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Title: Subtrajectory Clustering: Finding Set Covers for Set Systems of Subcurves
We study subtrajectory clustering under the Fréchet distance. Given one or more trajectories, the task is to split the trajectories into several parts, such that the parts have a good clustering structure. We approach this problem via a new set cover formulation, which we think provides a natural formalization of the problem as it is studied in many applications. Given a polygonal curve P with n vertices in fixed dimension, integers k, ℓ ≥ 1, and a real value Δ > 0, the goal is to find k center curves of complexity at most ℓ such that every point on P is covered by a subtrajectory that has small Fréchet distance to one of the k center curves (≤ Δ). In many application scenarios, one is interested in finding clusters of small complexity, which is controlled by the parameter ℓ. Our main result is a bicriterial approximation algorithm: if there exists a solution for given parameters k, ℓ, and Δ, then our algorithm finds a set of k' center curves of complexity at most ℓ with covering radius Δ' with k' in O(kℓ2 log (kℓ)), and Δ' ≤ 19Δ. Moreover, within these approximation bounds, we can minimize k while keeping the other parameters fixed. If ℓ is a constant independent of n, then, the approximation factor for the number of clusters k is O(log k) and the approximation factor for the radius Δ is constant. In this case, the algorithm has expected running time in Õ(km2 + mn) and uses space in O(n + m), where m=⌈L/Δ⌉ and L is the total arclength of the curve P.  more » « less
Award ID(s):
2106672 1907612
NSF-PAR ID:
10417669
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
Computing in geometry and topology
Volume:
2
Issue:
1
ISSN:
2750-7823
Page Range / eLocation ID:
1-48
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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