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Axis Parallel

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Mohammad Reza Kazemi – 1st expert on this subject based on the ideXlab platform

  • computing the smallest color spanning Axis Parallel square
    International Symposium on Algorithms and Computation, 2013
    Co-Authors: Payam Khanteimouri, Ali Mohades, Mohammad Ali Abam, Mohammad Reza Kazemi

    Abstract:

    For a given set of n colored points with k colors in the plane, we study the problem of computing the smallest color-spanning AxisParallel square. First, for a dynamic set of colored points on the real line, we propose a dynamic structure with O(log2 n) update time per insertion and deletion for maintaining the smallest color-spanning interval. Next, we use this result to compute the smallest color-spanning square. Although we show there could be Ω(kn) minimal color-spanning squares, our algorithm runs in O(nlog2 n) time and O(n) space.

  • ISAAC – Computing the Smallest Color-Spanning AxisParallel Square
    Algorithms and Computation, 2013
    Co-Authors: Payam Khanteimouri, Ali Mohades, Mohammad Ali Abam, Mohammad Reza Kazemi

    Abstract:

    For a given set of n colored points with k colors in the plane, we study the problem of computing the smallest color-spanning AxisParallel square. First, for a dynamic set of colored points on the real line, we propose a dynamic structure with O(log2 n) update time per insertion and deletion for maintaining the smallest color-spanning interval. Next, we use this result to compute the smallest color-spanning square. Although we show there could be Ω(kn) minimal color-spanning squares, our algorithm runs in O(nlog2 n) time and O(n) space.

Micha Sharir – 2nd expert on this subject based on the ideXlab platform

  • finding Axis Parallel rectangles of fixed perimeter or area containing the largest number of points
    Computational Geometry: Theory and Applications, 2019
    Co-Authors: Haim Kaplan, Micha Sharir

    Abstract:

    Abstract Let P be a set of n points in the plane in general position, and consider the problem of finding an AxisParallel rectangle with a given perimeter, or area, or diagonal, that encloses the maximum number of points of P. We present an exact algorithm that finds such a rectangle in O ( n 5 / 2 log ⁡ n ) time, and, for the case of a fixed perimeter or diagonal, we also obtain (i) an improved exact algorithm that runs in O ( n k 3 / 2 log ⁡ k ) time, and (ii) an approximation algorithm that finds, in O ( n + n k e 5 log 5 / 2 ⁡ n k log ⁡ ( 1 e log ⁡ n k ) ) time, a rectangle of the given perimeter that contains at least ( 1 − e ) k points of P, where k is the optimum value. We then show how to turn this algorithm into one that finds, for a given k, an AxisParallel rectangle of smallest perimeter (or area, or diagonal) that contains k points of P. We obtain the first subcubic algorithms for these problems, significantly improving the current state of the art.

  • small size eps nets for Axis Parallel rectangles and boxes
    SIAM Journal on Computing, 2010
    Co-Authors: Boris Aronov, Esther Ezra, Micha Sharir

    Abstract:

    We show the existence of $\varepsilon$-nets of size $O\left(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon}\right)$ for planar point sets and AxisParallel rectangular ranges. The same bound holds for points in the plane and “fat” triangular ranges and for point sets in $\boldsymbol{R}^3$ and AxisParallel boxes; these are the first known nontrivial bounds for these range spaces. Our technique also yields improved bounds on the size of $\varepsilon$-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of $\varepsilon$-nets of size $O\left(\frac{1}{\varepsilon}\log\log\log\frac{1}{\varepsilon}\right)$ for the dual range space of “fat” regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Bronnimann and Goodrich or of Even, Rawitz, and Shahar, we obtain improved approximation factors (computable in expected polynomial time by a randomized algorithm) for the hitting set or the set cover problems associated with the corresponding range spaces.

  • online conflict free coloring for halfplanes congruent disks and Axis Parallel rectangles
    ACM Transactions on Algorithms, 2009
    Co-Authors: Ke Chen, Haim Kaplan, Micha Sharir

    Abstract:

    We present randomized algorithms for online conflict-free coloring (CF in short) of points in the plane, with respect to halfplanes, congruent disks, and nearly-equal AxisParallel rectangles. In all three cases, the coloring algorithms use O(log n) colors, with high probability. We also present a deterministic algorithm for online CF coloring of points in the plane with respect to nearly-equal AxisParallel rectangles, using O(log3n) colors. This is the first efficient (i.e, using polylog(n) colors) deterministic online CF coloring algorithm for this problem.

Gábor Tardos – 3rd expert on this subject based on the ideXlab platform

  • notes coloring Axis Parallel rectangles
    Journal of Combinatorial Theory Series A, 2010
    Co-Authors: János Pach, Gábor Tardos

    Abstract:

    For every k and r, we construct a finite family of AxisParallel rectangles in the plane such that no matter how we color them with k colors, there exists a point covered by precisely r members of the family, all of which have the same color. For r=2, this answers a question of S. Smorodinsky [S. Smorodinsky, On the chromatic number of some geometric hypergraphs, SIAM J. Discrete Math. 21 (2007) 676-687].

  • Coloring AxisParallel rectangles
    Journal of Combinatorial Theory. Series A, 2010
    Co-Authors: János Pach, Gábor Tardos

    Abstract:

    For every k and r, we construct a finite family of AxisParallel rectangles in the plane such that no matter how we color them with k colors, there exists a point covered by precisely r members of the family, all of which have the same color. For r=2, this answers a question of S. Smorodinsky [S. Smorodinsky, On the chromatic number of some geometric hypergraphs, SIAM J. Discrete Math. 21 (2007) 676-687]. © 2009 Elsevier Inc.

  • delaunay graphs of point sets in the plane with respect to Axis Parallel rectangles
    Random Structures and Algorithms, 2009
    Co-Authors: Xiaomin Chen, János Pach, Mario Szegedy, Gábor Tardos

    Abstract:

    Given a point set P in the plane, the Delaunay graph with respect to AxisParallel rectangles is a graph defined on the vertex set P, whose two points p, q is an element of P are connected by an edge if and only if there is a rectangle Parallel to the coordinate axes that contains p and q, but no other elements of P. The following question of Even et al. (SIAM J Comput 33 (2003) 94-136) was motivated by a frequency assignment problem in cellular telephone networks: Does there exist a constant c > 0 such that the Delaunay graph of any set of it points in general position in the plane contains an independent set of size at least cn? We answer this question in the negative, by proving that the largest independent set in a randomly and uniformly selected point set in the unit square is O(n log(2) log n/ log n), with probability tending to 1. We also show that our bound is not far from optimal, as the Delaunay graph of a uniform random set of it points almost surely has an independent set of size at least cn log log n/(log n log log log it).