Rectilinear and polygonal p-piercing and p-center problems
Proceedings of the twelfth annual symposium on Computational geometry
Rectilinear p-piercing problems
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Faster construction of planar two-centers
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Data structures for mobile data
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
More planar two-center algorithms
Computational Geometry: Theory and Applications
Approximate clustering via core-sets
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On Piercing Sets of Axis-Parallel Rectangles and Rings
ESA '97 Proceedings of the 5th Annual European Symposium on Algorithms
Deformable spanners and applications
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Discrete & Computational Geometry
Dense Point Sets Have Sparse Delaunay Triangulations or “... But Not Too Nasty”
Discrete & Computational Geometry
Kinetic maintenance of mobile k-centres on trees
Discrete Applied Mathematics
Algorithmica - Special Issue: Scandinavian Workshop on Algorithm Theory; Guest Editor: Joachim Gudmundsson
Approximation algorithm for the kinetic robust K-center problem
Computational Geometry: Theory and Applications
Kinetic convex hulls and delaunay triangulations in the black-box model
Proceedings of the twenty-seventh annual symposium on Computational geometry
Survey of clustering algorithms
IEEE Transactions on Neural Networks
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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We study two versions of the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P; the rectilinear 2-center problem correspondingly asks for two congruent axis-aligned squares of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_{max} on the maximum displacement of any point within one time step. We show how to maintain the rectilinear 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. For the Euclidean 2-center we give a similar result: we can maintain in amortized sub-linear time (again under certain assumptions on the distribution) a (1+ε)-approximation of the optimal 2-center. In many cases---namely when the distance between the centers of the disks is relatively large or relatively small---the solution we maintain is actually optimal. We also present results for the case where the maximum speed of the centers is bounded. We describe a simple scheme to maintain a 2-approximation of the rectilinear 2-center, and we provide a scheme which gives a better approximation factor depending on several parameters of the point set and the maximum allowed displacement of the centers. The latter result can be used to obtain a 2.29-approximation for the Euclidean 2-center; this is an improvement over the previously best known bound of 8/π approx 2.55. These algorithms run in amortized sub-linear time per time step, as before under certain assumptions on the distribution.