arXiv subjects

Anne Driemel

Publications and source records attributed to Anne Driemel.

At least 37 records · Page 2Linked to original sources

arXiv2022

Approximating Length-Restricted Means under Dynamic Time Warping

We study variants of the mean problem under the $p$-Dynamic Time Warping ($p$-DTW) distance, a popular and robust distance measure for sequential data. In our setting we are given a set of finite point sequences over an arbitrary metric space and we want to compute a mean point sequence of given length that minimizes the sum of $p$-DTW distances, each raised to the $q$\textsuperscript{th} power, between the input sequences and the mean sequence. In general, the problem is $\mathrm{NP}$-hard and known not to be fixed-parameter tractable in the number of sequences. On the positive side, we show that restricting the length of the mean sequence significantly reduces the hardness of the problem. We give an exact algorithm running in polynomial time for constant-length means. We explore various approximation algorithms that provide a trade-off between the approximation factor and the running time. Our approximation algorithms have a running time with only linear dependency on the number of input sequences. In addition, we use our mean algorithms to obtain clustering algorithms with theoretical guarantees.

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arXiv2022

Faster Approximate Covering of Subcurves under the Fréchet Distance

Subtrajectory clustering is an important variant of the trajectory clustering problem, where the start and endpoints of trajectory patterns within the collected trajectory data are not known in advance. We study this problem in the form of a set cover problem for a given polygonal curve: find the smallest number $k$ of representative curves such that any point on the input curve is contained in a subcurve that has Fréchet distance at most a given $Δ$ to a representative curve. We focus on the case where the representative curves are line segments and approach this NP-hard problem with classical techniques from the area of geometric set cover: we use a variant of the multiplicative weights update method which was first suggested by Brönniman and Goodrich for set cover instances with small VC-dimension. We obtain a bicriteria-approximation algorithm that computes a set of $O(k\log(k))$ line segments that cover a given polygonal curve of $n$ vertices under Fréchet distance at most $O(Δ)$. We show that the algorithm runs in $\widetilde{O}(k^2 n + k n^3)$ time in expectation and uses $ \widetilde{O}(k n + n^3)$ space. For two dimensional input curves that are $c$-packed, we bound the expected running time by $\widetilde{O}(k^2 c^2 n)$ and the space by $ \widetilde{O}(kn + c^2 n)$. In $\mathbb{R}^d$ the dependency on $n$ instead is quadratic. In addition, we present a variant of the algorithm that uses implicit weight updates on the candidate set and thereby achieves near-linear running time in $n$ without any assumptions on the input curve, while keeping the same approximation bounds. This comes at the expense of a small (polylogarithmic) dependency on the relative arclength.

cs.CG↗

arXiv2022

Minimum-Error Triangulations for Sea Surface Reconstruction

We apply state-of-the-art computational geometry methods to the problem of reconstructing a time-varying sea surface from tide gauge records. Our work builds on a recent article by Nitzke et al.~(Computers \& Geosciences, 157:104920, 2021) who have suggested to learn a triangulation $D$ of a given set of tide gauge stations. The objective is to minimize the misfit of the piecewise linear surface induced by $D$ to a reference surface that has been acquired with satellite altimetry. The authors restricted their search to k-order Delaunay ($k$-OD) triangulations and used an integer linear program in order to solve the resulting optimization problem. In geometric terms, the input to our problem consists of two sets of points in $\mathbb{R}^2$ with elevations: a set $\mathcal{S}$ that is to be triangulated, and a set $\mathcal{R}$ of reference points. Intuitively, we define the error of a triangulation as the average vertical distance of a point in $\mathcal{R}$ to the triangulated surface that is obtained by interpolating elevations of $\mathcal{S}$ linearly in each triangle. Our goal is to find the triangulation of $\mathcal{S}$ that has minimum error with respect to $\mathcal{R}$. In our work, we prove that the minimum-error triangulation problem is NP-hard and cannot be approximated within any multiplicative factor in polynomial time unless $P=NP$. At the same time we show that the problem instances that occur in our application (considering sea level data from several hundreds of tide gauge stations worldwide) can be solved relatively fast using dynamic programming when restricted to $k$-OD triangulations for $k\le 7$. In particular, instances for which the number of connected components of the so-called $k$-OD fixed-edge graph is small can be solved within few seconds.

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arXiv2022

On Computing the $k$-Shortcut Fréchet Distance

The Fréchet distance is a popular measure of dissimilarity for polygonal curves. It is defined as a min-max formulation that considers all direction-preserving continuous bijections of the two curves. Because of its susceptibility to noise, Driemel and Har-Peled introduced the shortcut Fréchet distance in 2012, where one is allowed to take shortcuts along one of the curves, similar to the edit distance for sequences. We analyse the parameterized version of this problem, where the number of shortcuts is bounded by a parameter $k$. The corresponding decision problem can be stated as follows: Given two polygonal curves $T$ and $B$ of at most $n$ vertices, a parameter $k$ and a distance threshold $δ$, is it possible to introduce $k$ shortcuts along $B$ such that the Fréchet distance of the resulting curve and the curve $T$ is at most $δ$? We study this problem for polygonal curves in the plane. We provide a complexity analysis for this problem with the following results: (i) assuming the exponential-time-hypothesis (ETH), there exists no algorithm with running time bounded by $n^{o(k)}$; (ii) there exists a decision algorithm with running time in $O(kn^{2k+2}\log n)$. In contrast, we also show that efficient approximate decider algorithms are possible, even when $k$ is large. We present a $(3+\varepsilon)$-approximate decider algorithm with running time in $O(k n^2 \log^2 n)$ for fixed $\varepsilon$. In addition, we can show that, if $k$ is a constant and the two curves are $c$-packed for some constant $c$, then the approximate decider algorithm runs in near-linear time.

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arXiv2022

On the Discrete Fréchet Distance in a Graph

The Fréchet distance is a well-studied similarity measure between curves that is widely used throughout computer science. Motivated by applications where curves stem from paths and walks on an underlying graph (such as a road network), we define and study the Fréchet distance for paths and walks on graphs. When provided with a distance oracle of $G$ with $O(1)$ query time, the classical quadratic-time dynamic program can compute the Fréchet distance between two walks $P$ and $Q$ in a graph $G$ in $O(|P| \cdot |Q|)$ time. We show that there are situations where the graph structure helps with computing Fréchet distance: when the graph $G$ is planar, we apply existing (approximate) distance oracles to compute a $(1+\varepsilon)$-approximation of the Fréchet distance between any shortest path $P$ and any walk $Q$ in $O(|G| \log |G| / \sqrt{\varepsilon} + |P| + \frac{|Q|}{\varepsilon } )$ time. We generalise this result to near-shortest paths, i.e. $κ$-straight paths, as we show how to compute a $(1+\varepsilon)$-approximation between a $κ$-straight path $P$ and any walk $Q$ in $O(|G| \log |G| / \sqrt{\varepsilon} + |P| + \frac{κ|Q|}{\varepsilon } )$ time. Our algorithmic results hold for both the strong and the weak discrete Fréchet distance over the shortest path metric in $G$. Finally, we show that additional assumptions on the input, such as our assumption on path straightness, are indeed necessary to obtain truly subquadratic running time. We provide a conditional lower bound showing that the Fréchet distance, or even its $1.01$-approximation, between arbitrary \emph{paths} in a weighted planar graph cannot be computed in $O((|P|\cdot|Q|)^{1-δ})$ time for any $δ> 0$ unless the Orthogonal Vector Hypothesis fails. For walks, this lower bound holds even when $G$ is planar, unit-weight and has $O(1)$ vertices.

cs.CG↗

arXiv2021

Subtrajectory Clustering: Finding Set Covers for Set Systems of Subcurves

We study subtrajectory clustering under the Fréchet distance. Given one or more trajectories, the task is to split the trajectories into several parts, such that the parts have a good clustering structure. We approach this problem via a new set cover formulation, which we think provides a natural formalization of the problem as it is studied in many applications. Given a polygonal curve $P$ with $n$ vertices in fixed dimension, integers $k$, $\ell \geq 1$, and a real value $Δ> 0$, the goal is to find $k$ center curves of complexity at most $\ell$ such that every point on $P$ is covered by a subtrajectory that has small Fréchet distance to one of the $k$ center curves ($\leq Δ$). In many application scenarios, one is interested in finding clusters of small complexity, which is controlled by the parameter $\ell$. Our main result is a bicriterial approximation algorithm: if there exists a solution for given parameters $k$, $\ell$, and $Δ$, then our algorithm finds a set of $k'$ center curves of complexity at most $\ell$ with covering radius $Δ'$ with $k' \in O( k \ell^2 \log (k \ell))$, and $Δ'\leq 19 Δ$. Moreover, within these approximation bounds, we can minimize $k$ while keeping the other parameters fixed. If $\ell$ is a constant independent of $n$, then, the approximation factor for the number of clusters $k$ is $O(\log k)$ and the approximation factor for the radius $Δ$ is constant. In this case, the algorithm has expected running time in $ \tilde{O}\left( k m^2 + mn\right)$ and uses space in $O(n+m)$, where $m=\lceil\frac{L}Δ\rceil$ and $L$ is the total arclength of the curve $P$.

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arXiv2021

Tight Bounds for Approximate Near Neighbor Searching for Time Series under the Fréchet Distance

We study the $c$-approximate near neighbor problem under the continuous Fréchet distance: Given a set of $n$ polygonal curves with $m$ vertices, a radius $δ> 0$, and a parameter $k \leq m$, we want to preprocess the curves into a data structure that, given a query curve $q$ with $k$ vertices, either returns an input curve with Fréchet distance at most $c\cdot δ$ to $q$, or returns that there exists no input curve with Fréchet distance at most $δ$ to $q$. We focus on the case where the input and the queries are one-dimensional polygonal curves -- also called time series -- and we give a comprehensive analysis for this case. We obtain new upper bounds that provide different tradeoffs between approximation factor, preprocessing time, and query time. Our data structures improve upon the state of the art in several ways. We show that for any $0 < \varepsilon \leq 1$ an approximation factor of $(1+\varepsilon)$ can be achieved within the same asymptotic time bounds as the previously best result for $(2+\varepsilon)$. Moreover, we show that an approximation factor of $(2+\varepsilon)$ can be obtained by using preprocessing time and space $O(nm)$, which is linear in the input size, and query time in $O(\frac{1}{\varepsilon})^{k+2}$, where the previously best result used preprocessing time in $n \cdot O(\frac{m}{\varepsilon k})^k$ and query time in $O(1)^k$. We complement our upper bounds with matching conditional lower bounds based on the Orthogonal Vectors Hypothesis. Interestingly, some of our lower bounds already hold for any super-constant value of $k$. This is achieved by proving hardness of a one-sided sparse version of the Orthogonal Vectors problem as an intermediate problem, which we believe to be of independent interest.

cs.CG↗

arXiv2021

$(2+ε)$-ANN for time series under the Fréchet distance

We study approximate-near-neighbor data structures for time series under the continuous Fréchet distance. For an attainable approximation factor $c>1$ and a query radius $r$, an approximate-near-neighbor data structure can be used to preprocess $n$ curves in $\mathbb{R}$ (aka time series), each of complexity $m$, to answer queries with a curve of complexity $k$ by either returning a curve that lies within Fréchet distance $cr$, or answering that there exists no curve in the input within distance $r$. In both cases, the answer is correct. Our first data structure achieves a $(5+ε)$ approximation factor, uses space in $n\cdot \mathcal{O}\left({ε^{-1}}\right)^{k} + \mathcal{O}(nm)$ and has query time in $\mathcal{O}\left(k\right)$. Our second data structure achieves a $(2+ε)$ approximation factor, uses space in $n\cdot \mathcal{O}\left(\frac{m}{kε}\right)^{k} + \mathcal{O}(nm)$ and has query time in $\mathcal{O}\left(k\cdot 2^k\right)$. Our third positive result is a probabilistic data structure based on locality-sensitive hashing, which achieves space in $\mathcal{O}(n\log n+nm)$ and query time in $\mathcal{O}(k\log n)$, and which answers queries with an approximation factor in $\mathcal{O}(k)$. All of our data structures make use of the concept of signatures, which were originally introduced for the problem of clustering time series under the Fréchet distance. In addition, we show lower bounds for this problem. Consider any data structure which achieves an approximation factor less than $2$ and which supports curves of arclength up to $L$ and answers the query using only a constant number of probes. We show that under reasonable assumptions on the word size any such data structure needs space in $L^{Ω(k)}$.

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arXiv2020

Approximating $(k,\ell)$-Median Clustering for Polygonal Curves

In 2015, Driemel, Krivošija and Sohler introduced the $(k,\ell)$-median problem for clustering polygonal curves under the Fréchet distance. Given a set of input curves, the problem asks to find $k$ median curves of at most $\ell$ vertices each that minimize the sum of Fréchet distances over all input curves to their closest median curve. A major shortcoming of their algorithm is that the input curves are restricted to lie on the real line. In this paper, we present a randomized bicriteria-approximation algorithm that works for polygonal curves in $\mathbb{R}^d$ and achieves approximation factor $(1+ε)$ with respect to the clustering costs. The algorithm has worst-case running-time linear in the number of curves, polynomial in the maximum number of vertices per curve, i.e. their complexity, and exponential in $d$, $\ell$, $ε$ and $δ$, i.e., the failure probability. We achieve this result through a shortcutting lemma, which guarantees the existence of a polygonal curve with similar cost as an optimal median curve of complexity $\ell$, but of complexity at most $2\ell-2$, and whose vertices can be computed efficiently. We combine this lemma with the superset-sampling technique by Kumar et al. to derive our clustering result. In doing so, we describe and analyze a generalization of the algorithm by Ackermann et al., which may be of independent interest.

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arXiv2020

Spatio-Temporal Top-k Similarity Search for Trajectories in Graphs

We study the problem of finding the $k$ most similar trajectories to a given query trajectory. Our work is inspired by the work of Grossi et al. [6] that considers trajectories as walks in a graph. Each visited vertex is accompanied by a time-interval. Grossi et al. define a similarity function that captures temporal and spatial aspects. We improve this similarity function to derive a new spatio-temporal distance function for which we can show that a specific type of triangle inequality is satisfied. This distance function is the basis for our index structures, which can be constructed efficiently, need only linear memory, and can quickly answer queries for the top-$k$ most similar trajectories. Our evaluation on real-world and synthetic data sets shows that our algorithms outperform the baselines with respect to indexing time by several orders of magnitude while achieving similar or better query time and quality of results.

cs.DS↗

arXiv2020

On the hardness of computing an average curve

We study the complexity of clustering curves under $k$-median and $k$-center objectives in the metric space of the Fréchet distance and related distance measures. Building upon recent hardness results for the minimum-enclosing-ball problem under the Fréchet distance, we show that also the $1$-median problem is NP-hard. Furthermore, we show that the $1$-median problem is W[1]-hard with the number of curves as parameter. We show this under the discrete and continuous Fréchet and Dynamic Time Warping (DTW) distance. This yields an independent proof of an earlier result by Bulteau et al. from 2018 for a variant of DTW that uses squared distances, where the new proof is both simpler and more general. On the positive side, we give approximation algorithms for problem variants where the center curve may have complexity at most $\ell$ under the discrete Fréchet distance. In particular, for fixed $k,\ell$ and $\varepsilon$, we give $(1+\varepsilon)$-approximation algorithms for the $(k,\ell)$-median and $(k,\ell)$-center objectives and a polynomial-time exact algorithm for the $(k,\ell)$-center objective.

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arXiv2019

The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

The Vapnik-Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set $X$ is a set of polygonal curves in $\mathbb{R}^d$ and the sets $\mathcal{R}$ are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

cs.CG↗

arXiv2019

Sublinear data structures for short Fréchet queries

We study metric data structures for curves in doubling spaces, such as trajectories of moving objects in Euclidean $\mathbb{R}^d$, where the distance between two curves is measured using the discrete Fréchet distance. We design data structures in an \emph{asymmetric} setting where the input is a curve (or a set of $n$ curves) each of complexity $m$ and the queries are with curves of complexity $k\ll m$. We show that there exist approximate data structures that are independent of the input size $N = d \cdot n \cdot m$ and we study how to maintain them dynamically if the input is given in the stream. Concretely, we study two types of data structures: (i) distance oracles, where the task is to store a compressed version of the input curve, which can be used to answer queries for the distance of a query curve to the input curve, and (ii) nearest-neighbor data structures, where the task is to preprocess a set of input curves to answer queries for the input curve closest to the query curve. In both cases we are interested in approximation. For curves embedded in Euclidean $\mathbb{R}^d$ with constant $d$, our distance oracle uses space in $\mathcal{O}((k \log(ε^{-1}) ε^{-d})^k)$ ($ε$ is the precision parameter). The oracle performs $(1+ε)$-approximate queries in time in $\mathcal{O}(k^2)$ and is deterministic. We show how to maintain this distance oracle in the stream using polylogarithmic additional memory. In the stream, we can dynamically answer distance queries to the portion of the stream seen so far in $\mathcal{O}(k^4 \log^2 m)$ time. We apply our techniques to the second problem, approximate near neighbor (ANN) data structures, and achieve an exponential improvement in the dependency on the complexity of the input curves compared to the state of the art.

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arXiv2019

FRESH: Fréchet Similarity with Hashing

This paper studies the $r$-range search problem for curves under the continuous Fréchet distance: given a dataset $S$ of $n$ polygonal curves and a threshold $r>0$, construct a data structure that, for any query curve $q$, efficiently returns all entries in $S$ with distance at most $r$ from $q$. We propose FRESH, an approximate and randomized approach for $r$-range search, that leverages on a locality sensitive hashing scheme for detecting candidate near neighbors of the query curve, and on a subsequent pruning step based on a cascade of curve simplifications. We experimentally compare \fresh to exact and deterministic solutions, and we show that high performance can be reached by suitably relaxing precision and recall.

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arXiv2018

Probabilistic embeddings of the Fréchet distance

The Fréchet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric space. However, its inherent complexity poses considerable computational challenges in practice. To address this problem we study distortion of the probabilistic embedding that results from projecting the curves to a randomly chosen line. Such an embedding could be used in combination with, e.g. locality-sensitive hashing. We show that in the worst case and under reasonable assumptions, the discrete Fréchet distance between two polygonal curves of complexity $t$ in $\mathbb{R}^d$, where $d\in\lbrace 2,3,4,5\rbrace$, degrades by a factor linear in $t$ with constant probability. We show upper and lower bounds on the distortion. We also evaluate our findings empirically on a benchmark data set. The preliminary experimental results stand in stark contrast with our lower bounds. They indicate that highly distorted projections happen very rarely in practice, and only for strongly conditioned input curves. Keywords: Fréchet distance, metric embeddings, random projections

cs.CG↗

arXiv2018

Approximating $(k,\ell)$-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 Fréchet 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 Fréchet 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 Fréchet distance. In the case of the discrete Fréchet 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 Fréchet 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.

cs.CG↗

arXiv2017

On the complexity of range searching among curves

Modern tracking technology has made the collection of large numbers of densely sampled trajectories of moving objects widely available. We consider a fundamental problem encountered when analysing such data: Given $n$ polygonal curves $S$ in $\mathbb{R}^d$, preprocess $S$ into a data structure that answers queries with a query curve $q$ and radius $ρ$ for the curves of $S$ that have \Frechet distance at most $ρ$ to $q$. We initiate a comprehensive analysis of the space/query-time trade-off for this data structuring problem. Our lower bounds imply that any data structure in the pointer model model that achieves $Q(n) + O(k)$ query time, where $k$ is the output size, has to use roughly $Ω\left((n/Q(n))^2\right)$ space in the worst case, even if queries are mere points (for the discrete \Frechet distance) or line segments (for the continuous \Frechet distance). More importantly, we show that more complex queries and input curves lead to additional logarithmic factors in the lower bound. Roughly speaking, the number of logarithmic factors added is linear in the number of edges added to the query and input curve complexity. This means that the space/query time trade-off worsens by an exponential factor of input and query complexity. This behaviour addresses an open question in the range searching literature: whether it is possible to avoid the additional logarithmic factors in the space and query time of a multilevel partition tree. We answer this question negatively. On the positive side, we show we can build data structures for the \Frechet distance by using semialgebraic range searching. Our solution for the discrete \Frechet distance is in line with the lower bound, as the number of levels in the data structure is $O(t)$, where $t$ denotes the maximal number of vertices of a curve. For the continuous \Frechet distance, the number of levels increases to $O(t^2)$.

cs.CG↗

arXiv2017

Locality-sensitive hashing of curves

We study data structures for storing a set of polygonal curves in ${\rm R}^d$ such that, given a query curve, we can efficiently retrieve similar curves from the set, where similarity is measured using the discrete Fréchet distance or the dynamic time warping distance. To this end we devise the first locality-sensitive hashing schemes for these distance measures. A major challenge is posed by the fact that these distance measures internally optimize the alignment between the curves. We give solutions for different types of alignments including constrained and unconstrained versions. For unconstrained alignments, we improve over a result by Indyk from 2002 for short curves. Let $n$ be the number of input curves and let $m$ be the maximum complexity of a curve in the input. In the particular case where $m \leq \fracα{4d} \log n$, for some fixed $α>0$, our solutions imply an approximate near-neighbor data structure for the discrete Fréchet distance that uses space in $O(n^{1+α}\log n)$ and achieves query time in $O(n^α\log^2 n)$ and constant approximation factor. Furthermore, our solutions provide a trade-off between approximation quality and computational performance: for any parameter $k \in [m]$, we can give a data structure that uses space in $O(2^{2k}m^{k-1} n \log n + nm)$, answers queries in $O( 2^{2k} m^{k}\log n)$ time and achieves approximation factor in $O(m/k)$.

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