Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition

TL;DR

This paper introduces efficient approximation algorithms for hierarchical clustering and pants decomposition in metric, Euclidean, and hyperbolic spaces, optimizing sums of spanning tree lengths and perimeters with applications to geometric partitioning.

Contribution

It presents novel constant factor approximation algorithms for clustering and pants decomposition in various geometric spaces, unifying methods for Euclidean and hyperbolic geometries.

Findings

Algorithms achieve constant factor approximation guarantees.

Provides Euclidean square pants decomposition with minimal total length.

Extends to hyperbolic spaces, combining with tree clustering techniques.

Abstract

We provide efficient constant factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces.