arXiv subjects

Frederik Brüning

Publications and source records attributed to Frederik Brüning.

4 recordsLinked to original sources

arXiv2026

Solving Proximity Problems for Polygonal Shapes via Polynomials

We study range spaces, where the ground set consists of either polygonal curves in $\mathbb{R}^d$ or polygonal regions in the plane that may contain holes and the ranges are balls defined by distance measures, such as the Hausdorff distance, the Fréchet distance and the dynamic time warping distance. These range spaces appear in various applications like classification, simplification, range searching, density estimation and clustering when the instances are trajectories, time series or polygons. We show for the Fréchet distance of polygonal curves and the Hausdorff distance of polygonal curves and planar polygonal regions that the VC-dimension is upper-bounded by $O(dk\log(km))$, where $k$ is the complexity of the center of a ball, $m$ is the complexity of each polygonal curve or region in the ground set, and $d$ is the ambient dimension. For $d \geq 4$ this bound is tight in each of the parameters $d,k$ and $m$ separately. For a variant of dynamic time warping distance of polygonal curves using squared distances, our analysis directly yields an upper-bound of $O(\min(dk^2\log(m),dkm\log(k)))$. We also obtain exact solutions that are hitherto unknown for simplification, range searching, nearest neighbor search, and distance oracle.

cs.CG↗

arXiv2024

On the number of iterations of the DBA algorithm

The DTW Barycenter Averaging (DBA) algorithm is a widely used algorithm for estimating the mean of a given set of point sequences. In this context, the mean is defined as a point sequence that minimises the sum of dynamic time warping distances (DTW). The algorithm is similar to the $k$-means algorithm in the sense that it alternately repeats two steps: (1) computing an optimal assignment to the points of the current mean, and (2) computing an optimal mean under the current assignment. The popularity of DBA can be attributed to the fact that it works well in practice, despite any theoretical guarantees to be known. In our paper, we aim to initiate a theoretical study of the number of iterations that DBA performs until convergence. We assume the algorithm is given $n$ sequences of $m$ points in $\mathbb{R}^d$ and a parameter $k$ that specifies the length of the mean sequence to be computed. We show that, in contrast to its fast running time in practice, the number of iterations can be exponential in $k$ in the worst case - even if the number of input sequences is $n=2$. We complement these findings with experiments on real-world data that suggest this worst-case behaviour is likely degenerate. To better understand the performance of the algorithm on non-degenerate input, we study DBA in the model of smoothed analysis, upper-bounding the expected number of iterations in the worst case under random perturbations of the input. Our smoothed upper bound is polynomial in $k$, $n$ and $d$, and for constant $n$, it is also polynomial in $m$. For our analysis, we adapt the set of techniques that were developed for analysing $k$-means and observe that this set of techniques is not sufficient to obtain tight bounds for general $n$.

cs.CG↗

arXiv2022

Faster Approximate Covering of Subcurves under the Fréchet Distance

Subtrajectory clustering is an important variant of the trajectory clustering problem, where the start and endpoints of trajectory patterns within the collected trajectory data are not known in advance. We study this problem in the form of a set cover problem for a given polygonal curve: find the smallest number $k$ of representative curves such that any point on the input curve is contained in a subcurve that has Fréchet distance at most a given $Δ$ to a representative curve. We focus on the case where the representative curves are line segments and approach this NP-hard problem with classical techniques from the area of geometric set cover: we use a variant of the multiplicative weights update method which was first suggested by Brönniman and Goodrich for set cover instances with small VC-dimension. We obtain a bicriteria-approximation algorithm that computes a set of $O(k\log(k))$ line segments that cover a given polygonal curve of $n$ vertices under Fréchet distance at most $O(Δ)$. We show that the algorithm runs in $\widetilde{O}(k^2 n + k n^3)$ time in expectation and uses $ \widetilde{O}(k n + n^3)$ space. For two dimensional input curves that are $c$-packed, we bound the expected running time by $\widetilde{O}(k^2 c^2 n)$ and the space by $ \widetilde{O}(kn + c^2 n)$. In $\mathbb{R}^d$ the dependency on $n$ instead is quadratic. In addition, we present a variant of the algorithm that uses implicit weight updates on the candidate set and thereby achieves near-linear running time in $n$ without any assumptions on the input curve, while keeping the same approximation bounds. This comes at the expense of a small (polylogarithmic) dependency on the relative arclength.

cs.CG↗

arXiv2021

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 $k$, $\ell \geq 1$, and a real value $Δ> 0$, 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 Fréchet distance to one of the $k$ center curves ($\leq Δ$). 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 bicriterial approximation algorithm: if there exists a solution for given parameters $k$, $\ell$, and $Δ$, then our algorithm finds a set of $k'$ center curves of complexity at most $\ell$ with covering radius $Δ'$ with $k' \in O( k \ell^2 \log (k \ell))$, and $Δ'\leq 19 Δ$. 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 $Δ$ is constant. In this case, the algorithm has expected running time in $ \tilde{O}\left( k m^2 + mn\right)$ and uses space in $O(n+m)$, where $m=\lceil\frac{L}Δ\rceil$ and $L$ is the total arclength of the curve $P$.

cs.CG↗