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...oint x such that the Voronoi region of x is as large as possible. The area of Voronoi regions has been considered before in the context of games, such as the Voronoi game [1, 5] or the Hotelling game =-=[15]-=-. As far as we know, the only previous paper discussing maximizing the Voronoi region of a new point is by Dehne et al. [9], who show that the area function has only a single local maximum inside a re...

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..., and wish to find a set of points (guards) so that each point of P is seen by least one guard. This problem is NP-hard. Art-gallery problems have attracted a lot of research in the last thirty years =-=[16, 17]-=-. A natural heuristic for solving art-gallery problems is to use a greedy approach based on area: We first find a guard that maximizes the area seen, next find a guard that sees the maximal area not s...

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..., and wish to find a set of points (guards) so that each point of P is seen by least one guard. This problem is NP-hard. Art-gallery problems have attracted a lot of research in the last thirty years =-=[16, 17]-=-. A natural heuristic for solving art-gallery problems is to use a greedy approach based on area: We first find a guard that maximizes the area seen, next find a guard that sees the maximal area not s...

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...algorithm that finds a (1 − δ)-approximation in time O((n 2 /δ 4 ) log 3 (n/δ)) with high probability. We also show that approximating the largest visible polygon up to a constant factor is 3sum-hard =-=[11]-=-, implying that our algorithm is probably close to optimal as far as the dependency on n is concerned. Our second problem is motivated by the task of placing a new supermarket such that it takes over ...

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... P and Q under translations or rigid motions. The area of overlap (or the area of the symmetric difference) of two planar regions is a natural measure of their similarity that is insensitive to noise =-=[3, 7]-=-. Mount et al. [13] first studied the function mapping a translation vector to the area of overlap of a translated simple polygon P with another simple polygon Q, showing that it is continuous and pie...

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... P and Q under translations or rigid motions. The area of overlap (or the area of the symmetric difference) of two planar regions is a natural measure of their similarity that is insensitive to noise =-=[3, 7]-=-. Mount et al. [13] first studied the function mapping a translation vector to the area of overlap of a translated simple polygon P with another simple polygon Q, showing that it is continuous and pie...

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...ur task is indeed to find a point x such that the Voronoi region of x is as large as possible. The area of Voronoi regions has been considered before in the context of games, such as the Voronoi game =-=[1, 5]-=- or the Hotelling game [15]. As far as we know, the only previous paper discussing maximizing the Voronoi region of a new point is by Dehne et al. [9], who show that the area function has only a singl...

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...ations or rigid motions. The area of overlap (or the area of the symmetric difference) of two planar regions is a natural measure of their similarity that is insensitive to noise [3, 7]. Mount et al. =-=[13]-=- first studied the function mapping a translation vector to the area of overlap of a translated simple polygon P with another simple polygon Q, showing that it is continuous and piecewise polynomial o...

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...ea of overlap of two collections of triangles in time O(n 8 ). A (1 − ε)approximation algorithm for the case of convex polygons with running time O((log n)/ε + (1/ε) log(1/ε)) was given by Ahn et al. =-=[2]-=-. Finally, de Berg et al. [8] consider the case where P and Q are disjoint unions of m and n unit disks, with m ≤ n. They compute a (1 − ε)-approximation for the maximal area of overlap of P and Q und...

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...the context of games, such as the Voronoi game [1, 5] or the Hotelling game [15]. As far as we know, the only previous paper discussing maximizing the Voronoi region of a new point is by Dehne et al. =-=[9]-=-, who show that the area function has only a single local maximum inside a region where the set of Voronoi 1 Joint work with Alon Efrat and Sariel Har-Peled. An earlier version of this work has appear...

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...ur task is indeed to find a point x such that the Voronoi region of x is as large as possible. The area of Voronoi regions has been considered before in the context of games, such as the Voronoi game =-=[1, 5]-=- or the Hotelling game [15]. As far as we know, the only previous paper discussing maximizing the Voronoi region of a new point is by Dehne et al. [9], who show that the area function has only a singl...

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... first find a guard that maximizes the area seen, next find a guard that sees the maximal area not seen by the first guard, and so on until each point of P is seen by some guard. Ntafos and Tsoukalas =-=[14]-=- show how to find, for any δ > 0, a guard that sees an area of size (1 − δ)µopt. Their algorithm requires O(n 5 /δ 2 ) time in the worst case. We give a probabilistic algorithm that finds a (1 − δ)-ap...

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...ons of triangles in time O(n 8 ). A (1 − ε)approximation algorithm for the case of convex polygons with running time O((log n)/ε + (1/ε) log(1/ε)) was given by Ahn et al. [2]. Finally, de Berg et al. =-=[8]-=- consider the case where P and Q are disjoint unions of m and n unit disks, with m ≤ n. They compute a (1 − ε)-approximation for the maximal area of overlap of P and Q under translations in time O((nm...