Abstract Given a point set P in the Euclidean plane and a parameter t , we define an oriented t -spanner G as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest closed walk in G through those points is at most a factor t longer than the shortest cycle in the complete graph on P . We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a 1-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in $$\mathcal {O}(n^7)$$ time for n points, and a greedy algorithm that computes a 5-spanner in $$\mathcal {O}(n\log n)$$ time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in a plane oriented t -spanner with $$t=7.2 \cdot t_g$$ , where $$t_g$$ is an upper bound on the dilation of the greedy triangulation.

Map matching is a common task when analysing GPS tracks, such as vehicle trajectories. The goal is to match a recorded noisy polygonal curve to a path on the map, usually represented as a geometric graph. The Fréchet distance is a commonly used metric for curves, making it a natural fit. The map-matching problem is well-studied, yet until recently no-one tackled the data structure question: preprocess a given graph so that one can query the minimum Fréchet distance between all graph paths and a polygonal curve. Recently, Gudmundsson, Seybold, and Wong [Gudmundsson et al., 2023] studied this problem for arbitrary query polygonal curves and c-packed graphs. In this paper, we instead require the graphs to be λ-low-density t-spanners, which is significantly more representative of real-world networks. We also show how to report a path that minimises the distance efficiently rather than only returning the minimal distance, which was stated as an open problem in their paper.

Given a point set P in the Euclidean plane and a parameter t, we define an oriented t-spanner as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in G through those points is at most a factor t longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a 1-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in 𝒪(n⁸) time for n points, and a greedy algorithm that computes a 5-spanner in 𝒪(nlog n) time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented 𝒪(1)-spanner.

We consider the problem of computing the Fréchet distance between two curves for which the exact locations of the vertices are unknown. Each vertex may be placed in a given uncertainty region for that vertex, and the objective is to place vertices so as to minimise the Fréchet distance. This problem was recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal vertex placement at all. We present the first general algorithmic framework for this problem. We prove that it results in a polynomial-time algorithm for curves in 1D with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fréchet distance. We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than the regular Fréchet distance—and indeed we can easily prove that the problem is NP-hard in 2D—the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D.

In this article, we study a wide range of variants for computing the (discrete and continuous) Fréchet distance between uncertain curves. An uncertain curve is a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves. We prove that both problems are NP-hard for the Fréchet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Fréchet distance. In contrast, the lower bound (discrete [ 5 ] and continuous) Fréchet distance can be computed in polynomial time in some models. Furthermore, we show that computing the expected (discrete and continuous) Fréchet distance is #P-hard in some models. On the positive side, we present an FPTAS in constant dimension for the lower bound problem when Δ/δ is polynomially bounded, where δ is the Fréchet distance and Δ bounds the diameter of the regions. We also show a near-linear-time 3-approximation for the decision problem on roughly δ-separated convex regions. Finally, we study the setting with Sakoe–Chiba time bands, where we restrict the alignment between the curves, and give polynomial-time algorithms for the upper bound and expected discrete and continuous Fréchet distance for uncertainty modelled as point sets.

Let P be a simple polygon with n vertices, and let A be a set of m points or line segments inside P. We develop data structures that can efficiently count the objects from A that are visible to a query point or a query segment. Our main aim is to obtain fast, O(polylog nm), query times, while using as little space as possible. In case the query is a single point, a simple visibility-polygon-based solution achieves O(log nm) query time using O(nm²) space. In case A also contains only points, we present a smaller, O(n + m^{2+ε} log n)-space, data structure based on a hierarchical decomposition of the polygon. Building on these results, we tackle the case where the query is a line segment and A contains only points. The main complication here is that the segment may intersect multiple regions of the polygon decomposition, and that a point may see multiple such pieces. Despite these issues, we show how to achieve O(log n log nm) query time using only O(nm^{2+ε} + n²) space. Finally, we show that we can even handle the case where the objects in A are segments with the same bounds.

We study the problem of polygonal curve simplification under uncertainty, where instead of a sequence of exact points, each uncertain point is represented by a region which contains the (unknown) true location of the vertex. The regions we consider are disks, line segments, convex polygons, and discrete sets of points. We are interested in finding the shortest subsequence of uncertain points such that no matter what the true location of each uncertain point is, the resulting polygonal curve is a valid simplification of the original polygonal curve under the Hausdorff or the Fréchet distance. For both these distance measures, we present polynomial-time algorithms for this problem.

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.

In this paper we study a wide range of variants for computing the (discrete and continuous) Fréchet distance between uncertain curves. We define an uncertain curve as a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves. We prove that both problems are NP-hard for the continuous Fréchet distance, and the upper bound problem remains hard for the discrete Fréchet distance. In contrast, the lower bound discrete Fréchet distance can be computed in polynomial time using dynamic programming. Furthermore, we show that computing the expected discrete or continuous Fréchet distance is #P-hard when the uncertainty regions are modelled as point sets or line segments. On the positive side, we argue that in any constant dimension there is a FPTAS for the lower bound problem when Δ/δ is polynomially bounded, where δ is the Fréchet distance and Δ bounds the diameter of the regions. We then argue there is a near-linear-time 3-approximation for the decision problem when the regions are convex and roughly δ-separated. Finally, we study the setting with Sakoe–Chiba bands, restricting the alignment of the two curves, and give polynomial-time algorithms for upper bound and expected (discrete) Fréchet distance for point-set-modelled uncertainty regions.