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Dispersion in disks ∗
, 2011
"... We present three new approximation algorithms with improved constant ratios for selecting n points in n disks such that the minimum pairwise distance among the points is maximized. (1) A very simple O(n log n)time algorithm with ratio 0.511 for disjoint unit disks. (2) An LPbased algorithm with ra ..."
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We present three new approximation algorithms with improved constant ratios for selecting n points in n disks such that the minimum pairwise distance among the points is maximized. (1) A very simple O(n log n)time algorithm with ratio 0.511 for disjoint unit disks. (2) An LPbased algorithm with ratio 0.707 for disjoint disks of arbitrary radii that uses a linear number of variables and constraints, and runs in polynomial time. (3) A hybrid algorithm with ratio either 0.4487 or 0.4674 for (not necessarily disjoint) unit disks that uses an algorithm of Cabello in combination with either the simple O(n log n)time algorithm or the LPbased algorithm. The LP algorithm can be extended for disjoint balls of arbitrary radii in R d, for any (fixed) dimension d, while preserving the features of the planar algorithm. The algorithm introduces a novel technique which combines linear programming and projections for approximating Euclidean distances. The previous best approximation ratio for dispersion in disjoint disks, even when all disks have the same radius, was 1/2. Our results give a positive answer to an open question raised by Cabello, who asked whether the ratio 1/2 could be improved.
1.1 Introduction to Movement Problems
, 2011
"... Consider a scenario where a group of automated robots with limited mobility, energy, and so on need to reorganize their joint position in order to form a reliable radio network connecting two locations. Since resources are scarce, we wish to minimize the movement required to establish the radio netw ..."
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Consider a scenario where a group of automated robots with limited mobility, energy, and so on need to reorganize their joint position in order to form a reliable radio network connecting two locations. Since resources are scarce, we wish to minimize the movement required to establish the radio network. This,