Faster Approximate Covering of Subcurves under the Fréchet Distance

by   Frederik Brüning, et al.

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 O(k^2 n + k n^3) time in expectation and uses O(k n + n^3) space. For two dimensional input curves that are c-packed, we bound the expected running time by O(k^2 c^2 n) and the space by O(kn + c^2 n). In ℝ^d the dependency on n instead is quadratic. 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.


page 1

page 2

page 3

page 4


Subtrajectory Clustering: Finding Set Covers for Set Systems of Subcurves

We study subtrajectory clustering under the Fréchet distance. Given one ...

Finding Complex Patterns in Trajectory Data via Geometric Set Cover

Clustering trajectories is a central challenge when confronted with larg...

Curve Simplification and Clustering under Fréchet Distance

We present new approximation results on curve simplification and cluster...

Optimal Distributed Covering Algorithms

We present a time-optimal deterministic distributed algorithm for approx...

Progressive Simplification of Polygonal Curves

Simplifying polygonal curves at different levels of detail is an importa...

A smaller cover for closed unit curves

Forty years ago Schaer and Wetzel showed that a 1/π×1/2π√(π^2-4) rectang...

On Vietoris-Rips complexes of planar curves

A Vietoris--Rips complex is a way to thicken a (possibly discrete) metri...

Please sign up or login with your details

Forgot password? Click here to reset