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Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points

Karapınar Erdal, Romaguera Salvador, Tirado Pedro

Demonstratio Mathematica

2022

Vol. 55, nr 1

939--951

Involving w-distances we prove a fixed point theorem of Caristi-type in the realm of (non-necessarily T1) quasi-metric spaces. With the help of this result, a characterization of quasi-metric completeness is obtained. Our approach allows us to retrieve several key examples occurring in various fields of mathematics and computer science and that are modeled as non- T1 quasi-metric spaces. As an application, we deduce a characterization of complete G -metric spaces in terms of a weak version of Caristi’s theorem that involves a G-metric version of w-distances.

Indexing Schemes for Similarity Search: an Illustrated Paradigm

Pestov V., Stojmirović A.

Fundamenta Informaticae

2006

Vol. 70, nr 4

367-385

We suggest a variation of the Hellerstein-Koutsoupias-Papadimitriou indexability model for datasets equipped with a similarity measure, with the aim of better understanding the structure of indexing schemes for similarity-based search and the geometry of similarity workloads. This in particular provides a unified approach to a great variety of schemes used to index into metric spaces and facilitates their transfer to more general similarity measures such as quasi-metrics. We discuss links between performance of indexing schemes and high-dimensional geometry. The concepts and results are illustrated on a very large concrete dataset of peptide fragments equipped with a biologically significant similarity measure.