Given a set P of n points in the plane and a multiset W of k weights with k leq n, we assign a weight in W to a point in P to minimize the maximum weighted distance from the weighted center of P to any point in P. In this paper, we give two algorithms which take O(k^2 n^2 log^4 n) time and O(k^5 n log^4 k + kn log^3 n) time, respectively. For a constant k, the second algorithm takes only O(n log^3 n) time, which is near-linear.
@InProceedings{oh_et_al:LIPIcs.ISAAC.2016.58, author = {Oh, Eunjin and Ahn, Hee-Kap}, title = {{Assigning Weights to Minimize the Covering Radius in the Plane}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {58:1--58:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.58}, URN = {urn:nbn:de:0030-drops-68275}, doi = {10.4230/LIPIcs.ISAAC.2016.58}, annote = {Keywords: Weighted center, facility location, weight assignment, combinatorial op- timization, computational geometry} }
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