Abstract
We consider the problem of fitting an n× n distance matrix M by a tree metric T. We give a factor O( min {n 1/p,(klogn)1/p}) approximation algorithm for finding the closest ultrametric T under the L p norm, i.e. T minimizes ||T,M|| p . Here, k is the number of distinct distances in M. Combined with the results of [1], our algorithms imply the same factor approximation for finding the closest tree metric under the same norm. In [1], Agarwala et al. present the first approximation algorithm for this problem under L ∞ . Ma et al. [2] present approximation algorithms under the L p norm when the original distances are not allowed to contract and the output is an ultrametric. This paper presents the first algorithms with performance guarantees under L p (p<∞) in the general setting.
We also consider the problem of finding an ultrametric T that minimizes L relative: the sum of the factors by which each input distance is stretched. For the latter problem, we give a factor O(log2 n) approximation.
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Harb, B., Kannan, S., McGregor, A. (2005). Approximating the Best-Fit Tree Under L p Norms. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_11
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DOI: https://doi.org/10.1007/11538462_11
Publisher Name: Springer, Berlin, Heidelberg
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