Fitting Distances by Tree Metrics Minimizing the Total Error within a Constant Factor

Abstract

We consider the numerical taxonomy problem of fitting a positive distance function $D:\left(\genfrac{}{}{0ex}{}{S}{2}\right)\to {\mathbb{R}}_{>0}$ by a tree metric. We want a tree $T$ with positive edge weights and including $S$ among the vertices so that their distances in $T$ match those in $D$. A nice application is in evolutionary biology where the tree $T$ aims to approximate the branching process leading to the observed distances in $D$ [Cavalli-Sforza and Edwards 1967]. We consider the total error, that is the sum of distance errors over all pairs of points. We present a deterministic polynomial time algorithm minimizing the total error within a constant factor. We can do this both for general trees, and for the special case of ultrametrics with a root having the same distance to all vertices in $S$. The problems are APX-hard, so a constant factor is the best we can hope for in polynomial time. The best previous approximation factor was $O((\log n)(\log \log n))$ by Ailon and Charikar [2005] who wrote "Determining whether an $O(1)$ approximation can be obtained is a fascinating question".

46 pages, Accepted to FOCS 2021 (Full version)

References in corpus (6)

• From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering
• Hierarchical Clustering via Spreading Metrics
• Hierarchical Clustering for Euclidean Data
• Fitting Distances by Tree Metrics Minimizing the Total Error within a Constant Factor
• Bisect and Conquer: Hierarchical Clustering via Max-Uncut Bisection
• On Efficient Low Distortion Ultrametric Embedding

Cited by in corpus (1)

• Fitting Distances by Tree Metrics Minimizing the Total Error within a Constant Factor