervals under insertion of new intervals, even in a semi-online model, in which some coloring decisions can be delayed. This implies that a wide range of sweeping plane algorithms cannot guarantee any bound even for special cases of the octant problem.

show abstract

“…The case of not necessarily centrally symmetric polygons P was settled in [GV11]. In this subsection, we sketch the proof in the special case when P is a triangle, which already contains most of the key ideas of the general argument.…”

Section: Decomposition To ω(M) Parts For Trianglesmentioning

confidence: 99%

“…In [P86], it was shown that, for any centrally symmetric convex open polygon P , the parameter m k (P ) exists and is bounded by an exponentially fast growing function of k. In [TT07], a similar result was established for open triangles, and in [PT10] for open convex polygons. However, all these results were improved to the optimal linear bound in a series of papers by Pach and Tóth [PT07], Aloupis et al [Al10], and Gibson and Varadarajan [GV11]. Theorem 1.7.…”

mentioning

confidence: 92%

See 1 more Smart Citation

Survey on Decomposition of Multiple Coverings

Pach

Pálvölgyi

Tóth

2013

Bolyai Society Mathematical Studies

View full text Add to dashboard Cite

)>IJH=?J The study of multiple coverings was initiated by Davenport and L. Fejes Tóth more than 50 years ago. In 1980 and 1986, the rst named author published the rst papers about decomposability of multiple coverings. It was discovered much later that, besides its theoretical interest, this area has practical applications to sensor networks. Now there is a lot of activity in this eld with several breakthrough results, although, many basic questions are still unsolved. In this survey, we outline the most important results, methods, and questions.1 Cover-decomposability and the sensor cover problem De nition 1.1. A planar set P is said to be cover-decomposable if there exists a (minimal) constant m = m(P ) such that every m-fold covering of the plane with translates of P can be decomposed into two coverings.Note that the above term is slightly misleading: we decompose (partition) not the set P , but a collection P of its translates. Such a partition is sometimes regarded a coloring of the members of P.

show abstract

Optimally Decomposing Coverings with Translates of a Convex Polygon

Abstract: We show that any k-fold covering using translates of an arbitrary convex polygon can be decomposed into Ω(k) covers. Such a decomposition can be computed using an efficient (polynomial-time) algorithm.

Cited by 19 publications

References 11 publications

Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

Making Octants Colorful and Related Covering Decomposition Problems

Survey on Decomposition of Multiple Coverings

Contact Info

Product

Resources

About