The Frechet distance is often used to measure distances between paths, with applications in areas ranging from map matching to GPS trajectory analysis to hand- writing 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.
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Path-Connectivity of Fréchet Spaces of Graphs
We examine topological properties of spaces of paths and graphs
mapped to $\R^d$ under the Fr\'echet distance. We show that these spaces
are path-connected if the map is
either continuous or an immersion. If the map is an embedding, we show
that the space of paths is path-connected, while the space of graphs only
maintains this property in dimensions four or higher.
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- NSF-PAR ID:
- 10351571
- Date Published:
- Journal Name:
- Young Researcher's Forum (CG Week)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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