Central Trajectories

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, where we show that an optimal central trajectory C representing n trajectories, each consisting of τ edges, has complexity Θ(τn) and can be computed in O(τn log n) time. We then consider trajectories in R with d ≥ 2, and show that the complexity of C is at most O(τn) and can be computed in O(τn) time. ∗Department of Information and Computing Sciences, Universiteit Utrecht, The Netherlands, {m.j.vankreveld|m.loffler|f.staals}@uu.nl ar X iv :1 50 1. 01 82 2v 1 [ cs .C G ] 8 J an 2 01 5