Chapter

Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques

Volume 3624 of the series Lecture Notes in Computer Science pp 123-133

Approximating the Best-Fit Tree Under L p Norms

  • Boulos HarbAffiliated withDepartment of Computer and Information Science, University of Pennsylvania
  • , Sampath KannanAffiliated withDepartment of Computer and Information Science, University of Pennsylvania
  • , Andrew McGregorAffiliated withDepartment of Computer and Information Science, University of Pennsylvania

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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.