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# Faster Approximate Covering of Subcurves Under the Fréchet Distance

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## Cite As

Frederik Brüning, Jacobus Conradi, and Anne Driemel. Faster Approximate Covering of Subcurves Under the Fréchet Distance. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.28

## Abstract

Subtrajectory clustering is an important variant of the trajectory clustering problem, where the start and endpoints of trajectory patterns within the collected trajectory data are not known in advance. We study this problem in the form of a set cover problem for a given polygonal curve: find the smallest number k of representative curves such that any point on the input curve is contained in a subcurve that has Fréchet distance at most a given Δ to a representative curve. We focus on the case where the representative curves are line segments and approach this NP-hard problem with classical techniques from the area of geometric set cover: we use a variant of the multiplicative weights update method which was first suggested by Brönniman and Goodrich for set cover instances with small VC-dimension. We obtain a bicriteria-approximation algorithm that computes a set of O(klog(k)) line segments that cover a given polygonal curve of n vertices under Fréchet distance at most O(Δ). We show that the algorithm runs in Õ(k² n + k n³) time in expectation and uses Õ(k n + n³) space. For input curves that are c-packed and lie in the plane, we bound the expected running time by Õ(k² c² n) and the space by Õ(kn + c² n). In addition, we present a variant of the algorithm that uses implicit weight updates on the candidate set and thereby achieves near-linear running time in n without any assumptions on the input curve, while keeping the same approximation bounds. This comes at the expense of a small (polylogarithmic) dependency on the relative arclength.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• Clustering
• Set cover
• Fréchet distance
• Approximation algorithms

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## References

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