Any metric d:X×X on a set X must satisfy the triangle inequality:


An ultrametric must additionally satisfy a stronger version of the triangle inequality:


Here is an example of an ultrametric on a space with 5 points, labelled a,b,c,d,e:


In the table above, an entry n in the for element x and the for element y indicates that d(x,y)=n, where d is the ultrametric. By symmetry of the ultrametric (d(x,y)=d(y,x)), the above table yields all values of d(x,y) for all x,y{a,b,c,d,e}.

The ultrametric condition is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the ultrametric three point condition:

(x,y,z)x,y,z can be renamed such that d(x,z)d(x,y)=d(y,z)

Ultrametrics can be used to model bifurcating hierarchical systems.  The distancePlanetmathPlanetmath between nodes in a weight-balanced binary tree is an ultrametric. Similarly, an ultrametric can be modelled by a weight-balanced binary tree, although the choice of tree is not necessarily unique.  Tree models of ultrametrics are sometimes called ultrametric trees.

The metrics induced by non-Archimedean valuations are ultrametrics.

Title ultrametric
Canonical name Ultrametric
Date of creation 2013-03-22 13:28:28
Last modified on 2013-03-22 13:28:28
Owner Koro (127)
Last modified by Koro (127)
Numerical id 21
Author Koro (127)
Entry type Definition
Classification msc 54E35
Related topic MetricSpace
Related topic ValuationMathworldPlanetmathPlanetmath
Related topic UltrametricSpace