Metric retractions and similarity detecting algorithms

Let $\mathcal{X}=\langle X, \varrho\rangle$ be a metric space, let $A\subseteq X$ and let $\varepsilon$ be a positive real number. The similarity detecting problem is to find all $a\in A$ for which $\varrho(a, x)\leq\varepsilon$ where $x\in X$ is a given input. In this work we study the similarity detecting problem with the additional assumption that $\mathcal{X}$ is an ultrametric space of finite spectrum; these assumptions seem to be natural from the point of view of practical applications. We establish model theoretical results for ultrametric spaces. More concretely, we provide sufficient conditions for the existence of metric retractions for certain ultrametric spaces. Based on these theoretical results, we propose a similarity detecting algorithm for ultrametric spaces. The time complexity of our algorithm will be discussed, as well.