Finite metric spaces--combinatorics, geometry and algorithms

TL;DR

This paper reviews recent advances in the study of finite metric spaces, highlighting their importance in data analysis, graph theory, algorithms, and geometric structures, and exploring their diverse applications.

Contribution

It provides a comprehensive overview of recent developments connecting finite metric spaces with combinatorics, geometry, and algorithms, emphasizing their broad relevance.

Findings

Finite metric spaces encode significant information in graphs.

Submetrics of geometric objects reflect properties of the underlying structures.

Recent advances have expanded understanding of metric spaces in various fields.

Abstract

Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information. Metric spaces also come up in many recent advances in the theory of algorithms. Finally, finite submetrics of classical geometric objects such as normed spaces or manifolds reflect many important properties of the underlying structure. In this paper we review some of the recent advances in this area.