16 research outputs found

    Polychromatic Coloring for Half-Planes

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    We prove that for every integer kk, every finite set of points in the plane can be kk-colored so that every half-plane that contains at least 2k−12k-1 points, also contains at least one point from every color class. We also show that the bound 2k−12k-1 is best possible. This improves the best previously known lower and upper bounds of 43k\frac{4}{3}k and 4k−14k-1 respectively. We also show that every finite set of half-planes can be kk colored so that if a point pp belongs to a subset HpH_p of at least 3k−23k-2 of the half-planes then HpH_p contains a half-plane from every color class. This improves the best previously known upper bound of 8k−38k-3. Another corollary of our first result is a new proof of the existence of small size \eps-nets for points in the plane with respect to half-planes.Comment: 11 pages, 5 figure

    Octants are cover-decomposable into many coverings

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    We prove that octants are cover-decomposable into multiple coverings, i.e., for any k there is an m(k)m(k) such that any m(k)m(k)-fold covering of any subset of the space with a finite number of translates of a given octant can be decomposed into k coverings. As a corollary, we obtain that any m(k)m(k)-fold covering of any subset of the plane with a finite number of homothetic copies of a given triangle can be decomposed into k coverings. Previously only some weaker bounds were known for related problems [20]

    Colorful Strips

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    Given a planar point set and an integer kk, we wish to color the points with kk colors so that any axis-aligned strip containing enough points contains all colors. The goal is to bound the necessary size of such a strip, as a function of kk. We show that if the strip size is at least 2k−12k{-}1, such a coloring can always be found. We prove that the size of the strip is also bounded in any fixed number of dimensions. In contrast to the planar case, we show that deciding whether a 3D point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. We also consider the problem of coloring a given set of axis-aligned strips, so that any sufficiently covered point in the plane is covered by kk colors. We show that in dd dimensions the required coverage is at most d(k−1)+1d(k{-}1)+1. Lower bounds are given for the two problems. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. Finally, we study a variant where strips are replaced by wedges

    More on Decomposing Coverings by Octants

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    In this note we improve our upper bound given earlier by showing that every 9-fold covering of a point set in the space by finitely many translates of an octant decomposes into two coverings, and our lower bound by a construction for a 4-fold covering that does not decompose into two coverings. The same bounds also hold for coverings of points in R2\R^2 by finitely many homothets or translates of a triangle. We also prove that certain dynamic interval coloring problems are equivalent to the above question

    Set It and Forget It: Approximating the Set Once Strip Cover Problem

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    We consider the Set Once Strip Cover problem, in which n wireless sensors are deployed over a one-dimensional region. Each sensor has a fixed battery that drains in inverse proportion to a radius that can be set just once, but activated at any time. The problem is to find an assignment of radii and activation times that maximizes the length of time during which the entire region is covered. We show that this problem is NP-hard. Second, we show that RoundRobin, the algorithm in which the sensors simply take turns covering the entire region, has a tight approximation guarantee of 3/2 in both Set Once Strip Cover and the more general Strip Cover problem, in which each radius may be set finitely-many times. Moreover, we show that the more general class of duty cycle algorithms, in which groups of sensors take turns covering the entire region, can do no better. Finally, we give an optimal O(n^2 log n)-time algorithm for the related Set Radius Strip Cover problem, in which all sensors must be activated immediately.Comment: briefly announced at SPAA 201

    Hitting sets and colorings of hypergraphs

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    In this paper we study the minimal size of edges in hypergraph families which guarantees the existence of a polychromatic coloring, that is, a kk-coloring of a vertex set such that every hyperedge contains a vertex of all kk color classes. We also investigate the connection of this problem with cc-shallow hitting sets: sets of vertices that intersect each hyperedge in at least one and at most cc vertices. We determine in some hypergraph families the minimal cc for which a cc-shallow hitting set exists. We also study this problem for a special hypergraph family, which is induced by arithmetic progressions with a difference from a given set. We show connections between some geometric hypergraph families and the latter, and prove relations between the set of differences and polychromatic colorability

    Proper Coloring of Geometric Hypergraphs

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    We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored so that anymember ofF that contains at leastm points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then such an m exists. We prove this in the special case when F is the family of all homothetic copies of a given convex polygon. We also study the problem in higher dimensions

    Indecomposable Coverings with Concave Polygons

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    We show that for any concave polygon that has no parallel sides and for any k, there is a k-fold covering of some point set by the translates of this polygon that cannot be decomposed into two coverings. Moreover, we give a complete classification of open polygons with this property. We also construct for any polytope (having dimension at least three) and for any k, a k-fold covering of the space by its translates that cannot be decomposed into two coverings

    Maximizing Barrier Coverage Lifetime with Mobile Sensors

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    Sensor networks are ubiquitously used for detection and tracking and as a result covering is one of the main tasks of such networks. We study the problem of maximizing the coverage lifetime of a barrier by mobile sensors with limited battery powers, where the coverage lifetime is the time until there is a breakdown in coverage due to the death of a sensor. Sensors are first deployed and then coverage commences. Energy is consumed in proportion to the distance traveled for mobility, while for coverage, energy is consumed in direct proportion to the radius of the sensor raised to a constant exponent. We study two variants which are distinguished by whether the sensing radii are given as part of the input or can be optimized, the fixed radii problem and the variable radii problem. We design parametric search algorithms for both problems for the case where the final order of the sensors is predetermined and for the case where sensors are initially located at barrier endpoints. In contrast, we show that the variable radii problem is strongly NP-hard and provide hardness of approximation results for fixed radii for the case where all the sensors are initially co-located at an internal point of the barrier
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