Planar realizations of nonlinear Davenport-Schinzel sequences by segments
Discrete & Computational Geometry
Improved complexity bounds for center location problems on networks by using dynamic data structures
SIAM Journal on Discrete Mathematics
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
Smooth kinetic maintenance of clusters
Proceedings of the nineteenth annual symposium on Computational geometry
Kinetic data structures
Lowest common ancestors in trees and directed acyclic graphs
Journal of Algorithms
Competitive algorithms for maintaining a mobile center
Mobile Networks and Applications
Deformable spanners and applications
Computational Geometry: Theory and Applications
Geometric facility location under continuous motion: bounded-velocity approximations to the mobile euclidean k-centre and k-median problems
Nonlinearity Of Davenport-Schinzel Sequences And Of A Generalized Path Compression Scheme
SFCS '84 Proceedings of the 25th Annual Symposium onFoundations of Computer Science, 1984
Kinetic 2-centers in the black-box model
Proceedings of the twenty-ninth annual symposium on Computational geometry
| Hi-index | 0.04 |
Given a set P of points (clients) on a weighted tree T, a k-centre of P corresponds to a set of k points (facilities) on T such that the maximum graph distance between any client and its nearest facility is minimised. We consider the mobilek-centre problem on trees. 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, the motions of the corresponding 1-centre and 2-centre are piecewise linear; we derive a tight combinatorial bound of @Q(n) on the complexity of the motion of the 1-centre and corresponding bounds of O(n^2@a(n)) and @W(n^2) for a 2-centre, where @a(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(nlogn) and a 2-centre can be found in time O(n^2logn). These algorithms lend themselves to implementation within the framework of kinetic data structures. Finally, we examine properties of the mobile 1-centre on graphs and describe an optimal unit-velocity 2-approximation.