Figure 1: Clustering using the Fréchet distance (left), dynamic time warping (middle), and our new approach called continuous dynamic time warping (right), where the color denotes the clusters and the bold trajectories are the cluster centers. While the Fréchet distance shows a strong influence of outliers and dynamic time warping shows discretization issues, clustering via continuous dynamic time warping gives arguably the most natural results. Map data © OpenStreetMap contributors.

Solomon and Elkin [13] constructed a shortcutting scheme for weighted trees which results in a 1-spanner for the tree metric induced by the input tree. The spanner has logarithmic lightness, logarithmic diameter, a linear number of edges and bounded degree (provided the input tree has bounded degree). This spanner has been applied in a series of papers devoted to designing bounded degree, low-diameter, low-weight (1+ǫ)-spanners in Euclidean and doubling metrics. In this paper, we present a simple local routing algorithm for this tree metric spanner. The algorithm has a routing ratio of 1, is guaranteed to terminate after O(log n) hops and requires O(∆ log n) bits of storage per vertex where ∆ is the maximum degree of the tree on which the spanner is constructed. This local routing algorithm can be adapted to a local routing algorithm for a doubling metric spanner which makes use of the shortcutting scheme.

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We propose new query applications of the well known randomized incremental construction of the Trapezoidal Search DAG (TSD) on a set of n line segments in the plane, where queries are allowed to be any axis aligned window.We show that our algorithm reports the m trapezoids that are intersected by the query in O(m + log n) expected time, regardless of the spatial location of the segment set and the query. In case the query is a vertical segment, the query time bound reduces to O(k + log n) where k is the number of segments that are intersected. This improves on the query and space bound of the well known Segment Tree based approach, which is to date the theoretical bottleneck for optimal query time. In the case where the set of segments is a connected planar subdivision, this method can easily be extended to an algorithm which reports the k segments which intersect an axis aligned query window in O(k + log n) expected time.Our publicly available implementation handles degeneracies exactly, including segments with overlap and multi-intersections. Experiments show that the method is practical and provides more reliable query times in comparison to R-trees and the segment tree based data structure on real-world and synthetic data sets.