We present a simple and practical (1 + ε)-approximation algorithm for the Fréchet distance between two polygonal curves in R d . To analyze this algorithm we introduce a new realistic family of curves, c-packed curves, that is closed under simplification. We believe the notion of c-packed curves to be of independent interest. We show that our algorithm has near linear running time for c-packed polygonal curves, and similar results for other input models, such as low-density polygonal curves.

The similarity of two polygonal curves can be measured using the Fréchet distance. We introduce the notion of a more robust Fréchet distance, where one is allowed to shortcut between vertices of one of the curves. This is a natural approach for handling noise, in particular batched outliers. We compute a (3 + ε)-approximation to the minimum Fréchet distance over all possible such shortcuts, in near linear time, if the curve is c-packed and the number of shortcuts is either small or unbounded. To facilitate the new algorithm we develop several new tools: (a) a data structure for preprocessing a curve (not necessarily c-packed) that supports (1 + ε)-approximate Fréchet distance queries between a subcurve (of the original curve) and a line segment; (b) a near linear time algorithm that computes a permutation of the vertices of a curve, such that any prefix of 2k − 1 vertices of this permutation forms an optimal approximation (up to a constant factor) to the original curve compared to any polygonal curve with k vertices, for any k > 0; and (c) a data structure for preprocessing a curve that supports approximate Fréchet distance queries between a subcurve and query polygonal curve. The query time depends quadratically on the complexity of the query curve and only (roughly) logarithmically on the complexity of the original curve. To our knowledge, these are the first data structures to support these kind of queries efficiently.

We present a simple and practical (1 + ε)-approximation algorithm for the Fréchet distance between two polygonal curves in IR d . To analyze this algorithm we introduce a new realistic family of curves, c-packed curves, that is closed under simplification. We believe the notion of c-packed curves to be of independent interest. We show that our algorithm has near linear running time for c-packed polygonal curves, and similar results for other input models, such as low density polygonal curves.

The Fréchet distance is a popular distance measure for curves. We study the problem of clustering time series under the Fréchet distance. In particular, we give (1 + ε)-approximation algorithms for variations of the following problem with parameters k and ℓ. Given n univariate time series P , each of complexity at most m, we find k time series, not necessarily from P , which we call cluster centers and which each have complexity at most ℓ, such that (a) the maximum distance of an element of P to its nearest cluster center or (b) the sum of these distances is minimized. Our algorithms have running time near-linear in the input size for constant ε, k and ℓ. To the best of our knowledge, our algorithms are the first clustering algorithms for the Fréchet distance which achieve an approximation factor of (1 + ε) or better.

In this paper we address the problem of segmenting a trajectory such that each segment is in some sense homogeneous. We formally define different spatio-temporal criteria under which a trajectory can be homogeneous, including location, heading, speed, velocity, curvature, sinuosity, and\ud curviness. We present a framework that allows us to segment any trajectory into a minimum number of segments under any of these criteria, or any combination of these criteria.\ud In this framework, the segmentation problem can generally be solved in O(n log n) time, where n is the number of edges of the trajectory to be segmented.Postprint (published version