On Ultrametricity, Data Coding, and Computation

Abstract

The triangular inequality is a defining property of a metric space, while the stronger ultrametric inequality is a defining property of an ultrametric space. Ultrametric distance is defined from p-adic valuation. It is known that ultrametricity is a natural property of spaces in the sparse limit. The implications of this are discussed in this article. Experimental results are presented which quantify how ultrametric a given metric space is. We explore the practical meaningfulness of this property of a space being ultrametric. In particular, we examine the computational implications of widely prevalent and perhaps ubiquitous ultrametricity.