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
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.
- Title
- Approximating the Best-Fit Tree Under L p Norms
- Book Title
- Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques
- Book Subtitle
- 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th International Workshop on Randomization and Computation, RANDOM 2005, Berkeley, CA, USA, August 22-24, 2005. Proceedings
- Pages
- pp 123-133
- Copyright
- 2005
- DOI
- 10.1007/11538462_11
- Print ISBN
- 978-3-540-28239-6
- Online ISBN
- 978-3-540-31874-3
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- 3624
- Series ISSN
- 0302-9743
- Publisher
- Springer Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
- Additional Links
- Topics
- Industry Sectors
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- Editors
-
- Chandra Chekuri (16)
- Klaus Jansen (17)
- José D. P. Rolim (18)
- Luca Trevisan (19)
- Editor Affiliations
-
- 16. Dept. of Computer Science, University of Illinois
- 17. Institute for Computer Science, University of Kiel
- 18. Battelle Bâtiment A, Centre Universitaire d’Informatique
- 19. UC Berkeley
- Authors
-
- Boulos Harb (20)
- Sampath Kannan (20)
- Andrew McGregor (20)
- Author Affiliations
-
- 20. Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA, 19104, USA
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