Metric and Path-Connectedness Properties of the Frechet Distance for Paths and Graphs

TL;DR

This paper explores the topological properties of spaces of paths and graphs under the Frechet distance, focusing on their connectedness, to support future applications in graph analysis and related fields.

Contribution

It establishes foundational results on the path-connectedness of spaces of paths and graphs under the Frechet distance, extending understanding of their topological structure.

Findings

Spaces of paths under Frechet distance are path-connected.

Metric balls in these spaces are analyzed for connectedness.

Theoretical groundwork for practical applications in graph analysis.

Abstract

The Frechet distance is often used to measure distances between paths, with applications in areas ranging from map matching to GPS trajectory analysis to handwriting recognition. More recently, the Frechet distance has been generalized to a distance between two copies of the same graph embedded or immersed in a metric space; this more general setting opens up a wide range of more complex applications in graph analysis. In this paper, we initiate a study of some of the fundamental topological properties of spaces of paths and of graphs mapped to R^n under the Frechet distance, in an effort to lay the theoretical groundwork for understanding how these distances can be used in practice. In particular, we prove whether or not these spaces, and the metric balls therein, are path-connected.