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

Computing a Subtrajectory Cluster from c-packed Trajectories

We present a near-linear time approximation algorithm for the subtrajectory cluster problem of $c$-packed trajectories. The problem involves finding $m$ subtrajectories within a given trajectory $T$ such that their Fr\'echet distances are at most $(1 + \varepsilon)d$, and at least one subtrajectory must be of length~$l$ or longer. A trajectory $T$ is $c$-packed if the intersection of $T$ and any ball $B$ with radius $r$ is at most $c \cdot r$ in length. Previous results by Gudmundsson and Wong \cite{GudmundssonWong2022Cubicupperlower} established an $\Omega(n^3)$ lower bound unless the Strong Exponential Time Hypothesis fails, and they presented an $O(n^3 \log^2 n)$ time algorithm. We circumvent this conditional lower bound by studying subtrajectory cluster on $c$-packed trajectories, resulting in an algorithm with an $O((c^2 n/\varepsilon^2)\log(c/\varepsilon)\log(n/\varepsilon))$ time complexity

Approximating the Packedness of Polygonal Curves

In 2012 Driemel et al. \cite{DBLP:journals/dcg/DriemelHW12} introduced the concept of $c$-packed curves as a realistic input model. In the case when $c$ is a constant they gave a near linear time $(1+\varepsilon)$-approximation algorithm for computing the Fr\'echet distance between two $c$-packed polygonal curves. Since then a number of papers have used the model. In this paper we consider the problem of computing the smallest $c$ for which a given polygonal curve in $\mathbb{R}^d$ is $c$-packed. We present two approximation algorithms. The first algorithm is a $2$-approximation algorithm and runs in $O(dn^2 \log n)$ time. In the case $d=2$ we develop a faster algorithm that returns a $(6+\varepsilon)$-approximation and runs in $O((n/\varepsilon^3)^{4/3} polylog (n/\varepsilon)))$ time. We also implemented the first algorithm and computed the approximate packedness-value for 16 sets of real-world trajectories. The experiments indicate that the notion of $c$-packedness is a useful realistic input model for many curves and trajectories.Comment: A preliminary version to appear in ISAAC 202

Fine-grained complexity and algorithm engineering of geometric similarity measures

Point sets and sequences are fundamental geometric objects that arise in any application that considers movement data, geometric shapes, and many more. A crucial task on these objects is to measure their similarity. Therefore, this thesis presents results on algorithms, complexity lower bounds, and algorithm engineering of the most important point set and sequence similarity measures like the Fréchet distance, the Fréchet distance under translation, and the Hausdorff distance under translation. As an extension to the mere computation of similarity, also the approximate near neighbor problem for the continuous Fréchet distance on time series is considered and matching upper and lower bounds are shown.Punktmengen und Sequenzen sind fundamentale geometrische Objekte, welche in vielen Anwendungen auftauchen, insbesondere in solchen die Bewegungsdaten, geometrische Formen, und ähnliche Daten verarbeiten. Ein wichtiger Bestandteil dieser Anwendungen ist die Berechnung der Ähnlichkeit von Objekten. Diese Dissertation präsentiert Resultate, genauer gesagt Algorithmen, untere Komplexitätsschranken und Algorithm Engineering der wichtigsten Ähnlichkeitsmaße für Punktmengen und Sequenzen, wie zum Beispiel Fréchetdistanz, Fréchetdistanz unter Translation und Hausdorffdistanz unter Translation. Als eine Erweiterung der bloßen Berechnung von Ähnlichkeit betrachten wir auch das Near Neighbor Problem für die kontinuierliche Fréchetdistanz auf Zeitfolgen und zeigen obere und untere Schranken dafür

(1+ε)(1+\varepsilon)-ANN Data Structure for Curves via Subspaces of Bounded Doubling Dimension

We consider the $(1+\varepsilon)$-Approximate Nearest Neighbour (ANN) Problem for polygonal curves in $d$-dimensional space under the Fr\'echet distance and ask to what extent known data structures for doubling spaces can be applied to this problem. Initially, this approach does not seem viable, since the doubling dimension of the target space is known to be unbounded -- even for well-behaved polygonal curves of constant complexity in one dimension. In order to overcome this, we identify a subspace of curves which has bounded doubling dimension and small Gromov-Hausdorff distance to the target space. We then apply state-of-the-art techniques for doubling spaces and show how to obtain a data structure for the $(1+\varepsilon)$-ANN problem for any set of parametrized polygonal curves. The expected preprocessing time needed to construct the data-structure is $F(d,k,S,\varepsilon)n\log n$ and the space used is $F(d,k,S,\varepsilon)n$, with a query time of $F(d,k,S,\varepsilon)\log n + F(d,k,S,\varepsilon)^{-\log(\varepsilon)}$, where $F(d,k,S,\varepsilon)=O\left(2^{O(d)}k\Phi(S)\varepsilon^{-1}\right)^k$ and $\Phi(S)$ denotes the spread of the set of vertices and edges of the curves in $S$. We extend these results to the realistic class of $c$-packed curves and show improved bounds for small values of $c$.Comment: submitted to Computing in Geometry and Topolog

Approximability of the Discrete {Fr\'echet} Distance

<p>The Fréchet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al. [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds.</p><p>In this paper, we study the approximability of the discrete Fréchet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound showing that strongly subquadratic algorithms for the discrete Fréchet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399.</p><p>This raises the question of how well we can approximate the Fréchet distance (of two given $d$-dimensional point sequences of length $n$) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be $2^{\Theta(n)}$. Moreover, we design an $\alpha$-approximation algorithm that runs in time $O(n\log n + n^2/\alpha)$, for any $\alpha\in [1, n]$. Hence, an $n^\varepsilon$-approximation of the Fréchet distance can be computed in strongly subquadratic time, for any \varepsilon > 0.</p

Approximating (k,ℓ)(k,\ell)-center clustering for curves

The Euclidean $k$-center problem is a classical problem that has been extensively studied in computer science. Given a set $\mathcal{G}$ of $n$ points in Euclidean space, the problem is to determine a set $\mathcal{C}$ of $k$ centers (not necessarily part of $\mathcal{G}$) such that the maximum distance between a point in $\mathcal{G}$ and its nearest neighbor in $\mathcal{C}$ is minimized. In this paper we study the corresponding $(k,\ell)$-center problem for polygonal curves under the Fr\'echet distance, that is, given a set $\mathcal{G}$ of $n$ polygonal curves in $\mathbb{R}^d$, each of complexity $m$, determine a set $\mathcal{C}$ of $k$ polygonal curves in $\mathbb{R}^d$, each of complexity $\ell$, such that the maximum Fr\'echet distance of a curve in $\mathcal{G}$ to its closest curve in $\mathcal{C}$ is minimized. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension $2$ and higher. We show that, if $\ell$ is part of the input, then there is no polynomial-time approximation scheme unless $\mathsf{P}=\mathsf{NP}$. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fr\'echet distance. In the case of the discrete Fr\'echet distance on two-dimensional curves, we show hardness of approximation within a factor close to $2.598$. This result also holds when $k=1$, and the $\mathsf{NP}$-hardness extends to the case that $\ell=\infty$, i.e., for the problem of computing the minimum-enclosing ball under the Fr\'echet distance. Finally, we observe that a careful adaptation of Gonzalez' algorithm in combination with a curve simplification yields a $3$-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.Comment: 24 pages; results on minimum-enclosing ball added, additional author added, general revisio