arXiv subjects

Anne Driemel

Publications and source records attributed to Anne Driemel.

At least 19 recordsLinked to original sources

arXiv2026

The Quick Dog Jumps the Log

We give linear-time, and thus optimal, $(1+\varepsilon)$-approximation algorithms for numerous variants of the Frechet distance between $c$-packed curves (where $c \in O(1)$), removing an additional log factor that was present in previous algorithms. The key to our new algorithms is a linear-size approximation of the elevation function, which uses a decomposition of the domain into rectangles, and a careful implicit dynamic programming on this decomposition. The algorithm extends to the strong, weak, discrete, and continuous Frechet distances with a running time of roughly $O(cn/\varepsilon)$. The $c$-packedness assumption is used only in the analysis, and the algorithm is simple and should work efficiently for other inputs.

cs.CG↗

arXiv2026

Time Series Decomposition using the Fr\'echet Distance

In this paper, we introduce a new data analysis problem that aims to decompose a set of univariate time series into a small set of $k$ base curves of length at most $l$ such that the sum of Fr\'echet distances of the time series to a ``Fr\'echet combination'' of the base curves is minimized. Here, a Fr\'echet combination allows to combine individually scaled base curves using a $k$-dimensional traversal. We call the problem of finding a set of optimal base curves the Fr\'echet decomposition problem and we consider two variants: (a) the base curves can be arbitrary curves of bounded length and (b) the curves come from a given finite set of candidate curves. We think of the Fr\'echet decomposition problem as a Fr\'echet variant of principal component analysis. For the case of a single base curve we develop a $(1+\varepsilon)$-approximation algorithm for the Fr\'echet decomposition problem. Additionally we give an exact algorithm for the projection distance problem that asks to compute the distance of one given time series to a given set of $k$ base curves. This allows us to design an exact algorithm for the Fr\'echet decomposition problem for general $k$ when curves come from a fixed candidate set.

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arXiv2026

Near Linear Time Approximation Schemes for Clustering of Partially Doubling Metrics

Given a finite metric space $(X\cup Y, \mathbf{d})$ the $k$-median problem is to find a set of $k$ centers $C\subseteq Y$ that minimizes $\sum_{p\in X} \min_{c\in C} \mathbf{d}(p,c)$. In general metrics, the best polynomial time algorithm computes a $(2+\epsilon)$-approximation for arbitrary $\epsilon>0$ (Cohen-Addad et al. STOC 2025). However, if the metric is doubling, a near linear time $(1+\epsilon)$-approximation algorithm is known (Cohen-Addad et al. J. ACM 2021). We show that the $(1+\epsilon)$-approximation algorithm can be generalized to the case when either $X$ or $Y$ has bounded doubling dimension (but the other set not). The case when $X$ is doubling is motivated by the assumption that even though $X$ is part of a high-dimensional space, it may be that it is close to a low-dimensional structure. The case when $Y$ is doubling is motivated by specific clustering problems where the centers are low-dimensional. Specifically, our work in this setting implies the first near linear time approximation algorithm for the $(k,\ell)$-median problem under discrete Fr\'echet distance when $\ell$ is constant. We further introduce a novel complexity reduction for time series of real values that leads to a similar result for the case of discrete Fr\'echet distance. In order to solve the case when $Y$ has a bounded doubling dimension, we introduce a dimension reduction that replaces points from $X$ by sets of points in $Y$. To solve the case when $X$ has a bounded doubling dimension, we generalize Talwar's decomposition (Talwar STOC 2004) to our setting. The running time of our algorithms is $2^{2^t} \tilde O(n+m)$ where $t=O(\mathrm{ddim} \log \frac{\mathrm{ddim}}{\epsilon})$ and where $\mathrm{ddim}$ is the doubling dimension of $X$ (resp.\ $Y$). The results also extend to the metric facility location problem.

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arXiv2025

A near-linear time approximation scheme for $(k,\ell)$-median clustering under discrete Fr\'echet distance

A time series of complexity $m$ is a sequence of $m$ real valued measurements. The discrete Fr\'echet distance $d_{dF}(x,y)$ is a distance measure between two time series $x$ and $y$ of possibly different complexity. Given a set of $n$ time series represented as $m$-dimensional vectors over the reals, the $(k,\ell)$-median problem under discrete Fr\'echet distance aims to find a set $C$ of $k$ time series of complexity $\ell$ such that $$\sum_{x\in P} \min_{c\in C} d_{dF}(x,c)$$ is minimized. In this paper, we give the first near-linear time $(1+\varepsilon)$-approximation algorithm for this problem when $\ell$ and $\varepsilon$ are constants but $k$ can be as large as $\Omega(n)$. We obtain our result by introducing a new dimension reduction technique for discrete Fr\'echet distance and then adapt an algorithm of Cohen-Addad et al. (J. ACM 2021) to work on the dimension-reduced input. As a byproduct we also improve the best coreset construction for $(k,\ell)$-median under discrete Fr\'echet distance (Cohen-Addad et al., SODA 2025) and show that its size can be independent of the number of input time series \emph{ and } their complexity.

cs.DS↗

arXiv2025

Subtrajectory Clustering and Coverage Maximization in Cubic Time, or Better

Many application areas collect unstructured trajectory data. In subtrajectory clustering, one is interested to find patterns in this data using a hybrid combination of segmentation and clustering. We analyze two variants of this problem based on the well-known \textsc{SetCover} and \textsc{CoverageMaximization} problems. In both variants the set system is induced by metric balls under the Fr\'echet distance centered at polygonal curves. Our algorithms focus on improving the running time of the update step of the generic greedy algorithm by means of a careful combination of sweeps through a candidate space. In the first variant, we are given a polygonal curve $P$ of complexity $n$, distance threshold $\Delta$ and complexity bound $\ell$ and the goal is to identify a minimum-size set of center curves $\mathcal{C}$, where each center curve is of complexity at most $\ell$ and every point $p$ on $P$ is covered. A point $p$ on $P$ is covered if it is part of a subtrajectory $\pi_p$ of $P$ such that there is a center $c\in\mathcal{C}$ whose Fr\'echet distance to $\pi_p$ is at most $\Delta$. We present an approximation algorithm for this problem with a running time of $O((n^2\ell + \sqrt{k_\Delta}n^{5/2})\log^2n)$, where $k_\Delta$ is the size of an optimal solution. The algorithm gives a bicriterial approximation guarantee that relaxes the Fr\'echet distance threshold by a constant factor and the size of the solution by a factor of $O(\log n)$. The second problem variant asks for the maximum fraction of the input curve $P$ that can be covered using $k$ center curves, where $k\leq n$ is a parameter to the algorithm. Here, we show that our techniques lead to an algorithm with a running time of $O((k+\ell)n^2\log^2 n)$ and similar approximation guarantees. Note that in both algorithms $k,k_\Delta\in O(n)$ and hence the running time is cubic, or better if $k\ll n$.

cs.CG↗

arXiv2025

Range Counting Oracles for Geometric Problems

In this paper, we study estimators for geometric optimization problems in the sublinear geometric model. In this model, we have oracle access to a point set with size $n$ in a discrete space $[\Delta]^d$, where queries can be made to an oracle that responds to orthogonal range counting requests. The query complexity of an optimization problem is measured by the number of oracle queries required to compute an estimator for the problem. We investigate two problems in this framework, the Euclidean Minimum Spanning Tree (MST) and Earth Mover Distance (EMD). For EMD, we show the existence of an estimator that approximates the cost of EMD with $O(\log \Delta)$-relative error and $O(\frac{n\Delta}{s^{1+1/d}})$-additive error using $O(s\polylog \Delta)$ range counting queries for any parameter $s$ with $1\leq s \leq n$. Moreover, we prove that this bound is tight. For MST, we demonstrate that the weight of MST can be estimated within a factor of $(1 \pm \eps)$ using $\tilde{O}(\sqrt{n})$ range counting queries.

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arXiv2025

Property Testing of Curve Similarity

We propose sublinear algorithms for probabilistic testing of the discrete and continuous Fr\'echet distance - a standard similarity measure for curves. We assume the algorithm is given access to the input curves via a query oracle: a query returns the set of vertices of the curve that lie within a radius $\delta$ of a specified vertex of the other curve. The goal is to use a small number of queries to determine with constant probability whether the two curves are similar (i.e., their discrete Fr\'echet distance is at most $\delta$) or they are ''$\varepsilon$-far'' (for $0 < \varepsilon < 2$) from being similar, i.e., more than an $\varepsilon$-fraction of the two curves must be ignored for them to become similar. We present two algorithms which are sublinear assuming that the curves are $t$-approximate shortest paths in the ambient metric space, for some $t\ll n$. The first algorithm uses $O(\frac{t}{\varepsilon}\log\frac{t}{\varepsilon})$ queries and is given the value of $t$ in advance. The second algorithm does not have explicit knowledge of the value of $t$ and therefore needs to gain implicit knowledge of the straightness of the input curves through its queries. We show that the discrete Fr\'echet distance can still be tested using roughly $O(\frac{t^3+t^2\log n}{\varepsilon})$ queries ignoring logarithmic factors in $t$. Our algorithms work in a matrix representation of the input and may be of independent interest to matrix testing. Our algorithms use a mild uniform sampling condition that constrains the edge lengths of the curves, similar to a polynomially bounded aspect ratio. Applied to testing the continuous Fr\'echet distance of $t$-straight curves, our algorithms can be used for $(1+\varepsilon')$-approximate testing using essentially the same bounds as stated above with an additional factor of poly$(\frac{1}{\varepsilon'})$.

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arXiv2025

Transforming Dogs on the Line: On the Fr\'echet Distance Under Translation or Scaling in 1D

The Fr\'echet distance is a computational mainstay for comparing polygonal curves. The Fr\'echet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in space. It is defined as the minimum Fr\'echet distance that arises from allowing arbitrary translations of the input curves. This problem and numerous variants of the Fr\'echet distance under some transformations have been studied, with more work concentrating on the discrete Fr\'echet distance, leaving a significant gap between the discrete and continuous versions of the Fr\'echet distance under transformations. Our contribution is twofold: First, we present an algorithm for the Fr\'echet distance under translation on 1-dimensional curves of complexity n with a running time of $\mathcal{O}(n^{8/3} log^3 n)$. To achieve this, we develop a novel framework for the problem for 1-dimensional curves, which also applies to other scenarios and leads to our second contribution. We present an algorithm with the same running time of $\mathcal{O}(n^{8/3} \log^3 n)$ for the Fr\'echet distance under scaling for 1-dimensional curves. For both algorithms we match the running times of the discrete case and improve the previously best known bounds of $\tilde{\mathcal{O}}(n^4)$. Our algorithms rely on technical insights but are conceptually simple, essentially reducing the continuous problem to the discrete case across different length scales.

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arXiv2024

A faster algorithm for the Fr\'echet distance in 1D for the imbalanced case

The fine-grained complexity of computing the Fr\'echet distance has been a topic of much recent work, starting with the quadratic SETH-based conditional lower bound by Bringmann from 2014. Subsequent work established largely the same complexity lower bounds for the Fr\'echet distance in 1D. However, the imbalanced case, which was shown by Bringmann to be tight in dimensions $d\geq 2$, was still left open. Filling in this gap, we show that a faster algorithm for the Fr\'echet distance in the imbalanced case is possible: Given two 1-dimensional curves of complexity $n$ and $n^{\alpha}$ for some $\alpha \in (0,1)$, we can compute their Fr\'echet distance in $O(n^{2\alpha} \log^2 n + n \log n)$ time. This rules out a conditional lower bound of the form $O((nm)^{1-\epsilon})$ that Bringmann showed for $d \geq 2$ and any $\varepsilon>0$ in turn showing a strict separation with the setting $d=1$. At the heart of our approach lies a data structure that stores a 1-dimensional curve $P$ of complexity $n$, and supports queries with a curve $Q$ of complexity~$m$ for the continuous Fr\'echet distance between $P$ and $Q$. The data structure has size in $\mathcal{O}(n\log n)$ and uses query time in $\mathcal{O}(m^2 \log^2 n)$. Our proof uses a key lemma that is based on the concept of visiting orders and may be of independent interest. We demonstrate this by substantially simplifying the correctness proof of a clustering algorithm by Driemel, Krivo\v{s}ija and Sohler from 2015.

cs.CG↗

arXiv2024

On the number of iterations of the DBA algorithm

The DTW Barycenter Averaging (DBA) algorithm is a widely used algorithm for estimating the mean of a given set of point sequences. In this context, the mean is defined as a point sequence that minimises the sum of dynamic time warping distances (DTW). The algorithm is similar to the $k$-means algorithm in the sense that it alternately repeats two steps: (1) computing an optimal assignment to the points of the current mean, and (2) computing an optimal mean under the current assignment. The popularity of DBA can be attributed to the fact that it works well in practice, despite any theoretical guarantees to be known. In our paper, we aim to initiate a theoretical study of the number of iterations that DBA performs until convergence. We assume the algorithm is given $n$ sequences of $m$ points in $\mathbb{R}^d$ and a parameter $k$ that specifies the length of the mean sequence to be computed. We show that, in contrast to its fast running time in practice, the number of iterations can be exponential in $k$ in the worst case - even if the number of input sequences is $n=2$. We complement these findings with experiments on real-world data that suggest this worst-case behaviour is likely degenerate. To better understand the performance of the algorithm on non-degenerate input, we study DBA in the model of smoothed analysis, upper-bounding the expected number of iterations in the worst case under random perturbations of the input. Our smoothed upper bound is polynomial in $k$, $n$ and $d$, and for constant $n$, it is also polynomial in $m$. For our analysis, we adapt the set of techniques that were developed for analysing $k$-means and observe that this set of techniques is not sufficient to obtain tight bounds for general $n$.

cs.CG↗

arXiv2024

Range Reporting for Time Series via Rectangle Stabbing

We study the Fr\'echet queries problem. It is a data structure problem, where we are given a set $S$ of $n$ polygonal curves and a distance threshold $\rho$. The data structure should support queries with a polygonal curve $q$ for the elements of $S$, for which the continuous Fr\'echet distance to $q$ is at most $\rho$. Afshani and Driemel in 2018 studied this problem for two-dimensional polygonal curves and gave upper and lower bounds on the space-query time tradeoff. We study the case that the ambient space of the curves is one-dimensional and show an intimate connection to the well-studied rectangle stabbing problem. Here, we are given a set of hyperrectangles as input and a query with a point $q$ should return all input rectangles that contain this point. Using known data structures for rectangle stabbing or orthogonal range searching this directly leads to a data structure with $\mathcal{O}(n \log ^{t-1} n)$ storage and $\mathcal{O}(\log^{t-1} n+k)$ query time, where $k$ denotes the output size and $t$ can be chosen as the maximum number of vertices of either (a) the stored curves or (b) the query curves. The resulting bounds improve upon the bounds by Afshani and Driemel in both the storage and query time. In addition, we show that known lower bounds for rectangle stabbing and orthogonal range reporting with dimension parameter $d= \lfloor t/2 \rfloor$ can be applied to our problem via reduction. .

cs.CG↗

arXiv2024

Revisiting the Fr\'echet distance between piecewise smooth curves

Since its introduction to computational geometry by Alt and Godau in 1992, the Fr\'echet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal curves, with the notable exception of an algorithm for the decision problem for planar piecewise smooth curves due to Rote (2007). We present an algorithm for the decision problem for piecewise smooth curves that is both conceptually simpler and naturally extends to the first algorithm for the problem for piecewise smooth curves in $\mathbb{R}^d$. We assume that the algorithm is given two continuous curves, each consisting of a sequence of $m$, resp.\ $n$, smooth pieces, where each piece belongs to a sufficiently well-behaved class of curves, such as the set of algebraic curves of bounded degree. We introduce a decomposition of the free space diagram into a controlled number of pieces that can be used to solve the decision problem similarly to the polygonal case, in $O(mn)$ time, leading to a computation of the Fr\'echet distance that runs in $O(mn\log(mn))$ time. Furthermore, we study approximation algorithms for piecewise smooth curves that are also $c$-packed for some fixed value $c$. We adapt the existing framework for $(1+\epsilon)$-approximations and show that an approximate decision can be computed in $O(cn/\epsilon)$ time for any $\epsilon > 0$.

cs.CG↗

arXiv2023

Finding Complex Patterns in Trajectory Data via Geometric Set Cover

Clustering trajectories is a central challenge when faced with large amounts of movement data such as GPS data. We study a clustering problem that can be stated as a geometric set cover problem: Given a polygonal curve of complexity $n$, find the smallest number $k$ of representative trajectories of complexity at most $l$ such that any point on the input trajectories lies on a subtrajectory of the input that has Fr\'echet distance at most $\Delta$ to one of the representative trajectories. In previous work, Br\"uning et al.~(2022) developed a bicriteria approximation algorithm that returns a set of curves of size $O(kl\log(kl))$ which covers the input with a radius of $11\Delta$ in time $\widetilde{O}((kl)^2n + kln^3)$, where $k$ is the smallest number of curves of complexity $l$ needed to cover the input with a radius of $\Delta$. The representative trajectories computed by this algorithm are always line segments. In the applications however, one is usually interested in more complex representative curves which consist of several edges. We present a new approach that builds upon previous work computing a set of curves of size $O(k\log(n))$ in time $\widetilde{O}(l^2n^4 + kln^4)$ with the same distance guarantee of $11\Delta$, where each curve may consist of curves of complexity up to the given complexity parameter~$l$. We conduct experiments on tracking data of ocean currents and full body motion data suggesting its validity as a tool for analyzing large spatio-temporal data sets.

cs.CG↗

arXiv2023

Solving Proximity Problems for Polygonal Shapes via Polynomials

We study range spaces, where the ground set consists of either polygonal curves in $\mathbb{R}^d$ or polygonal regions in the plane that may contain holes and the ranges are balls defined by distance measures, such as the Hausdorff distance, the Fr\'echet distance and the dynamic time warping distance. These range spaces appear in various applications like classification, simplification, range searching, density estimation and clustering when the instances are trajectories, time series or polygons. We show for the Fr\'echet distance of polygonal curves and the Hausdorff distance of polygonal curves and planar polygonal regions that the VC-dimension is upper-bounded by $O(dk\log(km))$, where $k$ is the complexity of the center of a ball, $m$ is the complexity of each polygonal curve or region in the ground set, and $d$ is the ambient dimension. For $d \geq 4$ this bound is tight in each of the parameters $d,k$ and $m$ separately. For a variant of dynamic time warping distance of polygonal curves using squared distances, our analysis directly yields an upper-bound of $O(\min(dk^2\log(m),dkm\log(k)))$. We also obtain exact solutions that are hitherto unknown for simplification, range searching, nearest neighbor search, and distance oracle.

cs.CG↗

arXiv2023

$(1+\varepsilon)$-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 $c$.

cs.CG↗

arXiv2022

Learning Depth Vision-Based Personalized Robot Navigation From Dynamic Demonstrations in Virtual Reality

For the best human-robot interaction experience, the robot's navigation policy should take into account personal preferences of the user. In this paper, we present a learning framework complemented by a perception pipeline to train a depth vision-based, personalized navigation controller from user demonstrations. Our virtual reality interface enables the demonstration of robot navigation trajectories under motion of the user for dynamic interaction scenarios. The novel perception pipeline enrolls a variational autoencoder in combination with a motion predictor. It compresses the perceived depth images to a latent state representation to enable efficient reasoning of the learning agent about the robot's dynamic environment. In a detailed analysis and ablation study, we evaluate different configurations of the perception pipeline. To further quantify the navigation controller's quality of personalization, we develop and apply a novel metric to measure preference reflection based on the Fr\'echet Distance. We discuss the robot's navigation performance in various virtual scenes and demonstrate the first personalized robot navigation controller that solely relies on depth images. A supplemental video highlighting our approach is available online.

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arXiv2022

Analysis of a Greedy Heuristic for the Labeling of a Map with a Time-Window Interface

In this paper, we analyze the approximation quality of a greedy heuristic for automatic map labeling. As input, we have a set of events, each associated with a label at a fixed position, a timestamp, and a weight. Let a time-window labeling be a selection of these labels such that all corresponding timestamps lie in a queried time window and no two labels overlap. A solution to the time-window labeling problem consists of a data structure that encodes a time-window labeling for each possible time window; when a user specifies a time window of interest using a slider interface, we query the data structure for the corresponding labeling. We define the quality of a time-window labeling solution as the sum of the weights of the labels in each time-window labeling, integrated over all time windows. We aim at maximizing the quality under the condition that a label may never disappear when the user shrinks the time window. In this paper, we analyze how well a greedy heuristic approximates the maximum quality that can be realized under this condition. On the one hand, we present an instance with square labels of equal size and equal weight for which the greedy heuristic fails to find a solution of at least 1/4 of the quality of an optimal solution. On the other hand, we prove that the greedy heuristic does guarantee a solution with at least 1/8 of the quality of an optimal solution. In the case of disk-shaped labels of equal size and equal weight, the greedy heuristic gives a solution with at least 1/10 of the quality of an optimal solution. If the labels are squares or disks of equal size and the maximum weight divided by the minimum weight is at most b, then the greedy heuristic has approximation ratio Theta(log b).

cs.CG↗

arXiv2022

Pattern matching under DTW distance

In this work, we consider the problem of pattern matching under the dynamic time warping (DTW) distance motivated by potential applications in the analysis of biological data produced by the third generation sequencing. To measure the DTW distance between two strings, one must "warp" them, that is, double some letters in the strings to obtain two equal-lengths strings, and then sum the distances between the letters in the corresponding positions. When the distances between letters are integers, we show that for a pattern P with m runs and a text T with n runs: 1. There is an O(m + n)-time algorithm that computes all locations where the DTW distance from P to T is at most 1; 2. There is an O(kmn)-time algorithm that computes all locations where the DTW distance from P to T is at most k. As a corollary of the second result, we also derive an approximation algorithm for general metrics on the alphabet.

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