Abstract
The standard dynamic programming solution to finding k-medians on a line with n nodes requires O(kn 2) time. Dynamic programming speed-up techniques, e.g., use of the quadrangle inequality or properties of totally monotone matrices, can reduce this to O(kn) time but these techniques are inherently static. The major result of this paper is to show that we can maintain the dynamic programming speedup in an online setting where points are added from left to right on a line. Computing the new k-medians after adding a new point takes only O(k) amortized time and O(k log n) worst case time (simultaneously). Using similar techniques, we can also solve the online k -coverage with uniform coverage on a line problem with the same time bounds.
This work partially supported by Hong Kong RGC grants HKUST6010/01E, HKUST6162/00E, HKUST6082/01E and HKUST6206/02. The authors would like to thank Gerhard Trippen for his help in proofreading and latexing the figures.
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© 2004 Springer-Verlag Berlin Heidelberg
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Fleischer, R., Golin, M.J., Zhang, Y. (2004). Online Maintenance of k-Medians and k-Covers on a Line. In: Hagerup, T., Katajainen, J. (eds) Algorithm Theory - SWAT 2004. SWAT 2004. Lecture Notes in Computer Science, vol 3111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27810-8_10
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DOI: https://doi.org/10.1007/978-3-540-27810-8_10
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