Subtrajectory Clustering: Finding Set Covers for Set Systems of Subcurves
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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 tri-criterial 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' ∈ O( k ℓ^2 log (k ℓ)), ℓ'≤ 2ℓ, and Δ'≤ 19 Δ. 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( k m^2 + mn) and uses space in O(n+m), where m=⌈L/Δ⌉ and L is the total arclength of the curve P. For the important case of clustering with line segments (ℓ=2) we obtain bi-criteria approximation algorithms, where the approximation criteria are the number of clusters and the radius of the clustering.
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