Embedding ultrametrics into low-dimensional spaces

We study the problem of minimum-distortion embedding of ultrametrics into the plane and higher dimensional spaces. Ultrametrics are a natural class of metrics that frequently occur in applications involving hierarchical clustering. Low-distortion embeddings of ultrametrics into the plane help visualizing complex structures they often represent. Given an ultrametric, a natural question is whether we can efficiently find an optimal-distortion embedding of this ultrametric into the plane, and if not, whether we can design an efficient algorithm that produces embeddings with near-optimal distortion. We show that the problem of finding minimum-distortion embedding of ultrametrics into the plane is NP-hard, and thus approximation algorithms are called for. Given an input ultrametric M, let c denote the minimum distortion achievable by any embedding of M into the plane. Our main result is a linear-time algorithm that produces an O(c3 )-distortion embedding. This result can be generalized...

• Type: Conference
• Year: 2006
• Where: COMPGEOM
• Authors: Mihai Badoiu, Julia Chuzhoy, Piotr Indyk, Anastasios Sidiropoulos
• Keywords: Algorithms, COMPGEOM 2006, Embedding, Minimum-distortion Embedding