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.

Given two polygons of complexities \(n\) and \(m\) respectively, a fundamental problem in shape matching and geometric similarity is to compute their maximum area overlap under translation. For general simple polygons, the best-known algorithm runs in \(\mathcal{O}((nm)^2 \log(nm))\) time [Mount, Silverman, Wu ’96]. In a recent breakthrough that received the SoCG Best Paper Award 2025, Chan and Hair gave a linear-time algorithm for the special case when both polygons are convex. A key challenge in computational geometry is to design improved algorithms for other natural classes of polygons. We address this by presenting an \(\mathcal{O}((nm)^{3/2} \log(nm))\)-time algorithm for the case when both polygons are orthogonal, probably the most popular class of polygons besides convex and simple ones. This is the first algorithm for polygon overlap on orthogonal polygons that is faster than the almost 30 years old algorithm for general simple polygons.

We present the winning implementation of the Seventh Computational Geometry Challenge (CG:SHOP 2025). The task in this challenge was to find non-obtuse triangulations for given planar regions, respecting a given set of constraints consisting of extra vertices and edges that must be part of the triangulation. The goal was to minimize the number of introduced Steiner points. Our approach is to maintain a constrained Delaunay triangulation, for which we repeatedly remove, relocate, or add Steiner points. We use local search to choose the action that improves the triangulation the most, until the resulting triangulation is non-obtuse.

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.

Union volume estimation is a classical algorithmic problem. Given a family of objects O₁,…,O_n ⊂ ℝ^d, we want to approximate the volume of their union. In the special case where all objects are boxes (also called hyperrectangles) this is known as Klee’s measure problem. The state-of-the-art (1+ε)-approximation algorithm [Karp, Luby, Madras '89] for union volume estimation as well as Klee’s measure problem in constant dimension d uses a total of O(n/ε²) queries of three types: (i) determine the volume of O_i; (ii) sample a point uniformly at random from O_i; and (iii) ask whether a given point is contained in O_i. First, we show that if an algorithm learns about the objects only through these types of queries, then Ω(n/ε²) queries are necessary. In this sense, the complexity of [Karp, Luby, Madras '89] is optimal. Our lower bound holds even if the objects are equiponderous axis-aligned polygons in ℝ², if the containment query allows arbitrary (not necessarily sampled) points, and if the algorithm can spend arbitrary time and space examining the query responses. Second, we provide a more efficient approximation algorithm for Klee’s measure problem, which improves the running time from O(n/ε²) to O((n+1/ε²) ⋅ log^{O(d)} (n)). We circumvent our lower bound by exploiting the geometry of boxes in various ways: (1) We sort the boxes into classes of similar shapes after inspecting their corner coordinates. (2) With orthogonal range searching, we show how to sample points from the union of boxes in each class, and how to merge samples from different classes. (3) We bound the amount of wasted work by arguing that most pairs of classes have a small intersection.

We study the problem of computing the Fréchet distance between two polygonal curves under transformations. First, we consider translations in the Euclidean plane. Given two curves $π$ and $σ$ of total complexity $n$ and a threshold $δ\geq 0$, we present an $\tilde{\mathcal{O}}(n^{7 + \frac{1}{3}})$ time algorithm to determine whether there exists a translation $t \in \mathbb{R}^2$ such that the Fréchet distance between $π$ and $σ+ t$ is at most $δ$. This improves on the previous best result, which is an $\mathcal{O}(n^8)$ time algorithm. We then generalize this result to any class of rationally parameterized transformations, which includes translation, rotation, scaling, and arbitrary affine transformations. For a class $\mathcal T$ of rationally parametrized transformations with $k$ degrees of freedom, we show that one can determine whether there is a transformation $τ\in \mathcal T$ such that the Fréchet distance between $π$ and $τ(σ)$ is at most $δ$ in $\tilde{\mathcal{O}}(n^{3k+\frac{4}{3}})$ time.

This paper introduces $k$-Dynamic Time Warping ($k$-DTW), a novel dissimilarity measure for polygonal curves. $k$-DTW has stronger metric properties than Dynamic Time Warping (DTW) and is more robust to outliers than the Fréchet distance, which are the two gold standards of dissimilarity measures for polygonal curves. We show interesting properties of $k$-DTW and give an exact algorithm as well as a $(1+\varepsilon)$-approximation algorithm for $k$-DTW by a parametric search for the $k$-th largest matched distance. We prove the first dimension-free learning bounds for curves and further learning theoretic results. $k$-DTW not only admits smaller sample size than DTW for the problem of learning the median of curves, where some factors depending on the curves' complexity $m$ are replaced by $k$, but we also show a surprising separation on the associated Rademacher and Gaussian complexities: $k$-DTW admits strictly smaller bounds than DTW, by a factor $\tilde\Omega(\sqrt{m})$ when $k\ll m$. We complement our theoretical findings with an experimental illustration of the benefits of using $k$-DTW for clustering and nearest neighbor classification.

Federated Learning (FL) enables collaborative training of machine learning models across distributed clients without sharing raw data, ostensibly preserving data privacy. Nevertheless, recent studies have revealed critical vulnerabilities in FL, showing that a malicious central server can manipulate model updates to reconstruct clients’ private training data. Existing data reconstruction attacks have important limitations: they often rely on assumptions about the clients’ data distribution or their efficiency significantly degrades when batch sizes exceed just a few tens of samples. In this work, we introduce a novel data reconstruction attack that overcomes these limitations. Our method leverages a new geometric perspective on fully connected layers to craft malicious model parameters, enabling the perfect recovery of arbitrarily large data batches in classification tasks without any prior knowledge of clients’ data. Through extensive experiments on both image and tabular datasets, we demonstrate that our attack outperforms existing methods and achieves perfect reconstruction of data batches two orders of magnitude larger than the state of the art.

Computing shortest paths is one of the most fundamental algorithmic graph problems. It is known since decades that this problem can be solved in near-linear time if all weights are nonnegative. A recent break-through by [Bernstein, Nanongkai, Wulff-Nilsen '22] presented a randomized near-linear time algorithm for this problem. A subsequent improvement in [Bringmann, Cassis, Fischer '23] significantly reduced the number of logarithmic factors and thereby also simplified the algorithm. It is surprising and exciting that both of these algorithms are combinatorial and do not contain any fundamental obstacles for being practical. We launch the, to the best of our knowledge, first extensive investigation towards a practical implementation of [Bringmann, Cassis, Fischer '23]. To this end, we give an accessible overview of the algorithm, discussing what adaptions are necessary to obtain a fast algorithm in practice. We manifest these adaptions in an efficient implementation. We test our implementation on a benchmark data set that is adapted to be more difficult for our implementation in order to allow for a fair comparison. As in [Bringmann, Cassis, Fischer '23] as well as in our implementation there are multiple parameters to tune, we empirically evaluate their effect and thereby determine the best choices. Our implementation is then extensively compared to one of the state-of-the-art algorithms for this problem [Goldberg, Radzik '93]. On the hardest instance type, we are faster by up to almost two orders of magnitude.

Given a set of n points in the Euclidean plane, the k-MinSumRadius problem asks to cover this point set using k disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV’12]; however, the running time of this algorithm is O(n881), and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the k-MinSumRadius problem is that of small k. For the 2-MinSumRadius problem, a near-quadratic time algorithm with expected running time O(n2 log2 n log2 log n) was given over 30 years ago [Eppstein’92]. We present the first improvement of this result, namely, a near-linear time algorithm to compute the 2-MinSumRadius that runs in expected O(n log2 n log2 log n) time. We generalize this result to any constant dimension d, for which we give an O(n2−1/(⌈d/2⌉+1)+ε) time algorithm. Additionally, we give a near-quadratic time algorithm for 3-MinSumRadius in the plane that runs in expected O(n2 log2 n log2 log n) time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution.

We devise a polynomial-time algorithm for partitioning a simple polygon P into a minimum number of star-shaped polygons. The question of whether such an algorithm exists has been open for more than four decades [Avis and Toussaint, Pattern Recognit., 1981] and it has been repeated frequently, for example in O’Rourke’s famous book [Art Gallery Theorems and Algorithms, 1987]. In addition to its strong theoretical motivation, the problem is also motivated by practical domains such as CNC pocket milling, motion planning, and shape parameterization.

This work describes the winning implementation of the CG:SHOP 2023 Challenge. The topic of the Challenge was the convex cover problem: given a polygon P (with holes), find a minimum-cardinality set of convex polygons whose union equals P. We use a three-step approach: (1) Create a suitable partition of P. (2) Compute a visibility graph of the pieces of the partition. (3) Solve a vertex clique cover problem on the visibility graph, from which we then derive the convex cover. This way we capture the geometric difficulty in the first step and the combinatorial difficulty in the third step.

Hyperbolicity is a graph parameter related to how much a graph resembles a tree with respect to distances. Its computation is challenging as the main approaches consist in scanning all quadruples of the graph or using fast matrix multiplication as building block, both are not practical for large graphs. In this paper, we propose and evaluate an approach that uses a hierarchy of distance-k dominating sets to reduce the search space. This technique, compared to the previous best practical algorithms, enables us to compute the hyperbolicity of graphs with unprecedented size (up to a million nodes).

The Dynamic Time Warping (DTW) distance is a popular measure of similarity for a variety of sequence data. For comparing polygonal curves π, σ in ℝ^d, it provides a robust, outlier-insensitive alternative to the Fréchet distance. However, like the Fréchet distance, the DTW distance is not invariant under translations. Can we efficiently optimize the DTW distance of π and σ under arbitrary translations, to compare the curves' shape irrespective of their absolute location? There are surprisingly few works in this direction, which may be due to its computational intricacy: For the Euclidean norm, this problem contains as a special case the geometric median problem, which provably admits no exact algebraic algorithm (that is, no algorithm using only addition, multiplication, and k-th roots). We thus investigate exact algorithms for non-Euclidean norms as well as approximation algorithms for the Euclidean norm. For the L₁ norm in ℝ^d, we provide an 𝒪(n^{2(d+1)})-time algorithm, i.e., an exact polynomial-time algorithm for constant d. Here and below, n bounds the curves' complexities. For the Euclidean norm in ℝ², we show that a simple problem-specific insight leads to a (1+ε)-approximation in time 𝒪(n³/ε²). We then show how to obtain a subcubic 𝒪̃(n^{2.5}/ε²) time algorithm with significant new ideas; this time comes close to the well-known quadratic time barrier for computing DTW for fixed translations. Technically, the algorithm is obtained by speeding up repeated DTW distance estimations using a dynamic data structure for maintaining shortest paths in weighted planar digraphs. Crucially, we show how to traverse a candidate set of translations using space-filling curves in a way that incurs only few updates to the data structure. We hope that our results will facilitate the use of DTW under translation both in theory and practice, and inspire similar algorithmic approaches for related geometric optimization problems.

We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in $d$-dimensional Euclidean space, such as balls, segments, or hypercubes, and whose edges correspond to pairs of intersecting shapes. The diameter of a graph is the largest distance realized by a pair of vertices in the graph. Computing the diameter in near-quadratic time is possible in several classes of intersection graphs [Chan and Skrepetos 2019], but it is not at all clear if these algorithms are optimal, especially since in the related class of planar graphs the diameter can be computed in $\widetilde{\mathcal{O}}(n^{5/3})$ time [Cabello 2019, Gawrychowski et al. 2021]. In this work we (conditionally) rule out sub-quadratic algorithms in several classes of intersection graphs, i.e., algorithms of running time $\mathcal{O}(n^{2-δ})$ for some $δ>0$. In particular, there are no sub-quadratic algorithms already for fat objects in small dimensions: unit balls in $\mathbb{R}^3$ or congruent equilateral triangles in $\mathbb{R}^2$. For unit segments and congruent equilateral triangles, we can even rule out strong sub-quadratic approximations already in $\mathbb{R}^2$. It seems that the hardness of approximation may also depend on dimensionality: for axis-parallel unit hypercubes in~$\mathbb{R}^{12}$, distinguishing between diameter 2 and 3 needs quadratic time (ruling out $(3/2-\varepsilon)$- approximations), whereas for axis-parallel unit squares, we give an algorithm that distinguishes between diameter $2$ and $3$ in near-linear time. Note that many of our lower bounds match the best known algorithms up to sub-polynomial factors.

Dynamic Time Warping is arguably the most popular similarity measure for time series, where we define a time series to be a one-dimensional polygonal curve. The drawback of Dynamic Time Warping is that it is sensitive to the sampling rate of the time series. The Fréchet distance is an alternative that has gained popularity, however, its drawback is that it is sensitive to outliers. Continuous Dynamic Time Warping (CDTW) is a recently proposed alternative that does not exhibit the aforementioned drawbacks. CDTW combines the continuous nature of the Fréchet distance with the summation of Dynamic Time Warping, resulting in a similarity measure that is robust to sampling rate and to outliers. In a recent experimental work of Brankovic et al., it was demonstrated that clustering under CDTW avoids the unwanted artifacts that appear when clustering under Dynamic Time Warping and under the Fréchet distance. Despite its advantages, the major shortcoming of CDTW is that there is no exact algorithm for computing CDTW, in polynomial time or otherwise. In this work, we present the first exact algorithm for computing CDTW of one-dimensional curves. Our algorithm runs in time $O(n^5)$ for a pair of one-dimensional curves, each with complexity at most $n$. In our algorithm, we propagate continuous functions in the dynamic program for CDTW, where the main difficulty lies in bounding the complexity of the functions. We believe that our result is an important first step towards CDTW becoming a practical similarity measure between curves.

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 ≤ 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 · 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 < ∊ ≤ 1 an approximation factor of (1 + ∊) can be achieved within the same asymptotic time bounds as the previously best result for (2 + ∊). Moreover, we show that an approximation factor of (2 + ∊) can be obtained by using preprocessing time and space O(nm), which is linear in the input size, and query time in , where the previously best result used preprocessing time in 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.

We revisit a classical problem in computational geometry: Determine the largest copy of a simple polygon P that can be placed into the simple polygon Q. Despite significant effort studying a number of settings, known algorithms require high polynomial running times, even for the interesting case when either P or Q have constant size. (Barequet and Har-Peled, 2001) give a conditional lower bound of n2–o(1) under the 3SUM conjecture when P and Q are (convex) polygons with Θ(n) vertices each. This leaves open whether we can establish (1) hardness beyond quadratic time and (2) any superlinear bound for constant-sized P or Q. In this paper, we affirmatively answer these questions under the higher-order kSUM conjecture, proving natural hardness results that increase with each degree of freedom (scaling, x-translation, y-translation, rotation): (scaling, x-translation:) Finding the largest copy of P that can be x-translated into Q requires time n2–o(1) under the 3SUM conjecture, even for orthogonal (rectilinear) polygons P, Q with O(1) and n vertices, respectively. (scaling, x-translation, y-translation:) Finding the largest copy of P that can be arbitrarily translatedinto Q requires time n2–o(1) under the 4SUM conjecture, even for orthogonal polygons P, Q with O(1) and n vertices, respectively. This establishes the same lower bound under the assumption that Subset Sum cannot be solved in time O((2–∊)n/2) for any ∊ > 0. The above lower bounds are almost tight when one of the polygons is of constant size: Using an offline dynamic algorithm for maintaining the area of a union of rectangles due to Overmars and Yap, we obtain an -time algorithm for orthogonal polygons P, Q with p and q vertices, respectively. This matches the lower bounds up to an n1/2+o(1)-factor when P, Q have O(1) and n vertices. (scaling, x-translation, y-translation, rotation:) Finally, finding the largest copy of P that can be arbitrarily rotated and translated into Q requires time n3–o(1) under the 5SUM conjecture. As in our reductions, each degree of freedom determines one summand in a kSUM instance, these lower bounds appear likely to be best possible under kSUM. We are not aware of any other such natural (degree of freedom + 1)-SUM hardness for a geometric optimization problem. Finally, we prove an additional tight OV hardness of the translations-only case.

Abstract The 2-Choices dynamics is a process that models voting behavior on networks and works as follows: Each agent initially holds either opinion blue or red; then, in each round, each agent looks at two random neighbors and, if the two have the same opinion, the agent adopts it. We study its behavior on a class of networks with core–periphery structure. Assume that a densely-connected subset of agents, the core, holds a different opinion from the rest of the network, the periphery. We prove that, depending on the strength of the cut between core and periphery, a phase-transition phenomenon occurs: Either the core’s opinion rapidly spreads across the network, or a metastability phase takes place in which both opinions coexist for superpolynomial time. The interest of our result, which we also validate with extensive experiments on real networks, is twofold. First, it sheds light on the influence of the core on the rest of the network as a function of its connectivity toward the latter. Second, it is one of the first analytical results which shows a heterogeneous behavior of a simple dynamics as a function of structural parameters of the network.

Abstract 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 $$\mathbb {R}^d$$ R d and the sets $$\mathcal {R}$$ 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 and lower 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.

The discrete Fréchet distance is a popular measure for comparing polygonal curves. An important variant is the discrete Fréchet distance under translation, which enables detection of similar movement patterns in different spatial domains. For polygonal curves of length n in the plane, the fastest known algorithm runs in time Õ( n 5 ) [12]. This is achieved by constructing an arrangement of disks of size Õ( n 4 ), and then traversing its faces while updating reachability in a directed grid graph of size N := Õ( n 5 ), which can be done in time Õ(√ N ) per update [27]. The contribution of this article is two-fold. First, although it is an open problem to solve dynamic reachability in directed grid graphs faster than Õ(√ N ), we improve this part of the algorithm: We observe that an offline variant of dynamic s - t -reachability in directed grid graphs suffices, and we solve this variant in amortized time Õ( N 1/3 ) per update, resulting in an improved running time of Õ( N 4.66 ) for the discrete Fréchet distance under translation. Second, we provide evidence that constructing the arrangement of size Õ( N 4 ) is necessary in the worst case by proving a conditional lower bound of n 4 - o(1) on the running time for the discrete Fréchet distance under translation, assuming the Strong Exponential Time Hypothesis.

Consider the natural question of how to measure the similarity of curves in the plane by a quantity that is invariant under translations of the curves. Such a measure is justified whenever we aim to quantify the similarity of the curves' shapes rather than their positioning in the plane, e.g., to compare the similarity of handwritten characters. Perhaps the most natural such notion is the (discrete) Fréchet distance under translation. Unfortunately, the algorithmic literature on this problem yields a very pessimistic view: On polygonal curves with $n$ vertices, the fastest algorithm runs in time $O(n^{4.667})$ and cannot be improved below $n^{4-o(1)}$ unless the Strong Exponential Time Hypothesis fails. Can we still obtain an implementation that is efficient on realistic datasets? Spurred by the surprising performance of recent implementations for the Fréchet distance, we perform algorithm engineering for the Fréchet distance under translation. Our solution combines fast, but inexact tools from continuous optimization (specifically, branch-and-bound algorithms for global Lipschitz optimization) with exact, but expensive algorithms from computational geometry (specifically, problem-specific algorithms based on an arrangement construction). We combine these two ingredients to obtain an exact decision algorithm for the Fréchet distance under translation. For the related task of computing the distance value up to a desired precision, we engineer and compare different methods. On a benchmark set involving handwritten characters and route trajectories, our implementation answers a typical query for either task in the range of a few milliseconds up to a second on standard desktop hardware. We believe that our implementation will enable the use of the Fréchet distance under translation in applications, whereas previous approaches would have been computationally infeasible.

Due to the massively increasing amount of available geospatial data and the need to present it in an understandable way, clustering this data is more important than ever. As clusters might contain a large number of objects, having a representative for each cluster significantly facilitates understanding a clustering. Clustering methods relying on such representatives are called center-based. In this work we consider the problem of center-based clustering of trajectories.

The Fréchet distance provides a natural and intuitive measure for the popular task of computing the similarity of two (polygonal) curves. While a simple algorithm computes it in near-quadratic time, a strongly subquadratic algorithm cannot exist unless the Strong Exponential Time Hypothesis fails. Still, fast practical implementations of the Fréchet distance, in particular for realistic input curves, are highly desirable. This has even lead to a designated competition, the ACM SIGSPATIAL GIS Cup 2017: Here, the challenge was to implement a near-neighbor data structure under the Fréchet distance. The bottleneck of the top three implementations turned out to be precisely the decision procedure for the Fréchet distance. In this work, we present a fast, certifying implementation for deciding the Fréchet distance, in order to (1) complement its pessimistic worst-case hardness by an empirical analysis on realistic input data and to (2) improve the state of the art for the GIS Cup challenge. We experimentally evaluate our implementation on a large benchmark consisting of several data sets (including handwritten characters and GPS trajectories). Compared to the winning implementation of the GIS Cup, we obtain running time improvements of up to more than two orders of magnitude for the decision procedure and of up to a factor of 30 for queries to the near-neighbor data structure.

Center-based clustering, in particular k-means clustering, is frequently used for point data. Its advantages include that the resulting clustering is often easy to interpret and that the cluster centers provide a compact representation of the data. Recent theoretical advances have been made in generalizing center-based clustering to trajectory data. Building upon these theoretical results, we present practical algorithms for center-based trajectory clustering.

The discrete Fréchet distance is a popular measure for comparing polygonal curves. An important variant is the discrete Fréchet distance under translation, which enables detection of similar movement patterns in different spatial domains. For polygonal curves of length n in the plane, the fastest known algorithm runs in time Õ(n5) [Ben Avraham, Kaplan, Sharir ′15]. This is achieved by constructing an arrangement of disks of size O(n4), and then traversing its faces while updating reachability in a directed grid graph of size N = O(n2), which can be done in time per update [Diks, Sankowski ′07]. The contribution of this paper is two-fold. First, although it is an open problem to solve dynamic reachability in directed grid graphs faster than , we improve this part of the algorithm: We observe that an offline variant of dynamic s-t-reachability in directed grid graphs suffices, and we solve this variant in amortized time Õ(N1/3) per update, resulting in an improved running time of Õ(n4.66 …) for the discrete Fréchet distance under translation. Second, we provide evidence that constructing the arrangement of size O(n4) is necessary in the worst case, by proving a conditional lower bound of n4–o(1) on the running time for the discrete Fréchet distance under translation, assuming the Strong Exponential Time Hypothesis.

We consider the problem of indexing massive trajectory data in an underlying road network. Our Pathfinder index structure is based on a state-of-the-art speed-up technique for shortest path planning and allows to both compress and access huge amounts of trajectory data. In a continent-sized network with more than 400 million nodes and almost a billion edges, Pathfinder allows to retrieve all trajectories within a given space-time cube in a few microseconds per reported trajectory. The applicability of Pathfinder is shown using both synthetic and real-world trajectory sets.

A grid maze is a binary matrix where fields containing a 0 are accessible while fields containing a 1 are blocked. A movement sequence consists of relative movements up, down, left, right – moving to a blocked field results in non-movement. The simultaneous maze solving problem asks for the shortest movement sequence starting in the upper left corner and visiting the lower right corner for all mazes of size n × m (for which a path from the upper left to the lower right corner exists at all). We present a theoretical problem analysis, including hardness results and a cubic upper bound on the sequence length. In addition, we describe several approaches to practically compute solving sequences and lower bounds despite the high combinatorial complexity of the problem.

Given a set of prioritized balls with fixed centers in ℝd whose radii grow linearly over time, we want to compute the elimination order of these balls assuming that when two balls touch, the one with lower priority is ‘crushed’. A straightforward algorithm has running time O(n2 log n) which we improve to expected O(Δdn(log n + Δd)) where Δ = rmax/rmin is the ratio between largest and smallest radius amongst the balls. For a natural application of this problem, namely drawing labels on the globe, we have Δ = O(1). An efficient implementation based on a spherical Delaunay triangulation allows to compute the elimination order for millions of labels on commodity Desktop hardware. Dealing with rounding error induced robustness issues turned out to be one of the major challenges in the implementation.

Compared to conventional cars, electric vehicles (EVs) still suffer from considerably shorter cruising ranges. Combined with the sparsity of battery loading stations, the complete transition to E-mobility still seems a long way to go. In this paper, we consider the problem of placing as few loading stations as possible so that on any shortest path there are sufficiently many not to run out of energy. We show how to model this problem and introduce heuristics which provide close-to-optimal solutions even in large road networks.

Compared to conventional cars, electric vehicles still suffer from a considerably shorter cruising range. Combined with the sparsity of battery loading stations, the complete transition to E-mobility still seems a long way to go. In this paper, we consider the problem of placing as few loading stations as possible such that on any shortest path there are enough to guarantee sufficient energy supply. This means, that EV owners no longer have to plan their trips ahead incorporating loading station locations, and are no longer forced to accept long detours to reach their destinations. We show how to model this problem and introduce heuristics which provide close-to-optimal solutions even in large road networks.

For a directed graph G with vertex set V we call a subset C ⊆ V a k-(All-)Path Cover if C contains a node from any path consisting of k nodes. This paper considers the problem of constructing small k -Path Covers in the context of road networks with millions of nodes and edges. In many application scenarios the set C and its induced overlay graph constitute a very compact synopsis of G which is the basis for the currently fastest data structure for personalized shortest path queries, visually pleasing overlays of subsampled paths, and efficient reporting, retrieval and aggregation of associated data in spatial network databases. Apart from a theoretical investigation of the problem, we provide efficient algorithms that produce very small k -Path Covers for large real-world road networks (with a posteriori guarantees via instance-based lower bounds).