Abstract
The maximum diameter color-spanning set problem (MaxDCS) is defined as follows: given n points with m colors, select m points with m distinct colors such that the diameter of the set of chosen points is maximized. In this paper, we design an optimal O(n log n) time algorithm using rotating calipers for MaxDCS in the plane. Our algorithm can also be used to solve the maximum diameter problem of imprecise points modeled as polygons. We also give an optimal algorithm for the all farthest foreign neighbor problem (AFFN) in the plane, and propose algorithms to answer the farthest foreign neighbor query (FFNQ) of colored sets in two- and three-dimensional space. Furthermore, we study the problem of computing the closest pair of color-spanning set (CPCS) in d-dimensional space, and remove the log m factor in the best known time bound if d is a constant.
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This research was supported by the International Science and Technology Cooperation Program of China under Grant No. 2010DFA92720, and the National Natural Science Foundation of China under Grant Nos. 11271351, 60928006, and 61379087. A preliminary version of the paper was published in the Proceedings of COCOA 2013.
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Fan, CL., Luo, J., Wang, WC. et al. On Some Proximity Problems of Colored Sets. J. Comput. Sci. Technol. 29, 879–886 (2014). https://doi.org/10.1007/s11390-014-1475-0
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DOI: https://doi.org/10.1007/s11390-014-1475-0