Abstract
We present a deterministic kinetic data structure for the facility location problem that maintains a subset of the moving points as facilities such that, at any point of time, the accumulated cost for the whole point set is at most a constant factor larger than the optimal cost. In our scenario, each point can change its status between client and facility and moves continuously along a known trajectory in a d-dimensional Euclidean space, where d is a constant.
Our kinetic data structure requires \(\mathcal{O}(n(\log^{d}(n)+\log (nR)))\) space in total, where \(R:=\frac{\max_{p_{i}\in\mathcal{P}}{f_{i}}\cdot\max_{p_{i}\in\mathcal{P}}{d_{i}}}{\min_{p_{i}\in\mathcal {P}}{f_{i}}\cdot\min_{p_{i}\in\mathcal{P}}{d_{i}}}\) , ℘={p 1,p 2,…,p n } is the set of given points, and f i , d i are the maintenance cost and the demand of a point p i , respectively. In case that each trajectory can be described by a bounded degree polynomial, we process \(\mathcal{O}(n^{2}\log^{2}(nR))\) events, each requiring \(\mathcal{O}(\log^{d+1}(n)\cdot\log(nR))\) time and \(\mathcal {O}(\log(nR))\) status changes.
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Partially supported by the EU within FP7-ICT-2007-1 under contract no. 215270 (FRONTS), DFG-project “Smart Teams” within the SPP 1183 “Organic Computing”, and DFG grant Me 872/8-3.
A preliminary version of this paper appeared in the Proceedings of the 11th Scandinavian Workshop on Algorithm Theory (SWAT), LNCS, vol. 5124, pp. 378–389, 2008.
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Degener, B., Gehweiler, J. & Lammersen, C. Kinetic Facility Location. Algorithmica 57, 562–584 (2010). https://doi.org/10.1007/s00453-008-9250-7
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DOI: https://doi.org/10.1007/s00453-008-9250-7