arXiv subjects

Lotte Blank

Publications and source records attributed to Lotte Blank.

8 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

Fine-Grained Complexity of Continuous Euclidean k-Center

In the (continuous) Euclidean $k$-center problem, given $n$ points in $\mathbb{R}^d$ and an integer $k$, the goal is to find $k$ center points in $\mathbb{R}^d$ that minimize the maximum Euclidean distance from any input point to its closest center. In this paper, we establish conditional lower bounds for this problem in constant dimensions in two settings. $\bullet$ Parameterized by $k$: Assuming the Exponential Time Hypothesis (ETH), we show that there is no $f(k)n^{o(k^{1-1/d})}$-time algorithm for the Euclidean $k$-center problem. This result shows that the algorithm of Agarwal and Procopiuc [SODA 1998; Algorithmica 2002] is essentially optimal. Furthermore, our lower bound rules out any $(1+\varepsilon)$-approximation algorithm running in time $(k/\varepsilon)^{o(k^{1-1/d})}n^{O(1)}$, thereby establishing near-optimality of the corresponding approximation scheme by the same authors. $\bullet$ Small $k$: Assuming the 3-SUM hypothesis, we prove that for any $\varepsilon>0$ there is no $O(n^{2-\varepsilon})$-time algorithm for the Euclidean $2$-center problem in $\mathbb{R}^3$. This settles an open question posed by Agarwal, Ben Avraham, and Sharir [SoCG 2010; Computational Geometry 2013]. In addition, under the same hypothesis, we prove that for any $\varepsilon > 0$, the Euclidean $6$-center problem in $\mathbb{R}^2$ also admits no $O(n^{2-\varepsilon})$-time algorithm. The technical core of all our proofs is a novel geometric embedding of a system of linear equations. We construct a point set where each variable corresponds to a specific collection of points, and the geometric structure ensures that a small-radius clustering is possible if and only if the system has a valid solution.

cs.CG↗

arXiv2026

Fr\'echet Distance in the Imbalanced Case

Given two polygonal curves $P$ and $Q$ defined by $n$ and $m$ vertices with $m\leq n$, we show that the discrete Fr\'echet distance in 1D cannot be approximated within a factor of $2-\varepsilon$ in $\mathcal{O}((nm)^{1-\delta})$ time for any $\varepsilon, \delta>0$ unless OVH fails. Using a similar construction, we extend this bound for curves in 2D under the continuous or discrete Fr\'echet distance and increase the approximation factor to $1+\sqrt{2}-\varepsilon$ (resp. $3-\varepsilon$) if the curves lie in the Euclidean space (resp. in the $L_\infty$-space). This strengthens the lower bound by Buchin, Ophelders, and Speckmann to the case where $m=n^{\alpha}$ for $\alpha\in(0,1)$ and increases the approximation factor of $1.001$ by Bringmann. For the discrete Fr\'echet distance in 1D, we provide an approximation algorithm with optimal approximation factor and almost optimal running time. Further, for curves in any dimension embedded in any $L_p$ space, we present a $(3+\varepsilon)$-approximation algorithm for the continuous and discrete Fr\'echet distance using $\mathcal{O}((n+m^2)\log n)$ time, which almost matches the approximation factor of the lower bound for the $L_\infty$ metric.

cs.CG↗

arXiv2025

The Expiration Streaming Model: Diameter, $k$-Center, Counting, Sampling, and Friends

An important thread in the study of data-stream algorithms focuses on settings where stream items are active only for a limited time. We introduce a new expiration model, where each item arrives with its own arbitrary expiration time. The special case where items expire in the order that they arrive, which we call consistent expirations, contains the classical sliding-window model of Datar, Gionis, Indyk, and Motwani [SICOMP 2002] and its timestamp-based variant of Braverman and Ostrovsky [FOCS 2007]. Our first set of results explores the expiration streaming model and presents algorithms for several fundamental problems, including approximate counting, uniform sampling, and weighted sampling by efficiently tracking active items without explicitly storing them all. Naturally, these algorithms have many immediate applications, e.g., to range counting. Our second and main set of results for the expiration model designs algorithms for the diameter and k-center problems, where items are points in a metric space. Our results significantly extend those known for the special case of sliding-window streams by Cohen-Addad, Schwiegelshohn, and Sohler [ICALP 2016], and obtain a strictly better approximation factor for the diameter in the important special case of high-dimensional Euclidean metrics. We develop new decomposition and coordination techniques along with a geometric dominance framework to filter out redundant points based on both temporal and spatial proximity.

cs.DS↗

arXiv2025

Bicriteria Polygon Aggregation with Arbitrary Shapes

We study the problem of aggregating a set of polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Equivalently, this can be seen as a fence enclosure problem, where the goal is to surround the polygons with a set of closed curves. Our objective is to minimize a weighted sum of the total area and the total perimeter of the regions, which naturally extends other fencing problems and has applications in geographical information systems. Previously, this objective was only studied in a restricted variant, in which the boundary curves of the regions must be selected from a fixed subdivision of the plane. It is natural to ask whether the problem is still tractable if this restriction is removed, allowing output regions to be bounded by arbitrary curves. We provide a positive answer in the form of an algorithm with runtime $\mathcal{\tilde{O}}(n^4)$, where $n$ is the number of input vertices. To achieve this, we fully characterize the optimal solutions by showing that their boundaries are composed of input edges and circular arcs of constant radius. Additionally, we consider the parametric problem, where for every weighting factor we seek a solution that is optimal for that trade-off of area and perimeter. We show that $\mathcal{O}(n^2)$ combinatorial solutions suffice to describe all optimal solutions across all weighting factors, and provide both an exact algorithm and an approximation scheme. To make the algorithms scalable in practice, we develop engineering techniques that exploit structural properties of the solutions. Our experimental evaluation on real-world data shows linear runtime in practice, even for the parametric variant.

cs.CG↗

arXiv2025

Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D

The Fréchet distance is a computational mainstay for comparing polygonal curves. The Fréchet 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 Fréchet distance that arises from allowing arbitrary translations of the input curves. This problem and numerous variants of the Fréchet distance under some transformations have been studied, with more work concentrating on the discrete Fréchet distance, leaving a significant gap between the discrete and continuous versions of the Fréchet distance under transformations. Our contribution is twofold: First, we present an algorithm for the Fréchet 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 Fréchet 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.

cs.CG↗

arXiv2024

A faster algorithm for the Fréchet distance in 1D for the imbalanced case

The fine-grained complexity of computing the Fréchet 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 Fréchet 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 Fréchet distance in the imbalanced case is possible: Given two 1-dimensional curves of complexity $n$ and $n^α$ for some $α\in (0,1)$, we can compute their Fréchet distance in $O(n^{2α} \log^2 n + n \log n)$ time. This rules out a conditional lower bound of the form $O((nm)^{1-ε})$ 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 Fréchet 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, Krivošija and Sohler from 2015.

cs.CG↗

arXiv2024

Range Reporting for Time Series via Rectangle Stabbing

We study the Fréchet queries problem. It is a data structure problem, where we are given a set $S$ of $n$ polygonal curves and a distance threshold $ρ$. The data structure should support queries with a polygonal curve $q$ for the elements of $S$, for which the continuous Fréchet distance to $q$ is at most $ρ$. 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↗