Finite Ultrametric Spaces and Computer Science

Abstract

The purpose of the paper is to describe a few properties of ultrametric spaces (in particular, of finite ones) and to demonstrate some applications of these properties to computer science.

A metric space (X, d) is called ultrametric [6] (or non-Archimedean [4], or isosceles [9]) if its metric satisfies the strong triangle axiom:

$$ d\left( {x,z} \right) \leqslant \max \left[ {d\left( {x,y} \right),d\left( {y,z} \right)} \right]. $$

(Δ)

This is usually called the Ultrametric Axiom. Ultrametric spaces were described up to homeomorphism in [3, 21], up to uniform equivalence in [10], and up to isometry in [9, 20]. A survey of their metric [9, 20], geometric [14, 20], uniform [10], and categorical [11–17] properties can be found in the literature. The theory of ultrametric spaces is closely connected with various branches of mathematics. These are number theory (rings Z _p and fields Q _p of p-adic numbers), algebra (non-Archimedean normed fields), real analysis (the Baire space ${B_{{\aleph _o}}}$), general topology (generalized Baire spaces B _τ ), p-adic analysis (field Ω), p-adic functional analysis (algebras of Ω-valued functions), lattice theory [17], Lebesgue measure theory [18], Euclidean geometry [14], category theory and topoi [13, 15, 16], and so on. These relations deal with infinite ultrametric spaces (mainly separable). For applications in computer science, finite spaces are of interest as well.