Distance measures for embedded graphs

We introduce new distance measures for comparing straight-line embedded graphs based on the Fréchet distance and the weak Fréchet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into acc...

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Bibliographic Details
Published in:Computational geometry : theory and applications 2021-04, Vol.95, p.101743, Article 101743
Main Authors: Akitaya, Hugo A., Buchin, Maike, Kilgus, Bernhard, Sijben, Stef, Wenk, Carola
Format: Article
Language:English
Subjects:
Embedded graphs
Fréchet distance
Graph comparison
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Summary:We introduce new distance measures for comparing straight-line embedded graphs based on the Fréchet distance and the weak Fréchet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into account. We present a general algorithmic approach for computing these graph distances. Although we show that deciding the distances is NP-hard for general embedded graphs, we prove that our approach yields polynomial time algorithms if the graphs are trees, and for the distance based on the weak Fréchet distance if the graphs are planar embedded and if the embedding meets a certain geometric restriction. Moreover, we prove that deciding the distances based on the Fréchet distance remains NP-hard for planar embedded graphs and show how our general algorithmic approach yields an exponential time algorithm and a polynomial time approximation algorithm for this case.
ISSN:0925-7721
DOI:10.1016/j.comgeo.2020.101743