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Given a graph with edge costs and vertex profits and given a budget B, the Orienteering Problem asks for a walk of cost at most B of maximum profit. Additionally, each profit may be given with a time window within it can be collected by the walk. While the Orienteering Problem and thus the version with time windows are NP-hard in general, it remains open on numerous special graph classes. Since in several applications, especially for planning a route from A to B with waypoints, the input graph can be restricted to tree-like or path-like structures, in this paper we consider orienteering on these graph classes. While the Orienteering Problem with time windows is NP-hard even on undirected paths and cycles, and remains so even if all profits must be collected, we show that for directed paths it can be solved efficiently, even if each profit can be collected in one of several time windows. The same case is shown to be NP-hard for directed cycles. Particularly interesting is the Orienteering Problem on a directed cycle with one time window per profit. We give an efficient algorithm for the case where all time windows are shorter than the length of the cycle, resulting in a 2-approximation for the general setting. We further develop a polynomial-time approximation scheme for this problem and give a polynomial algorithm for the case where all profits must be collected. For the Orienteering Problem with time windows for the edges, we give a quadratic time algorithm for undirected paths and observe that the problem is NP-hard for trees. In the variant without time windows, we show that on trees and thus on graphs with bounded tree-width the Orienteering Problem remains NP-hard. We present, however, an FPT algorithm to solve orienteering with unit profits that we then use to obtain a ($1+\varepsilon$)-approximation algorithm on graphs with arbitrary profits and bounded tree-width.

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 Fr\'echet 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 Fr\'echet distance between all graph paths and a polygonal curve. Recently, Gudmundsson, Seybold, and Wong [SODA 2023, arXiv:2211.02951] studied this problem for arbitrary query polygonal curves and $c$-packed graphs. In this paper, we instead require the graphs to be $\lambda$-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., Comment: This is an extended version of the article published in SoCG 2024, doi:10.4230/LIPIcs.SoCG.2024.27. 15 pages, 4 figures

Spanner constructions focus on the initial design of the network. However, networks tend to improve over time. In this paper, we focus on the improvement step. Given a graph and a budget $k$, which $k$ edges do we add to the graph to minimise its dilation? Gudmundsson and Wong [TALG'22] provided the first positive result for this problem, but their approximation factor is linear in $k$. Our main result is a $(2 \sqrt[r]{2} \ k^{1/r},2r)$-bicriteria approximation that runs in $O(n^3 \log n)$ time, for all $r \geq 1$. In other words, if $t^*$ is the minimum dilation after adding any $k$ edges to a graph, then our algorithm adds $O(k^{1+1/r})$ edges to the graph to obtain a dilation of $2rt^*$. Moreover, our analysis of the algorithm is tight under the Erd\H{o}s girth conjecture., Comment: To appear in ESA 2024

We study the problem of multi-agent online graph exploration, in which a team of k agents has to explore a given graph, starting and ending on the same node. The graph is initially unknown. Whenever a node is visited by an agent, its neighborhood and adjacent edges are revealed. The agents share a global view of the explored parts of the graph. The cost of the exploration has to be minimized, where cost either describes the time needed for the entire exploration (time model), or the length of the longest path traversed by any agent (energy model). We investigate graph exploration on cycles and tadpole graphs for 2-4 agents, providing optimal results on the competitive ratio in the energy model (1-competitive with two agents on cycles and three agents on tadpole graphs), and for tadpole graphs in the time model (1.5-competitive with four agents). We also show competitive upper bounds of 2 for the exploration of tadpole graphs with three agents, and 2.5 for the exploration of tadpole graphs with two agents in the time model., Comment: v2: Update Related Work, more detailed description of models in Motivation

Given a point set $P$ in the Euclidean plane and a parameter $t$, we define an \emph{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 $O(n^7)$ time for $n$ points, and a greedy algorithm that computes a $5$-spanner in $O(n\log n)$ 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 $t=19 \cdot t_g$, where $t_g$ is a upper bound on the dilation of the greedy triangulation., Comment: conference version: ESA '23

In this paper, we investigate crossing-free 3D morphs between planar straight-line drawings. We show that, for any two (not necessarily topologically equivalent) planar straight-line drawings of an $n$-vertex planar graph, there exists a piecewise-linear crossing-free 3D morph with $O(n^2)$ steps that transforms one drawing into the other. We also give some evidence why it is difficult to obtain a linear lower bound (which exists in 2D) for the number of steps of a crossing-free 3D morph.

We consider the unlabeled motion-planning problem of $m$ unit-disc robots moving in a simple polygonal workspace of $n$ edges. The goal is to find a motion plan that moves the robots to a given set of $m$ target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end. If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time $O(n \log n + mn + m^2)$ if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance $4$ apart, any two target positions are at least distance $4$ apart, and any pair of a start and a target positions is at least distance $3$ apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary., Comment: A shorter version of this paper appeared in the Proceedings of the 38th International Symposium on Computational Geometry (SoCG 2022)

We consider the following surveillance problem: Given a set $P$ of $n$ sites in a metric space and a set of $k$ robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency $L$ of a schedule is the maximum latency of any site, where the latency of a site $s$ is the supremum of the lengths of the time intervals between consecutive visits to $s$. When $k=1$ the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into $\ell$ groups, for some~$\ell \leq k$, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a $1+\varepsilon$ factor loss in the approximation factor and an $O\left(\left( k/\varepsilon \right)^k\right)$ factor loss in the runtime, for any $\varepsilon >0$. Our second main result shows that an optimal cyclic solution is a $2(1-1/k)$-approximation of the overall optimal solution. Note that for $k=2$ this implies that an optimal cyclic solution is optimal overall. The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a $(2(1-1/k)+\varepsilon)$-approximation of the optimal unrestricted solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting., Comment: This paper is accepted in the 38th International Symposium on Computational Geometry (SoCG 2022)

We consider distributed systems of identical autonomous computational entities, called robots, moving and operating in the plane in synchronous Look-Compute-Move (LCM) cycles. The algorithmic capabilities of these systems have been extensively investigated in the literature under four distinct models (OBLOT, FSTA, FCOM, LUMI), each identifying different levels of memory persistence and communication capabilities of the robots. Despite their differences, they all always assume that robots have unlimited amounts of energy. In this paper, we remove this assumption and start the study of the computational capabilities of robots whose energy is limited, albeit renewable. We first study the impact that memory persistence and communication capabilities have on the computational power of such energy-constrained systems of robots; we do so by analyzing the computational relationship between the four models under this energy constraint. We provide a complete characterization of this relationship. We then study the difference in computational power caused by the energy restriction and provide a complete characterization of the relationship between energy-constrained and unrestricted robots in each model. We prove that within LUMI there is no difference; an integral part of the proof is the design and analysis of an algorithm that in LUMI allows energy-constrained robots to execute correctly any protocol for robots with unlimited energy. We then show the (apparently counterintuitive) result that in all other models, the energy constraint actually provides the robots with a computational advantage., Comment: 44 pages, 7 figures

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 Fr\'echet 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 Fr\'echet 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 Fr\'echet 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., Comment: In SoCG 2022

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 number of objects from $A$ that are visible to a query point or a query segment. Our main aim is to obtain fast, $O(\mathop{\textrm{polylog}} nm$), 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 $O(nm^2)$ space. In case $A$ also contains only points, we present a smaller, $O(n + m^{2 + \varepsilon}\log n)$-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 $O(nm^{2 + \varepsilon} + n^2)$ space. Finally, we show that we can even handle the case where the objects in $A$ are segments with the same bounds., Comment: 27 pages, 13 figures

We study metrics that assess how close a triangulation is to being a Delaunay triangulation, for use in contexts where a good triangulation is desired but constraints (e.g., maximum degree) prevent the use of the Delaunay triangulation itself. Our near-Delaunay metrics derive from common Delaunay properties and satisfy a basic set of design criteria, such as being invariant under similarity transformations. We compare the metrics, showing that each can make different judgments as to which triangulation is closer to Delaunay. We also present a preliminary experiment, showing how optimizing for these metrics under different constraints gives similar, but not necessarily identical results, on random and constructed small point sets.

We consider the problem of computing the Fr\'echet 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 Fr\'echet 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 Fr\'echet distance. We also study the weak Fr\'echet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than the regular Fr\'echet 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 Fr\'echet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D., Comment: 27 pages, 12 figures. This is the full version of the paper to appear at WADS 2021

Exactly 20 years ago at MFCS, Demaine posed the open problem whether the game of Dots & Boxes is PSPACE-complete. Dots & Boxes has been studied extensively, with for instance a chapter in Berlekamp et al. "Winning Ways for Your Mathematical Plays", a whole book on the game "The Dots and Boxes Game: Sophisticated Child's Play" by Berlekamp, and numerous articles in the "Games of No Chance" series. While known to be NP-hard, the question of its complexity remained open. We resolve this question, proving that the game is PSPACE-complete by a reduction from a game played on propositional formulas., Comment: 18 pages, 13 figures

We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs, we are given a graph $G$ that is embedded in Euclidean space. The edges of $G$ need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex. Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this paper, we present polynomial-time algorithms for 1D instances of MSC-TE and MSC-BE, but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches, and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances.

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 Fr\'echet distance. For both these distance measures, we present polynomial-time algorithms for this problem., Comment: 25 pages, 5 figures

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 setting, the representative of a cluster is again a trajectory. To obtain a compact representation of the clusters and to avoid overfitting, we restrict the complexity of the representative trajectories by a parameter l. This restriction, however, makes discrete distance measures like dynamic time warping (DTW) less suited. There is recent work on center-based clustering of trajectories with a continuous distance measure, namely, the Fr\'echet distance. While the Fr\'echet distance allows for restriction of the center complexity, it can also be sensitive to outliers, whereas averaging-type distance measures, like DTW, are less so. To obtain a trajectory clustering algorithm that allows restricting center complexity and is more robust to outliers, we propose the usage of a continuous version of DTW as distance measure, which we call continuous dynamic time warping (CDTW). Our contribution is twofold: 1. To combat the lack of practical algorithms for CDTW, we develop an approximation algorithm that computes it. 2. We develop the first clustering algorithm under this distance measure and show a practical way to compute a center from a set of trajectories and subsequently iteratively improve it. To obtain insights into the results of clustering under CDTW on practical data, we conduct extensive experiments., Comment: 12 pages, 16 figures. This is the authors' version of the paper published in SIGSPATIAL 2020

We consider the problem of finding patrol schedules for $k$ robots to visit a given set of $n$ sites in a metric space. Each robot has the same maximum speed and the goal is to minimize the weighted maximum latency of any site, where the latency of a site is defined as the maximum time duration between consecutive visits of that site. The problem is NP-hard, as it has the traveling salesman problem as a special case (when $k=1$ and all sites have the same weight). We present a polynomial-time algorithm with an approximation factor of $O(k^2 \log \frac{w_{\max}}{w_{\min}})$ to the optimal solution, where $w_{\max}$ and $w_{\min}$ are the maximum and minimum weight of the sites respectively. Further, we consider the special case where the sites are in 1D. When all sites have the same weight, we present a polynomial-time algorithm to solve the problem exactly. If the sites may have different weights, we present a $12$-approximate solution, which runs in polynomial time when the number of robots, $k$, is a constant., Comment: Proceedings of the 14th International Workshop on the Algorithmic Foundations of Robotics (WAFR 20)

In this paper we study a wide range of variants for computing the (discrete and continuous) Fr\'echet 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 Fr\'echet distance, which are the minimum and maximum Fr\'echet distance for any realisations of the curves. We prove that both the upper and lower bound problems are NP-hard for the continuous Fr\'echet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Fr\'echet distance. In contrast, the lower bound (discrete and continuous) Fr\'echet distance can be computed in polynomial time. Furthermore, we show that computing the expected discrete Fr\'echet distance is #P-hard when the uncertainty regions are modelled as point sets or line segments. The construction also extends to show #P-hardness for computing the continuous Fr\'echet distance when regions are modelled as point sets. On the positive side, we argue that in any constant dimension there is a FPTAS for the lower bound problem when $\Delta / \delta$ is polynomially bounded, where $\delta$ is the Fr\'echet distance and $\Delta$ 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 $\delta$-separated. Finally, we also study the setting with Sakoe--Chiba time bands, where we restrict the alignment between the two curves, and give polynomial-time algorithms for upper bound and expected discrete and continuous Fr\'echet distance for uncertainty regions modelled as point sets., Comment: 48 pages, 11 figures. This is the full version of the paper to be published in ICALP 2020

We present a new game, Dots & Polygons, played on a planar point set. Players take turns connecting two points, and when a player closes a (simple) polygon, the player scores its area. We show that deciding whether the game can be won from a given state, is NP-hard. We do so by a reduction from vertex-disjoint cycle packing in cubic planar graphs, including a self-contained reduction from planar 3-Satisfiability to this cycle-packing problem. This also provides a simple proof of the NP-hardness of the related game Dots & Boxes. For points in convex position, we discuss a greedy strategy for Dots & Polygons., Comment: 9 pages, 9 figures, a shorter version of this paper will appear at the 29th International Computational Geometry Media Exposition at CG Week in 2020

Reliable spanners can withstand huge failures, even when a linear number of vertices are deleted from the network. In case of failures, a reliable spanner may have some additional vertices for which the spanner property no longer holds, but this collateral damage is bounded by a fraction of the size of the attack. It is known that $\Omega(n\log n)$ edges are needed to achieve this strong property, where $n$ is the number of vertices in the network, even in one dimension. Constructions of reliable geometric $(1+\varepsilon)$-spanners, for $n$ points in $\Re^d$, are known, where the resulting graph has $O( n \log n \log \log^{6}n )$ edges. Here, we show randomized constructions of smaller size spanners that have the desired reliability property in expectation or with good probability. The new construction is simple, and potentially practical -- replacing a hierarchical usage of expanders (which renders the previous constructions impractical) by a simple skip-list like construction. This results in a $1$-spanner, on the line, that has linear number of edges. Using this, we present a construction of a reliable spanner in $\Re^d$ with $O( n \log \log^{2} n \log \log \log n )$ edges.

We give a new combinatorial proof for the number of convex polyominoes whose minimum enclosing rectangle has given dimensions. We also count the subclass of these polyominoes that contain the lower left corner of the enclosing rectangle (directed polyominoes). We indicate how to sample random polyominoes in these classes. As a side result, we calculate the first and second moments of the number of common points of two monotone lattice paths between two given points., Comment: 18 pages, 8 figures

We study the complexity of clustering curves under $k$-median and $k$-center objectives in the metric space of the Fr\'echet distance and related distance measures. Building upon recent hardness results for the minimum-enclosing-ball problem under the Fr\'echet 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 Fr\'echet 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 Fr\'echet 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.

We consider paths with low \emph{exposure} to a 2D polygonal domain, i.e., paths which are seen as little as possible; we differentiate between \emph{integral} exposure (when we care about how long the path sees every point of the domain) and \emph{0/1} exposure (just counting whether a point is seen by the path or not). For the integral exposure, we give a PTAS for finding the minimum-exposure path between two given points in the domain; for the 0/1 version, we prove that in a simple polygon the shortest path has the minimum exposure, while in domains with holes the problem becomes NP-hard. We also highlight connections of the problem to minimum satisfiability and settle hardness of variants of planar min- and max-SAT.

We show how to construct $(1+\varepsilon)$-spanner over a set $P$ of $n$ points in $\mathbb{R}^d$ that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters $\vartheta,\varepsilon \in (0,1)$, the computed spanner $G$ has $ O\bigl(\varepsilon^{-c} \vartheta^{-6} n \log n (\log\log n)^6 \bigr) $ edges, where $c= O(d)$. Furthermore, for any $k$, and any deleted set $B \subseteq P$ of $k$ points, the residual graph $G \setminus B$ is $(1+\varepsilon)$-spanner for all the points of $P$ except for $(1+\vartheta)k$ of them. No previous constructions, beyond the trivial clique with $O(n^2)$ edges, were known such that only a tiny additional fraction (i.e., $\vartheta$) lose their distance preserving connectivity. Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one-dimensional construction in a black box fashion.

We show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fr\'echet distance between curves better than a factor $3$ unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than $3$. Our lower bound holds for the continuous, the discrete, and the weak discrete Fr\'echet distance even for curves in one dimension. Interestingly, the continuous weak Fr\'echet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fr\'echet distance in linear time.

Simplifying polygonal curves at different levels of detail is an important problem with many applications. Existing geometric optimization algorithms are only capable of minimizing the complexity of a simplified curve for a single level of detail. We present an $O(n^3m)$-time algorithm that takes a polygonal curve of n vertices and produces a set of consistent simplifications for m scales while minimizing the cumulative simplification complexity. This algorithm is compatible with distance measures such as the Hausdorff, the Fr\'echet and area-based distances, and enables simplification for continuous scaling in $O(n^5)$ time. To speed up this algorithm in practice, we present new techniques for constructing and representing so-called shortcut graphs. Experimental evaluation of these techniques on trajectory data reveals a significant improvement of using shortcut graphs for progressive and non-progressive curve simplification, both in terms of running time and memory usage., Comment: 20 pages, 20 figures

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 Fr\'echet 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 Fr\'echet 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 Fr\'echet distance. In the case of the discrete Fr\'echet 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 Fr\'echet 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., Comment: 24 pages; results on minimum-enclosing ball added, additional author added, general revision

A geometric $t$-spanner on a set of points in Euclidean space is a graph containing for every pair of points a path of length at most $t$ times the Euclidean distance between the points. Informally, a spanner is $\mathcal{O}(k)$-robust if deleting $k$ vertices only harms $\mathcal{O}(k)$ other vertices. We show that on any one-dimensional set of $n$ points, for any $\varepsilon>0$, there exists an $\mathcal{O}(k)$-robust $1$-spanner with $\mathcal{O}(n^{1+\varepsilon})$ edges. Previously it was only known that $\mathcal{O}(k)$-robust spanners with $\mathcal{O}(n^2)$ edges exists and that there are point sets on which any $\mathcal{O}(k)$-robust spanner has $\Omega(n\log{n})$ edges., Comment: 6 pages, 6 figures

We consider the problem of packing a family of disks "on a shelf", that is, such that each disk touches the $x$-axis from above and such that no two disks overlap. We prove that the problem of minimizing the distance between the leftmost point and the rightmost point of any disk is NP-hard. On the positive side, we show how to approximate this problem within a factor of 4/3 in $O(n \log n)$ time, and provide an $O(n \log n)$-time exact algorithm for a special case, in particular when the ratio between the largest and smallest radius is at most four., Comment: Appeared at Proc. 28th International Symposium on Algorithms and Computation (ISAAC 2017)

In a geometric $k$-clustering problem the goal is to partition a set of points in $\mathbb{R}^d$ into $k$ subsets such that a certain cost function of the clustering is minimized. We present data structures for orthogonal range-clustering queries on a point set $S$: given a query box $Q$ and an integer $k>2$, compute an optimal $k$-clustering for $S\setminus Q$. We obtain the following results. We present a general method to compute a $(1+\epsilon)$-approximation to a range-clustering query, where $\epsilon>0$ is a parameter that can be specified as part of the query. Our method applies to a large class of clustering problems, including $k$-center clustering in any $L_p$-metric and a variant of $k$-center clustering where the goal is to minimize the sum (instead of maximum) of the cluster sizes. We extend our method to deal with capacitated $k$-clustering problems, where each of the clusters should not contain more than a given number of points. For the special cases of rectilinear $k$-center clustering in $\mathbb{R}^1$, and in $\mathbb{R}^2$ for $k=2$ or 3, we present data structures that answer range-clustering queries exactly., Comment: 23 pages and 2 figures

Let $P$ be a set of $n$ points in the plane. We consider the problem of partitioning $P$ into two subsets $P_1$ and $P_2$ such that the sum of the perimeters of $\text{CH}(P_1)$ and $\text{CH}(P_2)$ is minimized, where $\text{CH}(P_i)$ denotes the convex hull of $P_i$. The problem was first studied by Mitchell and Wynters in 1991 who gave an $O(n^2)$ time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in $O(n \log^2 n)$ time and a $(1+\varepsilon)$-approximation algorithm running in $O(n + 1/\varepsilon^2\cdot\log^2(1/\varepsilon))$ time., Comment: This version is similar to one published in Discrete & Computational Geometry

We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity. Our first set of results is motivated by the Bitonic TSP problem: given a set of $n$ points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in $O(n^2)$ time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Frechet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic TSP in $O(n \log^2 n)$ time and its bottleneck version in $O(n \log^3 n)$ time. Our second set of results concerns the popular $k$-OPT heuristic for TSP in the graph setting. More precisely, we study the $k$-OPT decision problem, which asks whether a given tour can be improved by a $k$-OPT move that replaces $k$ edges in the tour by $k$ new edges. A simple algorithm solves $k$-OPT in $O(n^k)$ time for fixed $k$. For 2-OPT, this is easily seen to be optimal. For $k=3$ we prove that an algorithm with a runtime of the form $\tilde{O}(n^{3-\epsilon})$ exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. The results for $k=2,3$ may suggest that the actual time complexity of $k$-OPT is $\Theta(n^k)$. We show that this is not the case, by presenting an algorithm that finds the best $k$-move in $O(n^{\lfloor 2k/3 \rfloor + 1})$ time for fixed $k \geq 3$. This implies that 4-OPT can be solved in $O(n^3)$ time, matching the best-known algorithm for 3-OPT. Finally, we show how to beat the quadratic barrier for $k=2$ in two important settings, namely for points in the plane and when we want to solve 2-OPT repeatedly., Comment: Extended abstract appears in the Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We introduce the concept of compactly representing a large number of state sequences, e.g., sequences of activities, as a flow diagram. We argue that the flow diagram representation gives an intuitive summary that allows the user to detect patterns among large sets of state sequences. Simplified, our aim is to generate a small flow diagram that models the flow of states of all the state sequences given as input. For a small number of state sequences we present efficient algorithms to compute a minimal flow diagram. For a large number of state sequences we show that it is unlikely that efficient algorithms exist. More specifically, the problem is W[1]-hard if the number of state sequences is taken as a parameter. We thus introduce several heuristics for this problem. We argue about the usefulness of the flow diagram by applying the algorithms to two problems in sports analysis. We evaluate the performance of our algorithms on a football data set and generated data., Comment: 18 pages, 8 figures

Robust estimators, like the median of a point set, are important for data analysis in the presence of outliers. We study robust estimators for locationally uncertain points with discrete distributions. That is, each point in a data set has a discrete probability distribution describing its location. The probabilistic nature of uncertain data makes it challenging to compute such estimators, since the true value of the estimator is now described by a distribution rather than a single point. We show how to construct and estimate the distribution of the median of a point set. Building the approximate support of the distribution takes near-linear time, and assigning probability to that support takes quadratic time. We also develop a general approximation technique for distributions of robust estimators with respect to ranges with bounded VC dimension. This includes the geometric median for high dimensions and the Siegel estimator for linear regression., Comment: Full version of a paper to appear at SoCG 2018

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fr\'echet distance between surfaces. While the Fr\'echet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality.

A fundamental problem in wireless sensor networks is to connect a given set of sensors while minimizing the \emph{receiver interference}. This is modeled as follows: each sensor node corresponds to a point in $\mathbb{R}^d$ and each \emph{transmission range} corresponds to a ball. The receiver interference of a sensor node is defined as the number of transmission ranges it lies in. Our goal is to choose transmission radii that minimize the maximum interference while maintaining a strongly connected asymmetric communication graph. For the two-dimensional case, we show that it is NP-complete to decide whether one can achieve a receiver interference of at most $5$. In the one-dimensional case, we prove that there are optimal solutions with nontrivial structural properties. These properties can be exploited to obtain an exact algorithm that runs in quasi-polynomial time. This generalizes a result by Tan et al. to the asymmetric case., Comment: 15 pages, 5 figures

Let $p$ and $q$ be two imprecise points, given as probability density functions on $\mathbb R^2$, and let $\cal R$ be a set of $n$ line segments (obstacles) in $\mathbb R^2$. We study the problem of approximating the probability that $p$ and $q$ can see each other; that is, that the segment connecting $p$ and $q$ does not cross any segment of $\cal R$. To solve this problem, we approximate each density function by a weighted set of polygons; a novel approach to dealing with probability density functions in computational geometry.

The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it take Omega(n^2) time, limiting its applicability on large data sets. We observe that for many point sets, the greedy spanner has many `short' edges that can be determined locally and usually quickly, and few or no `long' edges that can usually be determined quickly using local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets: we give a geometric property that holds with high probability on such point sets. This property implies that if an edge set on these points has t-paths between pairs of points `close' to each other, then it has t-paths between all pairs of points. This characterization gives a O(n log^2 n log^2 log n) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give a O((n + |E|) log^2 n log log n) expected time algorithm on uniformly distributed points that determines if E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E., Comment: 16 pages,22 figures. Full version of the ESA 2014 publication with the same title