Online Maintenance of k-Medians and k-Covers on a Line

Authors:
Rudolf Fleischer;Mordecai J. Golin;Yan Zhang
Affiliations:
Shanghai Key Laboratory of Intelligent Information Processing, Department of Computer Science and Engineering, Fudan University, Shanghai, People’s Republic of China;Department of Computer Science, Hong Kong UST, Clear Water Bay, Kowloon, Hong Kong;Department of Computer Science, Hong Kong UST, Clear Water Bay, Kowloon, Hong Kong
Venue:
Algorithmica
Year:
2006

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Abstract

The standard dynamic programming solution to finding k-medians on a line with n nodes requires O(kn2) 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. However, these speed-up techniques are inherently static and cannot be used in an online setting, i.e., if we want to increase the size of the problem by one new point. Then, in the worst case, we could do no better than recalculating the solution to the entire problem from scratch in O(kn) time. The major result of this paper is to show that we can maintain the dynamic programming speed up 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.