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13
On the FermatWeber center of a convex object
 Comput. Geom. Theory Appl
, 2005
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Optimal freespace management and routingconscious dynamic placement for reconfigurable devices
 IEEE Transactions on Computers
, 2007
"... We describe algorithmic results on two crucial aspects of allocating resources on arraybased hardware devices with partial reconfigurability. By using methods from the field of computational geometry, we derive a method that allows correct maintainance of free and occupied space of a set of n recta ..."
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We describe algorithmic results on two crucial aspects of allocating resources on arraybased hardware devices with partial reconfigurability. By using methods from the field of computational geometry, we derive a method that allows correct maintainance of free and occupied space of a set of n rectangular modules in time O(nlog n); previous approaches needed a time of O(n 2) for correct results and O(n) for heuristic results. We also show a matching lower bound of Ω(n log n), so our approach is optimal. We also show that finding an optimal feasible communicationconscious placement (which minimizes the total weighted Manhattan distance between the new module and existing demand points) can be computed with Θ(nlog n). Both resulting algorithms are practically easy to implement and show convincing experimental behavior. ACM Classification: C.1.3: Reconfigurable Hardware; F.2.2.c: Geometrical problems and computations
Algorithms for some geometric facility location and path planning problems
, 2008
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Stochastic and Dynamic Routing Problems for multiple UAVs
"... Consider a routing problem for a team of vehicles in the plane: target points appear randomly over time in a bounded environment and must be visited by one of the vehicles. It is desired to minimize the expected system time for the targets, i.e., the expected time elapsed between the appearance of a ..."
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Consider a routing problem for a team of vehicles in the plane: target points appear randomly over time in a bounded environment and must be visited by one of the vehicles. It is desired to minimize the expected system time for the targets, i.e., the expected time elapsed between the appearance of a target point, and the instant it is visited. In this paper, such a routing problem is considered for a team of Uninhabited Aerial Vehicles (UAVs), modeled as vehicles moving with constant forward speed along paths of bounded curvature. distinct set of operating conditions. Three algorithms are presented, each designed for a Each is proven to provide a system time within a constant factor of the optimal when operating under the appropriate conditions. It is shown that the optimal routing policy depends on problem parameters such as the workload per vehicle and the vehicle density in the environment. Finally, there is discussion of a phase transition between two of the policies as the problem parameters are varied. In particular, for the case in which targets appear sporadically, a dimensionless parameter is identified which completely captures this phase transition and an estimate of the critical value of the parameter is provided.
The OneRound Voronoi Game Replayed
, 2004
"... We consider the oneround Voronoi game, where the first player (“White”, called “Wilma”) places a set of n points in a rectangular area of aspect ratio ρ ≤ 1, followed by the second player (“Black”, called “Barney”), who places the same number of points. Each player wins the fraction of A closest to ..."
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We consider the oneround Voronoi game, where the first player (“White”, called “Wilma”) places a set of n points in a rectangular area of aspect ratio ρ ≤ 1, followed by the second player (“Black”, called “Barney”), who places the same number of points. Each player wins the fraction of A closest to one of his points, and the goal is to win more than half of the total area. This problem has been studied by Cheong et al. who showed that for large enough n and ρ = 1, Barney has a strategy that guarantees a fraction of 1/2+α, for some small fixed α. We resolve a number of open problems raised by that paper. In particular, we give a precise characterization of the outcome of the game for optimal play: We show that Barney has a winning strategy for n ≥ 3 and ρ> √ 2/n, and for n = 2 and ρ> √ 3/2. Wilma wins in all remaining cases, i.e., for n ≥ 3 and ρ ≤ √ 2/n, for n = 2 and ρ ≤ √ 3/2, and for n = 1. We also discuss complexity aspects of the game on more general boards, by proving that for a polygon with holes, it is NPhard to maximize the area Barney can win against a given set of points by Wilma.
Integer Point Sets Minimizing Average Pairwise L1 Distance: What is the Optimal Shape of a Town?
, 2010
"... An ntown, n ∈ N, is a group of n buildings, each occupying a distinct position on a 2dimensional integer grid. If we measure the distance between two buildings along the axisparallel street grid, then an ntown has optimal shape if the sum of all pairwise Manhattan distances is minimized. This pr ..."
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An ntown, n ∈ N, is a group of n buildings, each occupying a distinct position on a 2dimensional integer grid. If we measure the distance between two buildings along the axisparallel street grid, then an ntown has optimal shape if the sum of all pairwise Manhattan distances is minimized. This problem has been studied for cities, i.e., the limiting case of very large n. For cities, it is known that the optimal shape can be described by a differential equation, for which no closedform solution is known. We show that optimal ntowns can be computed in O(n 7.5) time. This is also practically useful, as it allows us to compute optimal solutions up to n = 80.
Abstract The OneRound Voronoi Game Replayed
"... We consider the oneround Voronoi game, where the first player (“White”, called “Wilma”) places a set of n points in a rectangular area Q of aspect ratio ρ � 1, followed by the second player (“Black”, called “Barney”), who places the same number of points. Each player wins the fraction of Q closest ..."
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We consider the oneround Voronoi game, where the first player (“White”, called “Wilma”) places a set of n points in a rectangular area Q of aspect ratio ρ � 1, followed by the second player (“Black”, called “Barney”), who places the same number of points. Each player wins the fraction of Q closest to one of his points, and the goal is to win more than half of the total area. This problem has been studied by Cheong et al. who showed that for large enough n and ρ � 1, Barney has a strategy that guarantees a fraction of 1�2 α, for some small fixed α. We resolve a number of open problems raised by that paper. In particular, we give a precise characterization of the outcome of the game for optimal play: We show that ÔBarney has a winning strategy Ô for n � 3 and ρ � 2�n, and for n � 2 and ρ � 3�2. Wilma wins in all remaining cases, i.e., for n � 3 and ρ � Ô 2�n, for n � 2 and ρ � Ô 3�2, and for n � 1. We also discuss complexity aspects of the game on more general boards, by proving that for a polygon with holes, it is NPhard to maximize the area Barney can win against a given set of points by