Fast Frechet Distance Between Curves With Long Edges

Computing the Fr\'echet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fr\'echet distance computations become easier. Let $P$ and $Q$ be two polygonal curves in $\mathbb{R}^d$ with $n$ and $m$ vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fr\'echet distance between them: (1) a linear-time algorithm for deciding the Fr\'echet distance between two curves, (2) an algorithm that computes the Fr\'echet distance in $O((n+m)\log (n+m))$ time, (3) a linear-time $\sqrt{d}$-approximation algorithm, and (4) a data structure that supports $O(m\log^2 n)$-time decision queries, where $m$ is the number of vertices of the query curve and $n$ the number of vertices of the preprocessed curve

Progressive Simplification of Polygonal Curves

Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of detail. We present an $O(n^3m)$-time algorithm that takes a polygonal curve of n vertices and produces a set of consistent simplifications for m scales while minimizing the cumulative simplification complexity. This algorithm is compatible with distance measures such as the Hausdorff, the Fr\'echet and area-based distances, and enables simplification for continuous scaling in $O(n^5)$ time. To speed up this algorithm in practice, we present new techniques for constructing and representing so-called shortcut graphs. Experimental evaluation of these techniques on trajectory data reveals a significant improvement of using shortcut graphs for progressive and non-progressive curve simplification, both in terms of running time and memory usage.Comment: 20 pages, 20 figure

Faster Fr\'echet Distance Approximation through Truncated Smoothing

The Fr\'echet distance is a popular distance measure for curves. Computing the Fr\'echet distance between two polygonal curves of $n$ vertices takes roughly quadratic time, and conditional lower bounds suggest that even approximating to within a factor $3$ cannot be done in strongly-subquadratic time, even in one dimension. The current best approximation algorithms present trade-offs between approximation quality and running time. Recently, van der Horst $\textit{et al.}$ (SODA, 2023) presented an $O((n^2 / \alpha) \log^3 n)$ time $\alpha$-approximate algorithm for curves in arbitrary dimensions, for any $\alpha \in [1, n]$. Our main contribution is an approximation algorithm for curves in one dimension, with a significantly faster running time of $O(n \log^3 n + (n^2 / \alpha^3) \log^2 n \log \log n)$. Additionally, we give an algorithm for curves in arbitrary dimensions that improves upon the state-of-the-art running time by a logarithmic factor, to $O((n^2 / \alpha) \log^2 n)$. Both of our algorithms rely on a linear-time simplification procedure that in one dimension reduces the complexity of the reachable free space to $O(n^2 / \alpha)$ without making sacrifices in the asymptotic approximation factor.Comment: 27 pages, 11 figure

Visual analytics of delays and interaction in movement data

Maximilian Konzack, Tim Ophelders, Michel A. Westenberg and Kevin Buchin are supported by the Netherlands Organisation for Scientific Research (NWO) under grant no. 612.001.207 (Maximilian Konzack, Michel A. Westenberg and Kevin Buchin) and grant no. 639.023.208 (Tim Ophelders).The analysis of interaction between movement trajectories is of interest for various domains when movement of multiple objects is concerned. Interaction often includes a delayed response, making it difficult to detect interaction with current methods that compare movement at specific time intervals. We propose analyses and visualizations, on a local and global scale, of delayed movement responses, where an action is followed by a reaction over time, on trajectories recorded simultaneously. We developed a novel approach to compute the global delay in subquadratic time using a fast Fourier transform (FFT). Central to our local analysis of delays is the computation of a matching between the trajectories in a so-called delay space. It encodes the similarities between all pairs of points of the trajectories. In the visualization, the edges of the matching are bundled into patches, such that shape and color of a patch help to encode changes in an interaction pattern. To evaluate our approach experimentally, we have implemented it as a prototype visual analytics tool and have applied the tool on three bidimensional data sets. For this we used various measures to compute the delay space, including the directional distance, a new similarity measure, which captures more complex interactions by combining directional and spatial characteristics. We compare matchings of various methods computing similarity between trajectories. We also compare various procedures to compute the matching in the delay space, specifically the Fréchet distance, dynamic time warping (DTW), and edit distance (ED). Finally, we demonstrate how to validate the consistency of pairwise matchings by computing matchings between more than two trajectories.Publisher PDFPeer reviewe