## On the generalized fréchet distance and its applications

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2022
##### Publication type
Contribution to a conference collection
Published
##### Published in
SIGSPATIAL '22 : Proceedings of the 30th International Conference on Advances in Geographic Information Systems / Renz, Matthias; Sarwat, Mohamed (ed.). - New York, NY : ACM, 2022. - 35. - ISBN 978-1-4503-9529-8
##### Abstract
Measuring the similarity of spatio-temporal trajectories in a sensible fashion is an important building block for applications such as trajectory clustering or movement pattern analysis. However, typically employed similarity measures only take the spatial components of the trajectory into account, or are complicated combinations of different measures. In this paper we introduce the so called Generalized Fréchet distance, which extends the well-known Fréchet distance. For two polygonal curves of length n and m in d-dimensional space, the Generalized Fréchet distance enables an individual weighting of each dimension on the similarity value by using a convex function. This allows to integrate arbitrary data dimensions as e.g. temporal information in an elegant, flexible and application-aware manner. We study the Generalized Fréchet Distance for both the discrete and the continuous version of the problem, prove useful properties, and present efficient algorithms to compute the decision and optimization problem. In particular, we prove that for d ∈ O(1) the asymptotic running times of the optimization problem for the continuous version are O(nm log(nm)) under realistic assumptions, and O(nm) for the discrete version for arbitrary weight functions. Therefore the theoretical running times match those of the classical Fréchet distance. In our experimental evaluation, we demonstrate the usefulness of the Generalized Fréchet distance and study the practical behaviour of our algorithms. On sets of real-world trajectories, we confirm that the weighting of the spatial and temporal dimensions heavily impacts the relative similarity, and hence the ability to tailor the measure to the application is a useful tool.
##### Subject (DDC)
004 Computer Science
##### Keywords
Trajectory similarity, Fréchet distance, Algorithm analysis
##### Conference
SIGSPATIAL '22 : The 30th International Conference on Advances in Geographic Information Systems, Nov 1, 2022 - Nov 4, 2022, Seattle, Washington
##### Cite This
ISO 690GUTSCHLAG, Theodor, Sabine STORANDT, 2022. On the generalized fréchet distance and its applications. SIGSPATIAL '22 : The 30th International Conference on Advances in Geographic Information Systems. Seattle, Washington, Nov 1, 2022 - Nov 4, 2022. In: RENZ, Matthias, ed., Mohamed SARWAT, ed.. SIGSPATIAL '22 : Proceedings of the 30th International Conference on Advances in Geographic Information Systems. New York, NY:ACM, 35. ISBN 978-1-4503-9529-8. Available under: doi: 10.1145/3557915.3560970
BibTex
@inproceedings{Gutschlag2022gener-66286,
year={2022},
doi={10.1145/3557915.3560970},
title={On the generalized fréchet distance and its applications},
isbn={978-1-4503-9529-8},
publisher={ACM},
booktitle={SIGSPATIAL '22 : Proceedings of the 30th International Conference on Advances in Geographic Information Systems},
editor={Renz, Matthias and Sarwat, Mohamed},
author={Gutschlag, Theodor and Storandt, Sabine},
note={Funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - Project-ID 251654672 - TRR 161. Article Number: 35}
}

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<dcterms:abstract xml:lang="eng">Measuring the similarity of spatio-temporal trajectories in a sensible fashion is an important building block for applications such as trajectory clustering or movement pattern analysis. However, typically employed similarity measures only take the spatial components of the trajectory into account, or are complicated combinations of different measures. In this paper we introduce the so called Generalized Fréchet distance, which extends the well-known Fréchet distance. For two polygonal curves of length n and m in d-dimensional space, the Generalized Fréchet distance enables an individual weighting of each dimension on the similarity value by using a convex function. This allows to integrate arbitrary data dimensions as e.g. temporal information in an elegant, flexible and application-aware manner. We study the Generalized Fréchet Distance for both the discrete and the continuous version of the problem, prove useful properties, and present efficient algorithms to compute the decision and optimization problem. In particular, we prove that for d ∈ O(1) the asymptotic running times of the optimization problem for the continuous version are O(nm log(nm)) under realistic assumptions, and O(nm) for the discrete version for arbitrary weight functions. Therefore the theoretical running times match those of the classical Fréchet distance. In our experimental evaluation, we demonstrate the usefulness of the Generalized Fréchet distance and study the practical behaviour of our algorithms. On sets of real-world trajectories, we confirm that the weighting of the spatial and temporal dimensions heavily impacts the relative similarity, and hence the ability to tailor the measure to the application is a useful tool.</dcterms:abstract>
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##### Comment on publication
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 251654672 - TRR 161.