On Ultrametricity,Data Coding,and Computation

The triangular inequality is a defining property of a metric space, while thestronger ultrametric inequality is a defining property of an ultrametric space. Ultrametricdistance is defined from p-adic valuation. It is known that ultrametricity is a naturalproperty 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 spaceis. We explore the practical meaningfulness of this property of a space being ultrametric.In particular, we examine the computational implications of widely prevalent and perhapsubiquitous ultrametricity.