子轨迹聚类：求子曲线集合系统的集合覆盖
Subtrajectory Clustering: Finding Set Covers for Set Systems of Subcurves
摘要：
我们研究Fréchet距离下的子轨迹聚类。给定一个或多个轨迹，任务是将轨迹分成几个部分，使得这些部分具有良好的聚类结构。我们通过一个新的集合覆盖公式来处理这个问题，我们认为它提供了一个自然的形式化的问题，因为它在许多应用程序中进行了研究。给定一条多边形曲线$P$，其$n$顶点的维数为固定，整数为$k$、$\ell \geq 1$和一个实值$Δ> 0$，目标是找到复杂度至多为$\ell$的$k$中心曲线，使得$P$上的每一点都被一条子轨迹覆盖，该子轨迹与其中一条$k$中心曲线（$\leq Δ$）具有较小的Fréchet距离。在许多应用场景中，人们感兴趣的是找到由参数$\ell$控制的小复杂度的聚类。我们的主要结果是一个双准则近似算法：如果存在一个解决方案，给定的参数$k$，$\ell$，$Δ$，然后我们的算法发现一组$k'$中心曲线的复杂性最多$\ell$覆盖半径$Δ'$与$k' \in O( k \ell^2 \log (k \ell))$，和$Δ'\leq 19 Δ$。此外，在这些近似范围内，我们可以最小化$k$，同时保持其他参数固定。如果$\ell$是与$n$无关的常数，则簇数$k$的近似因子是$O(\log k)$，半径$Δ$的近似因子是常数。在这种情况下，算法在$ \tilde{O}\left( k m^2 + mn\right)$中具有预期运行时间，并在$O(n+m)$中使用空间，其中$m=\lceil\frac{L}Δ\rceil$和$L$是曲线$P$的总弧长。
英文摘要：
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 $P$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
