Coresets for $(k,l)$-Clustering under the Fréchet Distance

Abstract: Clustering is the task of partitioning a given set of geometric objects. This is thoroughly studied when the objects are points in the euclidean space. There are also several approaches for points in general metric spaces. In this thesis we consider clustering polygonal curves, i.e., curves composed of line segments, under the Fr\'echet distance. We obtain clusterings by minimizing an objective function, which yields a set of centers that induces a partition of the input. The objective functions we consider is the so called $(k,l)$-\textsc{center}, where we are to find the $k$ center-curves that minimize the maximum distance between any input-curve and a nearest center-curve and the so called $k$-\textsc{median}, where we are to find the $k$ center-curves that minimize the sum of the distances between the input-curves and a nearest center-curve. Given a set of $n$ polygonal curves, we are interested in reducing this set to an $\epsilon$-coreset, i.e., a notably smaller set of curves that has a very similar clustering-behavior. We develop a construction method for such $\epsilon$-coresets for the $(k,l)$-\textsc{center}, that yields $\epsilon$-coresets of size of a polynomial of $\frac{1}{\epsilon}$, in time linear in $n$ and a polynomial of $\frac{1}{\epsilon}$, for line segments. Also, we develop a construction technique for the $(k,l)$-\textsc{center} that yields $\epsilon$-coresets of size exponential in $m$ with basis $\frac{1}{\epsilon}$, in time sub-quadratic in $n$ and exponential in $m$ with basis $\frac{1}{\epsilon}$, for general polygonal curves. Finally, we develop a construction method for the $k$-\textsc{median}, that yields $\epsilon$-coresets of size polylogarithmic in $n$ and a polynomial of $\frac{1}{\epsilon}$, in time linear in $n$ and a polynomial of $\frac{1}{\epsilon}$.