We study subtrajectory clustering under the Frechet 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 $k$, $ell geq 1$, and a real value $Delta > 0$, the goal is to find $k$ center curves of complexity at most $ell$ such that every point on $P$ is covered by a subtrajectory that has small Frechet distance to one of the $k$ center curves ($leq Delta$). In many application scenarios, one is interested in finding clusters of small complexity, which is controlled by the parameter $ell$. Our main result is a tri-criterial approximation algorithm: if there exists a solution for given parameters $k$, $ell$, and $Delta$, then our algorithm finds a set of $k$ center curves of complexity at most $ell$ with covering radius $Delta$ with $k in O( k ell^2 log (k ell))$, $ellleq 2ell$, and $Deltaleq 19 Delta$. Moreover, within these approximation bounds, we can minimize $k$ while keeping the other parameters fixed. If $ell$ 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 $Delta$ is constant. In this case, the algorithm has expected running time in $ tilde{O}left( k m^2 + mnright)$ and uses space in $O(n+m)$, where $m=lceilfrac{L}{Delta}rceil$ and $L$ is the total arclength of the curve $P$. For the important case of clustering with line segments ($ell$=2) we obtain bi-criteria approximation algorithms, where the approximation criteria are the number of clusters and the radius of the clustering.
تحميل البحث