The Experts below are selected from a list of 12756 Experts worldwide ranked by ideXlab platform
Mohammad Reza Kazemi - One of the best experts on this subject based on the ideXlab platform.
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computing the smallest color spanning Axis Parallel square
International Symposium on Algorithms and Computation, 2013Co-Authors: Payam Khanteimouri, Ali Mohades, Mohammad Ali Abam, Mohammad Reza KazemiAbstract:For a given set of n colored points with k colors in the plane, we study the problem of computing the smallest color-spanning Axis-Parallel 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.
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ISAAC - Computing the Smallest Color-Spanning Axis-Parallel Square
Algorithms and Computation, 2013Co-Authors: Payam Khanteimouri, Ali Mohades, Mohammad Ali Abam, Mohammad Reza KazemiAbstract:For a given set of n colored points with k colors in the plane, we study the problem of computing the smallest color-spanning Axis-Parallel 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 - One of the best experts on this subject based on the ideXlab platform.
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finding Axis Parallel rectangles of fixed perimeter or area containing the largest number of points
Computational Geometry: Theory and Applications, 2019Co-Authors: Haim Kaplan, Micha SharirAbstract:Abstract Let P be a set of n points in the plane in general position, and consider the problem of finding an Axis-Parallel 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 Axis-Parallel 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.
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small size eps nets for Axis Parallel rectangles and boxes
SIAM Journal on Computing, 2010Co-Authors: Boris Aronov, Esther Ezra, Micha SharirAbstract: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 Axis-Parallel rectangular ranges. The same bound holds for points in the plane and “fat” triangular ranges and for point sets in $\boldsymbol{R}^3$ and Axis-Parallel 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.
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online conflict free coloring for halfplanes congruent disks and Axis Parallel rectangles
ACM Transactions on Algorithms, 2009Co-Authors: Ke Chen, Haim Kaplan, Micha SharirAbstract: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 Axis-Parallel 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 Axis-Parallel rectangles, using O(log3n) colors. This is the first efficient (i.e, using polylog(n) colors) deterministic online CF coloring algorithm for this problem.
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Binary Space Partitions for Axis-Parallel Segments, Rectangles, and Hyperrectangles
Discrete and Computational Geometry, 2004Co-Authors: Adrian Dumitrescu, Joseph S. B. Mitchell, Micha SharirAbstract:We provide a variety of new upper and lower bounds and simpler proof techniques for the efficient construction of binary space partitions (BSPs) of Axis-Parallel rectangles of various dimensions. (a) We construct a set of $n$ disjoint Axis-Parallel segments in the plane such that any binary space auto-partition has size at least $2n-o(n)$, almost matching an upper bound of d’Amore and Franciosa. (b) We establish a similar lower bound of $7n/3-o(n)$ for disjoint rectangles in the plane. (c) We simplify and improve BSP constructions of Paterson and Yao for disjoint segments in $\reals^d$ and disjoint rectangles in $\reals^3$. (d) We derive a worst-case bound of $\Theta(n^{5/3})$ for the size of BSPs of disjoint $2$-rectangles in $4$-space. (e) For disjoint $k$-rectangles in $d$-space, we prove the worst-case bound $\Theta(n^{d/(d-k)})$, for any $k
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Symposium on Computational Geometry - Binary space partitions for Axis-Parallel segments, rectangles, and hyperrectangles
Proceedings of the seventeenth annual symposium on Computational geometry - SCG '01, 2001Co-Authors: Adrian Dumitrescu, Joseph S. B. Mitchell, Micha SharirAbstract:We provide a variety of new results, including upper and lower bounds, as well as simpler proof techniques for the efficient construction of binary space partitions (BSP's) of Axis-Parallel segments, rectangles, and hyperrectangles. (a) A consequence of the analysis in \cite{dAF} is that any set of $n$ Axis-Parallel and pairwise-disjoint line segments in the plane admits a binary space partition of size at most $2n-1$. We establish a worst-case lower bound of $2n-o(n)$ for the size of such a BSP, thus showing that this bound is almost tight in the worst case. (b) We give an improved worst-case lower bound of $\frac{9}{4}n-o(n)$ on the size of a BSP for isothetic pairwise disjoint rectangles. (c) We present simple methods, with equally simple analysis, for constructing BSP's for Axis-Parallel segments in higher dimensions, simplifying the technique of \cite{PY2} and improving the constants. (d) We obtain an alternative construction (to that in \cite{PY2}) of BSP's for collections of Axis-Parallel rectangles in 3-space. (e) We present a construction of BSP's of size $O(n^{5/3})$ for $n$ Axis-Parallel pairwise disjoint 2-rectangles in $\reals^4$, and give a matching worst-case lower bound of $\Omega(n^{5/3})$ for the size of such a BSP. (f) We extend the results of \cite{PY2} to Axis-Parallel $k$-dimensional rectangles in $\reals^d$, for $k
Gábor Tardos - One of the best experts on this subject based on the ideXlab platform.
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notes coloring Axis Parallel rectangles
Journal of Combinatorial Theory Series A, 2010Co-Authors: János Pach, Gábor TardosAbstract:For every k and r, we construct a finite family of Axis-Parallel 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].
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Coloring Axis-Parallel rectangles
Journal of Combinatorial Theory. Series A, 2010Co-Authors: János Pach, Gábor TardosAbstract:For every k and r, we construct a finite family of Axis-Parallel 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.
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delaunay graphs of point sets in the plane with respect to Axis Parallel rectangles
Random Structures and Algorithms, 2009Co-Authors: Xiaomin Chen, János Pach, Mario Szegedy, Gábor TardosAbstract:Given a point set P in the plane, the Delaunay graph with respect to Axis-Parallel 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).
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delaunay graphs of point sets in the plane with respect to Axis Parallel rectangles
Symposium on Discrete Algorithms, 2008Co-Authors: Xiaomin Chen, János Pach, Mario Szegedy, Gábor TardosAbstract:Given a point set P in the plane, the Delaunay graph with respect to Axis-Parallel rectangles is a graph defined on the vertex set P, whose two points p,q ∈ 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. [ELRS03] 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 n 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 log2 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 n points almost surely has an independent set of size at least cn/ log n. We give two further applications of our methods. 1. We construct 2-dimensional n-element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o(n). This answers a question of Matousek and Přivětivý [MaP06] and improves a result of Křiž and Nesetřil [KrN91]. 2. For any positive integers c and d, we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an Axis-Parallel rectangle containing at least d points, all of which have the same color. This solves an old problem from [BrMP05].
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SODA - Delaunay graphs of point sets in the plane with respect to Axis-Parallel rectangles
2008Co-Authors: Xiaomin Chen, János Pach, Mario Szegedy, Gábor TardosAbstract:Given a point set P in the plane, the Delaunay graph with respect to Axis-Parallel rectangles is a graph defined on the vertex set P, whose two points p,q ∈ 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. [ELRS03] 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 n 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 log2 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 n points almost surely has an independent set of size at least cn/ log n. We give two further applications of our methods. 1. We construct 2-dimensional n-element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o(n). This answers a question of Matousek and Přivětivý [MaP06] and improves a result of Křiž and Nesetřil [KrN91]. 2. For any positive integers c and d, we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an Axis-Parallel rectangle containing at least d points, all of which have the same color. This solves an old problem from [BrMP05].
Adrian Dumitrescu - One of the best experts on this subject based on the ideXlab platform.
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On the Largest Empty Axis-Parallel Box Amidst n Points
Algorithmica, 2013Co-Authors: Adrian Dumitrescu, Minghui JiangAbstract:We give the first efficient (1− ε )-approximation algorithm for the following problem: Given an Axis-Parallel d -dimensional box R in ℝ^ d containing n points, compute a maximum-volume empty Axis-Parallel d-dimensional box contained in R . The minimum of this quantity over all such point sets is of the order $\Theta (\frac {1}{n} )$ . Our algorithm finds an empty Axis-aligned box whose volume is at least (1− ε ) of the maximum in O ((8 edε ^−2)^ d ⋅ n log^ d n ) time. No previous efficient exact or approximation algorithms were known for this problem for d ≥4. As the problem has been recently shown to be NP-hard in arbitrarily high dimensions ( i.e. , when d is part of the input), the existence of an efficient exact algorithm is unlikely. We also present a (1− ε )-approximation algorithm that, given an Axis-Parallel d -dimensional cube R in ℝ^ d containing n points, computes a maximum-volume empty Axis-Parallel hypercube contained in R . The minimum of this quantity over all such point sets is also shown to be of the order $\Theta (\frac{1}{n} )$ . A faster (1− ε )-approximation algorithm, with a milder dependence on d in the running time, is obtained in this case.
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on the largest empty Axis Parallel box amidst n points
Algorithmica, 2013Co-Authors: Adrian Dumitrescu, Minghui JiangAbstract:We give the first efficient (1−e)-approximation algorithm for the following problem: Given an Axis-Parallel d-dimensional box R in ℝ d containing n points, compute a maximum-volume empty Axis-Parallel d-dimensional box contained in R. The minimum of this quantity over all such point sets is of the order \(\Theta (\frac {1}{n} )\). Our algorithm finds an empty Axis-aligned box whose volume is at least (1−e) of the maximum in O((8ede −2) d ⋅nlog d n) time. No previous efficient exact or approximation algorithms were known for this problem for d≥4. As the problem has been recently shown to be NP-hard in arbitrarily high dimensions (i.e., when d is part of the input), the existence of an efficient exact algorithm is unlikely.
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on the largest empty Axis Parallel box amidst n points
arXiv: Computational Geometry, 2009Co-Authors: Adrian Dumitrescu, Minghui JiangAbstract:We give the first nontrivial upper and lower bounds on the maximum volume of an empty Axis-Parallel box inside an Axis-Parallel unit hypercube in $\RR^d$ containing $n$ points. For a fixed $d$, we show that the maximum volume is of the order $\Theta(\frac{1}{n})$. We then use the fact that the maximum volume is $\Omega(\frac{1}{n})$ in our design of the first efficient $(1-\eps)$-approximation algorithm for the following problem: Given an Axis-Parallel $d$-dimensional box $R$ in $\RR^d$ containing $n$ points, compute a maximum-volume empty Axis-Parallel $d$-dimensional box contained in $R$. The running time of our algorithm is nearly linear in $n$, for small $d$, and increases only by an $O(\log{n})$ factor when one goes up one dimension. No previous efficient exact or approximation algorithms were known for this problem for $d \geq 4$. As the problem has been recently shown to be NP-hard in arbitrary high dimensions (i.e., when $d$ is part of the input), the existence of efficient exact algorithms is unlikely. We also obtain tight estimates on the maximum volume of an empty Axis-Parallel hypercube inside an Axis-Parallel unit hypercube in $\RR^d$ containing $n$ points. For a fixed $d$, this maximum volume is of the same order order $\Theta(\frac{1}{n})$. A faster $(1-\eps)$-approximation algorithm, with a milder dependence on $d$ in the running time, is obtained in this case.
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Separating points by Axis-Parallel lines
International Journal of Computational Geometry and Applications, 2005Co-Authors: Gruia Călinescu, Adrian Dumitrescu, Howard KarloffAbstract:We study the problem of separating n points in the plane, no two of which have the same x- or y-coordinate, using a minimum number of vertical and horizontal lines avoiding the points, so that each cell of the subdivision contains at most one point. Extending previous NP-hardness results due to Freimer et al. we prove that this problem and some variants of it are APX-hard. We give a 2-approximation algorithm for this problem, and a d-approximation algorithm for the d-dimensional variant, in which the points are to be separated using Axis-Parallel hyperplanes. To this end, we reduce the point separation problem to the rectangle stabbing problem studied by Gaur et al. Their approximation algorithm uses LP-rounding. We present an alternative LP-rounding procedure which also works for the rectangle stabbing problem. We show that the integrality ratio of the LP is exactly 2.
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Binary Space Partitions for Axis-Parallel Segments, Rectangles, and Hyperrectangles
Discrete and Computational Geometry, 2004Co-Authors: Adrian Dumitrescu, Joseph S. B. Mitchell, Micha SharirAbstract:We provide a variety of new upper and lower bounds and simpler proof techniques for the efficient construction of binary space partitions (BSPs) of Axis-Parallel rectangles of various dimensions. (a) We construct a set of $n$ disjoint Axis-Parallel segments in the plane such that any binary space auto-partition has size at least $2n-o(n)$, almost matching an upper bound of d’Amore and Franciosa. (b) We establish a similar lower bound of $7n/3-o(n)$ for disjoint rectangles in the plane. (c) We simplify and improve BSP constructions of Paterson and Yao for disjoint segments in $\reals^d$ and disjoint rectangles in $\reals^3$. (d) We derive a worst-case bound of $\Theta(n^{5/3})$ for the size of BSPs of disjoint $2$-rectangles in $4$-space. (e) For disjoint $k$-rectangles in $d$-space, we prove the worst-case bound $\Theta(n^{d/(d-k)})$, for any $k
Payam Khanteimouri - One of the best experts on this subject based on the ideXlab platform.
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computing the smallest color spanning Axis Parallel square
International Symposium on Algorithms and Computation, 2013Co-Authors: Payam Khanteimouri, Ali Mohades, Mohammad Ali Abam, Mohammad Reza KazemiAbstract:For a given set of n colored points with k colors in the plane, we study the problem of computing the smallest color-spanning Axis-Parallel 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.
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ISAAC - Computing the Smallest Color-Spanning Axis-Parallel Square
Algorithms and Computation, 2013Co-Authors: Payam Khanteimouri, Ali Mohades, Mohammad Ali Abam, Mohammad Reza KazemiAbstract:For a given set of n colored points with k colors in the plane, we study the problem of computing the smallest color-spanning Axis-Parallel 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.
