Central trajectories

Abstract

$\newcommand{\c}{\mathcal{C}}\newcommand{\R}{\mathbb{R}}$An important task in trajectory analysis is clustering. The results of a clustering are often summarized by a single representative trajectory and an associated size of each cluster. We study the problem of computing a suitable representative of a set of similar trajectories. To this end we define a central trajectory $\c$, which consists of pieces of the input trajectories, switches from one entity to another only if they are within a small distance of each other, and such that at any time $t$, the point $\c(t)$ is as central as possible. We measure centrality in terms of  the radius of the smallest disk centered at $\c(t)$ enclosing all entities at time $t$, and discuss how the techniques can be adapted to other measures of centrality. We first study the problem in $\R^1$, where we show that an optimal central trajectory $\c$ representing $n$ trajectories, each consisting of $\tau$ edges, has complexity $\Theta(\tau n^2)$ and can be computed in $O(\tau n^2 \log n)$ time. We then consider trajectories in $\R^d$ with $d\geq 2$, and show that the complexity of $\c$ is at most $O(\tau n^{5/2})$ and can be computed in $O(\tau n^3)$ time.