A quasi-metric is a distance function which satisfies the triangle inequality
but is not symmetric: it can be thought of as an asymmetric metric. The central
result of this thesis, developed in Chapter 3, is that a natural correspondence
exists between similarity measures between biological (nucleotide or protein)
sequences and quasi-metrics.
Chapter 2 presents basic concepts of the theory of quasi-metric spaces and
introduces a new examples of them: the universal countable rational
quasi-metric space and its bicompletion, the universal bicomplete separable
quasi-metric space. Chapter 4 is dedicated to development of a notion of the
quasi-metric space with Borel probability measure, or pq-space. The main result
of this chapter indicates that `a high dimensional quasi-metric space is close
to being a metric space’.
Chapter 5 investigates the geometric aspects of the theory of database
similarity search in the context of quasi-metrics. The results about