Finding the upper envelope of n line segments in O(n log n) time
Information Processing Letters
A practical evaluation of kinetic data structures
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Kinetic data structures: a state of the art report
WAFR '98 Proceedings of the third workshop on the algorithmic foundations of robotics on Robotics : the algorithmic perspective: the algorithmic perspective
Data structures for mobile data
Journal of Algorithms
Mobile facility location (extended abstract)
DIALM '00 Proceedings of the 4th international workshop on Discrete algorithms and methods for mobile computing and communications
Maintaining approximate extent measures of moving points
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Kinetic data structures
Geometric facility location under continuous motion: bounded-velocity approximations to the mobile euclidean k-centre and k-median problems
Smooth kinetic maintenance of clusters
Computational Geometry: Theory and Applications - Special issue on the 19th annual symposium on computational geometry - SoCG 2003
Deformable spanners and applications
Computational Geometry: Theory and Applications
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Let C denote a set of n mobile clients, each of which follows a continuous trajectory on a weighted tree T. We establish tight bounds on the maximum relative velocity of the 1-centre and 2-centre of C. When each client in C moves with linear motion along a path on T we derive a tight bound of Θ(n) on the complexity of the motion of the 1-centre and corresponding bounds of O(n2 α(n)) and Ω(n2) for a 2- centre, where α(n) denotes the inverse Ackermann function. We describe efficient algorithms for calculating the trajectories of the 1-centre and 2- centre of C: the 1-centre can be found in optimal time O(n log n) when the distance function between mobile clients is known or O(n2) when the function must be calculated, and a 2-centre can be found in time O(n2 log n). These algorithms lend themselves to implementation within the framework of kinetic data structures, resulting in structures that are compact, efficient, responsive, and local.