Efficient Greedy Discrete Subtrajectory Clustering.

We cluster a set of trajectories T using subtrajectories of T . We require for a clustering C that any two subtrajectories (T [a, b], T [c, d]) in a cluster have disjoint intervals [a, b] and [c, d]. Clustering quality may be measured by the number of clusters, the number of vertices of T that are absent from the clustering, and by the Fréchet distance between subtrajectories in a cluster. A Δ-cluster of T is a cluster P of subtrajectories of T with a centre P ∈ P, where all subtrajectories in P have Fréchet distance at most Δ to P. Buchin, Buchin, Gudmundsson, Löffler and Luo present two O(n2 + nml)-time algorithms: SC(max, l, Δ, T) computes a single Δ-cluster where P has at least l vertices and maximises the cardinality m of P. SC(m, max, Δ, T) computes a single Δ-cluster where P has cardinality m and maximises the complexity l of P. In this paper, which is a mixture of algorithms engineering and theoretical insights, we use such maximum-cardinality clusters in a greedy clustering algorithm. We first provide an efficient implementation of SC(max, l, Δ, T) and SC(m, max, Δ, T) that significantly outperforms previous implementations. Next, we use these functions as a subroutine in a greedy clustering algorithm, which performs well when compared to existing subtrajectory clustering algorithms on real-world data. Finally, we observe that, for fixed Δ and T, these two functions always output a point on the Pareto front of some bivariate function θ(l,m). We design a new algorithm PSC(Δ, T) that in O(n2 log4 n) time computes a 2-approximation of this Pareto front. This yields a broader set of candidate clusters, with comparable quality to the output of the previous functions. We show that using PSC(Δ, T) as a subroutine improves the clustering quality and performance even further.