On the Approximability of Numerical Taxonomy (Fitting Distances by Tree Metrics)
SIAM Journal on Computing
Efficient algorithms for inverting evolution
Journal of the ACM (JACM)
Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications
SIAM Journal on Computing
Evolutionary Trees Can be Learned in Polynomial Time in the Two-State General Markov Model
SIAM Journal on Computing
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Clustering with Qualitative Information
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Fitting tree metrics: Hierarchical clustering and Phylogeny
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Deterministic Pivoting Algorithms for Constrained Ranking and Clustering Problems
Mathematics of Operations Research
Fitting Tree Metrics: Hierarchical Clustering and Phylogeny
SIAM Journal on Computing
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We consider the problem of fitting an n× n distance matrix M by a tree metric T. We give a factor O( min {n1/p,(klogn)1/p}) approximation algorithm for finding the closest ultrametric T under the Lp 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 Lp 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 Lp (p We also consider the problem of finding an ultrametric T that minimizes Lrelative: the sum of the factors by which each input distance is stretched. For the latter problem, we give a factor O(log2n) approximation.